First-Best Dynamic Assignment of Commuters with Endogenous Heterogeneities in a Corridor Network

Transcription

1 First-Best Dynamic Assignment of Commuters with Endogenous Heterogeneities in a Corridor Network ISTTT 22@Northwestern University Minoru Osawa, Haoran Fu, Takashi Akamatsu Tohoku University July 24, 2017

2 Background Departure-time choice equilibrium model for commute problems has long been studied. - Vickrey, 1969; Hendrickson & Kocur, 1981; etc.. In the long term, travel demand patterns vary over time because of commuters choices (e.g. residential locations and obs) and inevitably affect short-term dynamic. An effective framework incorporating the short term and long term is worth of studying. 2

3 Purpose Propose a framework that synthesizes long-term travel demand forecasting with short-term dynamic traffic assignment. Long-term equilibrium: residential location; ob Short-term equilibrium: departure time Investigate properties of equilibrium. Temporal/spatial patterns 3

4 Contributions Integration of dynamic bottleneck congestion and Urban Economics cf. Arnott, 1998; Gubins & Verhoef, 2014; Takayama & Kuwahara, 2016 In a corridor network with multiple bottlenecks cf. Shen & Zhang, 2009; Tian, et al., 2012; etc. With endogenous heterogeneities Policy implication for investment on bottleneck capacity expansions 4

5 Outline Model Settings: Network, Commuters Behavior Equilibria: Short-term and Long-term Integrated Model Analysis of Short-term Equilibrium Special Case I: Corridor without Heterogeneities Special Case II: Single Bottleneck with Heterogeneities General Case: Corridor with Heterogeneities Analysis of Long-term Equilibrium Spatial Sorting Patterns Policy Implications Concluding Remarks 5

6 Network A corridor with multiple bottlenecks residential location AI I bottleneck (BN) land supply A i A 2 A1 i 2 1 BN capacity labor demand L 1 L 2 L L J obs at CBD 6

7 Commuters Behavior Commuters choices In the long term - Residential location i with endogenous land rent - Job with endogenous wage w In the short term - Departure time (arrival time t at the CBD) with commuting cost C () t Commuters obective Maximize own utility (minimize own cost) r i max v ( t) max min C )}, t { w r i ( t t long-term short-term 7

8 Interactions between Short and Long Terms Feedback structure Long-term policies: land supply, labor demand, BN capacity { A } { L } { } i i Equilibrium commuting cost { i, } Long-term Equilibrium (location choice i and ob choice ) { Q i, } Travel demand Short-term Equilibrium (arrival-time choice t) Short-term policy: Travel Demand Management (TDM) 8

9 Short-term TDM Policy: TNP Scheme Definition A tradable network permit (TNP) is a right that allows its holder to pass through a pre-specified BN during a pre-specified time period. - Akamatsu, 2007; Wada & Akamatsu, 2013; Akamatsu & Wada, 2017 Assumptions A competitive market is assumed for trading permits. The number of issued permits is limited to be less than or equal to BN capacity. Queues of all BNs are eliminated. 9

10 Short-term Equilibrium under TDM Policy Optimal-choice conditions in short-term equilibrium equilibrium cost commuting cost arrival-flow rate C ( t) 0 if q C ( t) 0 if q value of time schedule delay ( t) 0 ( t) 0 C, ( t i ) [ s ( t ) d i ] p m( t ) Permit market clearing conditions supply (BN capacity) i,, t permit price free-flow travel time demand i q, ( ) 0 if ( ) 0 m m t p i i t i, t i q, ( ) 0 if () 0 m i m t pi t mi 10

11 Long-term Equilibrium Optimal choice conditions in long-term equilibrium equilibrium utility level Land/labor market clearing conditions supply demand land market demand labor market supply travel demand V w ri 0 if Qi, 0 i, V w ri 0 if Qi, 0 Q q ( )d t t t A Q Ai Q 0 if i i L Q L Q 0 if 0 if i 0 if i r r i w w i 11

12 Efficiency of The Integrated Equilibrium Proposition 1: The integrated equilibrium is socially optimal in the sense that it minimizes the total transportation cost. Equivalent linear programming problem (LP) total cost permit supply land supply labor demand min c ( t) q ( t)dt q0 i t s. t. q ( t) ( p ( t) ) i, t mi i t t m, i i q ( t)d t A ( r ) i i q ( t)d t L ( w ) i c ( t) [ s( t) di ] 12

13 Benders Decomposition of the Integrated Model The long-term problem (master problem) min Q0 i i s. t. Q A ( r ) i d Q i i Q L ( w ) z 1 The short-term problem (subproblem) z min s( t) q ( t)dt 1 i t q0 s. t. q ( t) ( p () t ) t mi q ( t)d t m, i i Q ( ) 13

14 Outline Model Settings: Network, Commuters Behavior Equilibria: Short-term and Long-term Integrated Model Analysis of Short-term Equilibrium Special Case I: Corridor without Heterogeneities Special Case II: Single Bottleneck with Heterogeneities General Case: Corridor with Heterogeneities Analysis of Long-term Equilibrium Spatial Sorting Patterns Policy Implications Concluding Remarks 14

15 Special Case I: Corridor without Heterogeneities Model settings Single ob class but multiple BNs AI Ai A 2 A1 L { } I Equivalent LP i 2 1 min c ( t) q ( t)dt q0 i i i s. t. q ( t) ( p ( t) ) i, t t mi i t m i i q ( t)d t A ( r ) i i i 15

16 Analytical Solution [Special Case I] Proposition 2-A (location-based temporal sorting pattern): Commuters residing closer to the CBD arrive at times in a time window closer to the desired arrival time i q () t i T 1 T 2 T 3 Q 1 Q 2 Equilibrium arrival-flow rates at the CBD Q 3 t st () p () t 3 p () t 2 p () t 1 T 3 T 2 T 1 Equilibrium Schedule commuting delay function costs T pt () tt 16

17 Special Case II: Single BN with Heterogeneities Model settings Single BN but multiple ob classes A Ai A 2 1 L 1 L 2 i 2 1 Equivalent LP L L J min c ( t) q ( t)dt q0 1, t 1, 1, 1, s. t. q ( t) ( p () t ) t t 1 q ( t)d t L ( w ) 1 17

18 Analytical Solution [Special Case II] Proposition 2-B (ob-based temporal sorting pattern): Commuters of higher value of time arrive at times closer to the desired arrival time. q 1, () t p () t 1 Q 1,2 Q 1,1 1,1 T 1,2 T 1,1 1 1,2 T 1,1 T 1,2 Equilibrium arrival-flow rates at the CBD t Equilibrium commuting costs and permit prices (isocost curve) t 18

19 General Case: Corridor with Heterogeneities Proposition 3 (temporal sorting pattern): Flows of commuters admit a combination of sorting patterns in Special Case I and II. 1 2 i q () t LT 2,2 2 T 2,1 L 1 T 1,2 T 1,1 Equilibrium arrival-flow rates at the CBD A 1 A 2 t m 2 2,1 1,1 2,2 1,2 p () m t T 2,2 T 2,1 T 1,2 T 1,1 Equilibrium commuting costs and permit prices (isocost curve) t 19

20 Outline Model Settings: Network, Commuters Behavior Equilibria: Short-term and Long-term Integrated Model Analysis of Short-term Equilibrium Special Case I: Corridor with Homogeneities Special Case II: Single Bottleneck with Heterogeneities General Case: Corridor with Heterogeneities Analysis of Long-term Equilibrium Spatial Sorting Patterns Policy Implications Concluding Remarks 20

21 The Long-term Problem Solving the short-term problem gives the equilibrium commuting cost given travel demand Q. Q Short-term Equilibrium ρ ( Q) for ρ Equivalent convex optimization problem for the long-term equilibrium ignored in a naive model z min 2 Q0 i i i, i 0Q i s. t. Q A ( r ) i d Q i i Q L ( w ) i i ρ ( ω)dω 21

22 Integrated Model vs. Naive Model Integrated model Naive model exogenously given and fixed ρ Long-term { Q, r, w ρ} Short-term { q, p, ρ Q} Q C N ρ Long-term { Q, r, w ρ} Short-term { q, p, ρ Q} Q N ρ, ρ Q, Q TNP scheme N : equilibrium commuting cost : travel demand : commuting cost N TNP scheme C 22

23 Spatial Sorting Patterns Integrated model vs. naive model Job index non-zero elements of N Q, Q Job index Location index i N 7 7 Integrated model Naive model Proposition 4: The location-ob distribution of commuters is more dispersed in the integrated model. 23

24 Long-term Policy: BN Capacity Expansions BN capacity expansions A long-term policy of road construction Self-financing principle Lemma: The optimal amount of investment on BN capacity expansions coincides with the revenue of permits. - Wada & Akamatsu, 2013; Akamatsu, 2017 Optimal amount of investment in total: R ( Q) pi ( t Q)d i t i t 24

25 Policy Implication for BN Capacity Expansions Integrated model BN capacity expansions Naive model BN capacity expansions ρ Long-term { Q, r, w ρ} Short-term { q, p, ρ Q} Q C N ρ Long-term { Q, r, w ρ} Short-term { q, p, ρ Q} Q N TNP scheme TNP scheme Proposition 5: The naive model overestimates optimal amount of total investment on BN capacity expansions, N i.e. R( Q ) R( Q ). 25

26 Concluding Remarks An integrated model effectively synthesizing the long term and short term is proposed. Analytical solutions for DSO problems in a corridor network are derived. Qualitative properties of solution in equilibrium are investigated. A model considering short and long terms separately misleads long-term policies, such as investment on BN capacity expansions. 26

27 Appendix: False BNs without Heterogeneities Definition A false bottleneck is a bottleneck whose permit price is always zero in equilibrium. False BNs do not affect essential properties, but only cause irrelevant freeness in the solution. Criterion to detect false BNs BN i is a false BN if and only if for some A nm n m m i. ni i A n st () p () t 3 overlap (if BN 3 is false) t 27

28 Appendix: Comparisons of Land Rents and Wages Land rent Wage 28

29 Optimal Amount of Investment on Each BN Examples Naive model Naive model Integrated model Integrated model Naive model Integrated model Naive model Integrated model 29