Economic Geography meets Hotelling: a home-sweet-home eect

Transcription

1 Economic Geography meets Hotelling: a home-sweet-home eect José Gaspar 1 Soa B.S.D. Castro 2 João Correia-da-Silva 3 1 ISEG, University of Lisbon, CPBS, Universidade Católica Portuguesa and CEF.UP. 2 CMUP and Faculty of Economics, University of Porto. 2 CEF.UP and Faculty of Economics, University of Porto. VI Regional Modelling Workshop, Seville October

2 Contents Motivation, objectives and background Framework General results (preliminary) Illustration of the HSH-eect in NEG models Probabilistic migration: a particular case of the HSH-eect Conclusions and future work

3 Motivation, objectives and background Motivation and objectives Motivation Regional characteristics are perceived dierently by each individual Industrialized regions fail to attract potential migrants from less developed regions Heterogeneity in preferences for location constitutes a dispersion force; Literature: Tabuchi and Thisse (2002), Mossay (2003), Redding (2016) Objectives General framework for NEG models with 2 regions We introduce consumer heterogeneity and study the impacts on the spatial distributions as a result of: market-factors that generate agglomeration (NEG specic) non-market factors (heterogeneity) that promote dispersion Uncover insights under a class of dierent well-known NEG models

4 Motivation, objectives and background Background Short-run (xed spatial distribution) general equilibrium from an NEG model: Increasing returns to scale at the rm level Inter-regional trade subject to export hurdles Inter-regional mobility of (at least some) production factors Heterogeneity à la Hotelling (1929): Regions are dierentiated just as two goods in the linear city model We discard amenities that produce regional asymmetries (i.e., no location is advantageous per se)

5 Framework A general framework Regions and agents: There are two countries, L and R A unit mass of agents with heterogeneous preferences regarding their country of residence Preferences are described by a parameter x, uniformly distributed along the interval [0, 1]. Agent of type x = 0 (x = 1) has strongest preference for country L (country R); Agent of type x = 1 is indierent 2 The two countries can be thought of as two points on opposite extremes of the line segment [0, 1]

6 Framework Utility: consumption goods and residential preferences Utility: For a type x agent, utilities from living in L and R are given, respectively, by: V L =U(h) t(x) V R =U(h) t(1 x) h [0, 1] is the fraction of workers residing in L U i denotes utility from (NEG specic) short-run equilibrium in region i {L, R} t(x) is the utility penalty associated with living in L (resp. t(1 x) in country R) U(h) and t(x) are C 2.

7 Framework The spatial economy and short-run equilibrium Spatial distribution: Agents with x [0, h] reside in country L and agents with x (h, 1] reside in country R The spatial distribution is completely described by h [0, 1]. Short-run equilibrium: h is xed and Ui is derived from short-run general equilibrium NEG models The utility levels for a borderline agent of type x = h are V (h) and V (1 h), where V (z) U(z) t(z).

8 Framework Migration and long-run equilibria Long-run equilibrium: Each agent resides in the country that provides him the highest utility level A spatial distribution h [0, 1] is a long-run equilibrium if no agent gains from migrating Given symmetry, we shall focus on the case whereby L is larger, i.e., h [ 1 2, 1]. We focus on long-run equilibria that are stable: Agglomeration, h = 1, is a LR equilibrium if V (1) > V (0). It is always stable. Dispersion, h [ 1 2, 1), is a LR equilibrium if V (h ) = V (1 h ). It is stable if V (h) V (1 h) < 0. Dispersion is symmetric if h = 1 and asymmetric otherwise. 2

9 Framework Existence and stability of LR equilibria Existence and the qualitative properties of LR equilibria depend on V (h) = V (h) V (1 h). We can write V U t, where U = U(h) U(1 h) and t = t(h) t(1 h). Spatial distribution is a result of two counteracting forces: Gains from NEG specic utility due to increasing returns promote agglomeration Hotelling penalty that the borderline agent suers from residing in a given country generates dispersion The less a consumer desires to live in a country, the higher the utility penalty dierential t(h): the home-sweet-home (HSH) eect

10 General results (preliminary) Main results in the general framework (1) Proposition If V (h) is convex for h [1/2, 1],then there exists a stable long-run equilibrium, which may be agglomeration, symmetric dispersion, or both. Figure: Stable equilibria from left to right: agglomeration; symmetric dispersion; both.

11 General results (preliminary) Main results in the general framework (2) Proposition If V (h) is concave for h [1/2, 1],then there exists a unique stable long-run equilibrium, which may be agglomeration, symmetric dispersion, or asymmetric dispersion. Figure: Stable equilibria from left to right: agglomeration; symmetric dispersion; asymmetric dispersion.

12 Illustration of the HSH-eect in NEG models The Püger (2004) model Quasi-linear translog utility:ui = µ ln M i + A i, M i is a CES composite of q(s) manufactures of quality s S and A i is the numeraire good Footloose cost structure for manufacturing rms: C i (q i ) = αw i + βq i, α is the xed input requirement (skilled workers h) and β is the variable input requirement (unskilled workers) Indirect utility U(h) is strictly concave, h [1/2, 1] Assuming t(h) is bounded in [0, 1]: (i) if t(h) is non-strictly convex, V (h) is concave = Proposition 2 holds (as in Püger, 2004). (ii) if t(h) is strictly concave, V (h) may become convex = Proposition 1 (as in CP, FKV and FE model).

13 Illustration of the HSH-eect in NEG models The Ottaviano (2001) model Footloose entrepreneur model log upper-tier utility: [ (Mi ) µ ( ) ] 1 µ Ai u i = ln + µ 1 µ Computing S-R indirect utility Ui (h), we have d 2 U(h) > 0, h [1/2, 1] = strictly convex dh 2 Assuming t(h) is bounded in [0, 1]: (i) if t(h) is non-strictly concave, V (h) is convex = Proposition 1 holds (same as in the CP, FKV and FE model). (ii) if t(h) is strictly convex, V (h) may become concave = Proposition 2 (as in Püger, 2004).

14 Illustration of the HSH-eect in NEG models The OTT model (Ottaviano et al, 2002) Quasi-linear utility with quadratic sub-utility: u i = α q i (s)ds β γ q i (s) 2 ds γ [ 2 q i (s)ds] + A i. s S 2 s S 2 s S Computing S-R indirect utility Ui (h), it is well known that U(h) is linear in h. Assuming t(h) is bounded in [0, 1]: (i) if t(h) is strictly convex, V (h) is convex = Proposition 1 (as in CP, FKV and FE model). (ii) if t(h) is strictly concave, V (h) is concave = Proposition 2 holds (as in Püger, 2004). (iii) if t(h) is linear, V (h) is linear = agglomeration or symmetric dispersion (borderline pitchfork)

15 Probabilistic migration: a particular case of the HSH-eect Heterogeneity in NEG literature Probabilistic migration in NEG Idiosyncratic preferences for residential location assumed to be an unobservable random utility component Typically following some extreme value distribution: Gumbel (extreme value type I) distribution, as in Tabuchi and Thisse, 2002; Murata, 2003; Akamatsu et al., 2012 Fréchet (extreme value type II) distribution, as in Ahlfeldt et al. 2015; Redding Heterogeneity in NEG typically modeled according to the (conditional) Logit model

16 Probabilistic migration: a particular case of the HSH-eect Gumbel distribution and the HSH-eect The probability of choosing L is given by: P L (h) = e U L(h)/µ e U L(h)/µ + e U R (h)/µ. (1) P(h) is the same as the borderline agent h in our model.therefore, using (1) yields: U L µ ln h = U R µ ln(1 h) U = µ [ln h ln(1 h)] t(h) }{{} HSH eect = Logit model yields a specic HSH eect

17 Probabilistic migration: a particular case of the HSH-eect Fréchet distribution and the HSH-eect The probability of choosing L is given by: Using (2) yields: U L (h) 1 µ P L (h) =, (2) U L (h) 1 µ + U R (h) 1 µ ln U L µ ln h U = ln U R µ ln(1 h) ln U = µ [ln h ln(1 h)] t(h). }{{} HSH eect = Same HSH eect as before, but with a positive monotonic transformation on U i.

18 Probabilistic migration: a particular case of the HSH-eect Example: the Murata (2003) model Spatial distributions isomorphic under Gumbel and Fréchet distributions (resp. linear SR utility and log SR in Murata, 2003) Supercritical pitchfork bifurcations: corresponds to concave V (h) (Prop. 2), but only Figures 2 and 3 since t(h) is not bounded at h = 1, the HSH tends to innity!

19 Probabilistic migration: a particular case of the HSH-eect Alternative: linear home-sweet-home eect Linear utility penalty t(h) = µx. The HSH eect is given by: t t(h) t(1 h) = µ(2h 1). (3) This relaxes the extreme convexity of t(h) due to extreme value type preferences. Utility for consumers at x = {0, 1} is bounded away from innity, allowing for agglomeration in either country.

20 Probabilistic migration: a particular case of the HSH-eect Spatial distributions in Murata (2003) with linear HSH eect Subcritical pitchfork bifurcations: Corresponds to convex V (h) (Prop. 1), any of the gures.

21 Conclusions and future work Main conclusions and future work Conclusions: We are able to account for the interaction between market factors and non market factors for a wide array of NEG models Extreme value distribution of unobservable (logit model) is restrictive for evaluating impacts of heterogeneity Dierent types of heterogeneity produce very dierent impacts on the spatial distribution of economic activities Future work: Consider multiple regions, dierent distribution of preferences, or both n-region NEG model with dierent consumer preferences Hotelling type preferences from Logit model in a simple equidistant/racetrack economy Inference of statistical distribution of idiosyncratic preferences towards residential choice

22 Thank you for your attention!