Geography and Path Dependence

Transcription

1 Geography and Path Dependence Treb Allen 1 Dave Donaldson 2 1 Dartmouth and NBER 2 MIT and NBER November 2017

2 Path Dependence and Economic Geography Evidence for agglomeration economies seems strong: Case studies (e.g. Silicon Valley) Direct estimates (e.g. Berlin Wall, Million Dollar Plants, TVA...) Long theoretical tradition highlights potential implications: Multiple (Pareto-ranked) steady-states in dynamic models Path dependence (i.e. dl t da t k 0, even though Yt A t k = 0 and k large) in dynamic models Potential for policies to promote movement to better steady-state Is path dependence empirically consequential? Perhaps not: Historical accident may simply choose from a set of roughly equivalent steady states (e.g. portages). Need a way of assessing quantitatively importance of path dependence in the real world.

3 Goals of this paper 1. Develop tools for the quantitative study of path dependence Tractable dynamic model suited to real geography (many regions, unrestricted trade and migration costs) Plausible parameter values under which model features multiple steady-states, yet equilibrium transition paths unique. 2. Estimate parameters using US spatial history (1850-present) Wide range of instruments possible: geography, lagged populations, lagged (and now obsolete) productivity/amenity shifters, responsiveness of economy to temporary shocks Today: Preliminary estimates based on one strategy 3. Answer counterfactual questions such as: How consequential is path dependence? How bad are chosen steady-states relative to best? What is the least cost route toward best steady state? Today: Preliminary results suggest path dependence is quite consequential.

4 Related literature Empirical evidence: For agglomeration economies: Ellison et. al. (2010), Greenstone et. al. (2010), Kline and Moretti (2014), Ahlfeldt et. al. (2015) Against path dependence: Davis and Weinstein (2002) For path dependence: Bleakley and Lin (2012) Quantitative economic geography models: Glaeser (2008), Allen and Arkolakis (2014), Ahlfeldt et. al. (2015) Dynamic migration models: Krugman (1991), Matsuyma (1991), Ottaviano (1999), Herrendorf et. al. (2000), Quantitative dynamic economic geography models: Desmet et. al. (2015), Caliendo et. al. (2015)

5 Model overview: Main ingredients Flexible bilateral migration and trade frictions, local characteristics. Incorporate real world geography. Armington trade, extreme value discrete choice migration. Convenient gravity equations for trade flows and migration. Overlapping generations. Straightforward characterization of dynamics. Productivity and amenity spillovers. Possibility of multiple steady states and path dependence.

6 Model setup: Geography N locations. Each location i {1,..., N} in each time period t {1,...} is endowed with: Technology for producing a differentiated good (Armington assumption). An innate productivity Ā it. An innate amenity ū it. All pairs of locations (i, j) are endowed with: A bilateral iceberg trade cost τ ijt 1. A bilateral iceberg migration cost µ ijt 1.

7 Model setup: Dynamics Agents live two periods ( childhood and adulthood ). Consider an agent who is an adult in period t: In period t 1, that agent is born where her parent lived. In period t, choose where to live (i.e. produce/consume). Gives birth to generation t + 1 in that location. Agents only produce/consume in adulthood, do not care about children. Let L it be adult population in location i in time t. The world s initial population {L i0 } is given exogenously.

8 Model setup: Production and Consumption Production Perfect competition, (adult) labor only factor of production. Quantity produced: Q it = ( ) Ā it L α1 it Lα2 it 1 L it, }{{} A it where α 1 and α 2 govern the strength of contemporaneous and historical productivity spillovers. Consumption Adults have CES preferences over differentiated varieties with EoS σ, earn wage w it, have price index P it. Welfare: ( ) w W it = ū it L β1 it it Lβ2 it 1, }{{} P it u it where β 1 and β 2 govern the strength of contemporaneous and historical of amenity spillovers.

9 Gravity Armington + consumer maximization yields gravity equation for trade: P it ( N k=1 X ijt = τ 1 σ ijt ( ) ) 1 1 σ 1 σ w τ kt ki A kt ( wit A it ) 1 σ P σ 1 jt w jt L jt. is Dixit-Stiglitz price index. Frechet + welfare maximization yield gravity equation for migration: L ijt = µ θ ijt Π θ it L it 1 W θ jt, ( N Π it k=1 (W kt/µ ik ) θ) 1 θ is expected utility of a child born in location i in year t 1.

10 Equilibrium conditions For any initial population {L i0 } and geography { Ā it, ū it, τ ijt, µ ijt }, an equilibrium is {L it, w it, W it, Π it } s.t. i, t: 1. Payments to labor are equal to total sales: w it L it = N j=1 X ijt 2. Trade is balanced: w it L it = N j=1 X jit 3. Contemporaneous population is equal to total immigration: L it = N j=1 L jit 4. Historical population is equal to total emigration: L it 1 = N j=1 L ijt

11 Equilibrium conditions + gravity Yields 4 N T equations for 4 N T unknowns: 1. Payments to labor are equal to total sales: wit σ L 1+α 1(1 σ) it = j (Āit L α 2 it 1ūjtL β 2 jt 1 τ ijt ) σ 1 W 1 σ jt wjt σ L 1+β 1(σ 1) jt 2. Trade is balanced: w 1 σ it L β 1(1 σ) it W σ 1 it = j (ūit L β 2 it 1ĀjtL α ) 2 σ 1 jt 1 w 1 σ jt L α 1(σ 1) jt τ jit 3. The population is equal to total immigration: L it W θ it = j µ θ jit Π θ jt L jt 1, 4. The population is equal to total emigration: Π θ it j µ θ ijt W jt θ

12 Existence and Uniqueness of an Equilibrium Define matrix A (α 1, β 1 ) θ(α 1σ+β 1 (σ 1)+1) (σ 1) σ+θ(1 (σ 1)α 1 β 1 ) θ σ(σ+θ(1 (σ 1)α 1 β 1 )) σ((σ 1)(1 (σ 1)α 1 β 1 )+σ(α 1 σ+β 1 (σ 1)+1)) σ+θ(1 (σ 1)α 1 β 1 ) θ(1 (σ 1)α 1 β 1 ) σ+θ(1 (σ 1)α 1 β 1 ). Proposition 1(a): For any initial population {L i0 } and geography {Āit > 0, ū it > 0, τ ijt = τ jit, µ ijt }, there exists a unique equilibrium if ρ (A (α 1, β 1 )) 1. This will occur as long as α 1 and β 1 are sufficiently small. Proof: System is a special case of the generalized gravity system considered by Allen, Arkolakis, Li On the Existence and Uniqueness of Trade Equilibria. Note: Result does not depend on values of α 2 and β 2 (since current generation takes L it 1 as given).

13 Existence and Uniqueness of an Equilibrium

14 Steady state equilibrium Consider a steady state with time-invariant geography { Ā i, ū i, τ ij, µ ij } and endogenous variables {L i, w i, W i, Π i }. Proposition 1(b): For any geography {Āi > 0, ū i > 0, τ ij = τ ji, µ ij = µ ji }, there exists a unique equilibrium if ρ (A (α 1 + α 2, β 1 + β 2 )) 1. Implication: if α 1 + α 2 > α 1 and/or β 1 + β 2 > β 1, can have unique transition path but multiple steady states. Well behaved path dependence!

15 Properties of the steady state Define steady state welfare Ω = E [max i (W i Π i ε i )]. In steady state, welfare is equalized across all locations: W i Π i L 1 θ i = Ω i {1,..., N} Equilibrium steady state distribution of population can be written as: γ ln L i = C + (1 σ) ln ū i + σ ln Ā i + (1 σ) ln Π i ln P i, where γ 1 θ (1 σ) σ σ 1 ( σ + 1) β + σ α. Implication: More people will live in high Ā i, high ū i, high Π i, and low P i places, with elasticities governed by strength of spillovers.

16 Steady state on a plane: no trade or migration costs

17 Steady state on a plane: only trade costs

18 Steady state on a plane: only migration costs

19 Steady state on a plane: both trade and migration costs

20 Colonizing a plane

21 Colonizing a plane

22 Colonizing a plane

23 Colonizing a plane

24 Colonizing a plane

25 Colonizing a plane

26 Colonizing a plane

27 Colonizing a plane

28 Colonizing a plane

29 Colonizing a plane

30 Path dependence Suppose ρ (A (α 1, β 1 )) 1 but ρ (A (α 1 + α 2, β 1 + β 2 )) > 1. Then initial distribution of labor {L i0 } will determine which steady state the economy converges toward. Simple example: Two identical locations separated by trade costs with production externalities.

31 Path dependence: Possible steady states

32 Path dependence Suppose ρ (A (α 1, β 1 )) 1 but ρ (A (α 1 + α 2, β 1 + β 2 )) > 1. Then initial distribution of labor {L i0 } will determine which steady state the economy converges toward. Simple example: Two identical locations separated by trade costs with production externalities. Now consider the transition path from just to the left and just to the right of the homogeneous steady state.

33 Path dependence: Dynamics

34 Path dependence: Dynamics

35 Path dependence: Dynamics

36 Path dependence: Dynamics

37 Path dependence: Dynamics

38 Path dependence: Dynamics

39 Path dependence: Dynamics

40 Path dependence: Dynamics

41 Path dependence: Dynamics

42 Path dependence Suppose ρ (A (α 1, β 1 )) 1 but ρ (A (α 1 + α 2, β 1 + β 2 )) > 1. Then initial distribution of labor {L i0 } will determine which steady state the economy converges toward. Simple example: Two identical locations separated by trade costs with production externalities. Now consider the transition path from just to the left and just to the right of the homogeneous steady state. Small differences in initial condition can result in very different steady states. We can iterate across all possible {L i0 } to construct the basins of attraction for each steady state.

43 Path dependence: Basins of attraction

44 Path dependence: heterogeneous steady states In previous example, the two stable steady states had identical welfare implications. But similar intuition holds when the steady states are associated with different welfare levels. Consider example where left region has higher amenity than right region.

45 Path dependence: Possible steady states

46 Path dependence: Dynamics

47 Path dependence: Dynamics

48 Path dependence: Dynamics

49 Path dependence: Dynamics

50 Path dependence: Dynamics

51 Path dependence: Dynamics

52 Path dependence: Dynamics

53 Path dependence: Dynamics

54 Path dependence: Dynamics

55 Path dependence: Basins of attraction

56 Path dependence: heterogeneous steady states In previous example, the two stable steady states had identical welfare implications. But similar intuition holds when the steady states are associated with different welfare levels. Consider example where left region has higher amenity than right region. Implication: Initial population could cause world to converge to bad steady state......but the good steady state has larger basin of attraction.

57 Data L ijt : Decennial US Census from 1790-present: data (from 5% sample) on population by county of current residence and state of birth (and age) Manipulate this (with assumptions...) to get proxy for L ijt (and hence L jt i L ijt) Could probably do better with individual-level Census records (tracking people/families across waves) w it : Decennial US Census from 1850-present (except 1960): data on total agricultural and manufacturing output That plus Cobb-Douglas production function identifies Y it = w it L it X ijt : From 1997 Commodity Flow Survey.

58 Estimation Six key elasticities: (α 1, α 2, β 1, β 2, σ, θ) New estimation procedure with several advantages: Only need a little bilateral trade and migration data. Simultaneously estimates all six elasticities. Implemented using two 2SLS regressions: Transparent identification assumptions. Residuals from the regressions are Ā it and ū it. Step #1: Recover trade and migration costs from gravity equations: ln X ijt = (1 σ) κ t ln dist ij + γ it + δ jt + ε ijt ln L ijt = θλ t ln dist ij + ρ it + π jt + ν ijt

59 Estimation Step #1: Recovering trade and migration costs Migration Gravity over Time Migration gravity coefficient Decade Trade elasticity (using 1997 CFS data): -1.2.

60 Estimation Step #2: Model inversion Define T ij ˆτ 1 σ ij, M ij ˆµ θ ij, p it w it A it, and Y it w it L it. Re-write equilibrium conditions as follows: Goods market clearing: p σ 1 it P σ 1 it Labor market clearing: ( W θ = j = j it ) 1 = j Π θ it = i ( ) Yjt T ij P σ 1 jt Y it ( ) 1 T ji p σ 1 jt M ji L jt 1 L it M ij W θ jt ( Π θ jt ) 1 Proposition 2: Given observed {Y it, L it, L it 1 }, there exists unique (to-scale) { p σ 1 it, P σ 1 it, Wit θ, it} Πθ. Note: Does not require knowledge of trade or migration elasticities.

61 Estimation Step #3: IV to recover model elasticities Step #3: Regress inverted p σ 1 it ( ln and W θ it on L it, L it 1, and w it : ln ( p σ 1 ) it = (σ 1) ln wit + α 1 (1 σ) ln L it + α 2 (1 σ) ln L it 1 + (1 σ) ln Ā it ) ( Wit θ = θ ln w it / ( ) P 1 σ ) 1 1 σ it + β 1 θ ln L it + β 2 θ ln L it 1 + θ ln ū it Issue: Residuals ln Ā it and ln ū it correlated with endogenous outcomes. Need instruments. For now: use equilibrium {ln w it, ln L it, ln L it 1 } simulated from model where geography only depends on observables and L it 2. Work in progress: Use amenity shifters that don t affect productivities to IV equation #1 / productivity shifters that don t affect amenities to IV equation #2.

62 Table: Estimated model elasticities (preliminary) Trade regression Migration regression (1) (2) (3) (4) OLS IV OLS IV Log wage 0.968*** *** (0.002) (2.071) Log real wage (given *** *** estimate of trade elasticity) (0.001) (2.190) Log population 0.967*** 0.680*** 0.982*** *** (0.002) (0.222) (0.002) (0.203) Log lagged *** 1.141*** *** 1.024*** population (0.001) (0.113) (0.001) (0.081) Elasticity of 0.032*** *** substitution (σ) (0.002) (2.071) Migration elasticity *** *** (θ) (0.001) (2.190) Contemporaneous *** 0.065*** productivity spillover (α 1) (0.003) (0.018) Lagged productivity 0.007*** 0.109*** spillover (α 2) (0.001) (0.015) Contemporaneous *** *** amenity spillover (β 1) ( ) (0.012) Lagged amenity 6.720*** 0.077*** spillover (β 2) (2.442) (0.009) State-year FE Yes Yes Yes Yes Cell FE Yes Yes Yes Yes R-squared Observations Note: unique equilibrium but path dependence (!)

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78 Quantifying Path Dependence: Three questions 1. How do initial conditions shape the distribution of economic activity today? Illustrative example. 2. How do initial conditions affect the welfare of residents today? Monte Carlo approach. 3. What are the policy implications of path dependence? Work in progress.

79 Path dependence and the distribution of economic activity What would the U.S. look like if it had been colonized elsewhere? Not as academic a question as you might think: France attempted to settle North America from the west in the late 1700s. In 1785, Louis XVI sent an expedition to the Pacific Northwest under Jean-Francois de Galaup, Compte de La Perouse, On August 14 of 1785, Jefferson wrote: [The French] give out that the object is merely for the improvement of our knowledge... Their loading and some other circumstances appear to me to indicate some other design; perhaps that of colonising on the West coast of America, or perhaps to establish one or more factories there for the fur trade Perouse successfully scouted the PNW but was lost at sea on the way back to France. What would have happened had Perouse made it back?

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95 The welfare implications of path dependence How does the evolution of welfare depend on initial conditions? Monte Carlo procedure: Simulate evolution of U.S. economy B times. For each b {1,..., B}: Randomly draw from the observed initial distribution of populations (in 1840) and re-assign to different locations. In each year t {1850,..., 2000}, calculate expected welfare E [ ] Wit b L b it i L ln ( Wit b Π b itlit) b. Work in progress: B = 30. N = 579 (lower resolution).

96 1850 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

97 1860 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

98 1870 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

99 1880 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

100 1890 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

101 1900 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

102 1910 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

103 1920 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

104 1930 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

105 1940 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

106 1950 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

107 1970 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

108 1980 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

109 1990 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

110 2000 Bootstrap 1 Bootstrap 2 Bootstrap 3 Bootstrap 4

111 Distribution of population across simulations Substantial heterogeneity in spatial distribution across simulations..

112 Simulated Welfare Distribution of welfare across simulations Year but the welfare implications are similar...

113 Simulated Welfare Distribution of welfare across simulations versus reality Actual welfare Year... and much higher than the actual distribution of spatial activity!

114 Summary of the project Path dependence in economic geography: Is plausibly real Well explored in important theoretical literature (for simple cases) Goal is to develop a tractable way of exploring this in the data Replace static models featuring multiple equilibria with dynamic models featuring multiple steady-states (and hence path-dependence) Yet try to retain features of static models that make quantitative work feasible (identification results on fundamentals, algorithms for finding equilibria) Then take framework to the data for estimation and quantification of importance of path dependence

115 Next steps Improve estimation (more, better instruments) Extend counterfactuals How to get from current steady state to best steady state? How did the particular realization of productivity and amenity shocks shape the economic geography today?