Spatial Equilibrium and Search Frictions - an Application to the New York City Taxi Market∗ Ida Johnssonz March 18, 2018 Abstract This paper uses a dynamic spatial equilibrium model to analyze the effect of matching frictions and pricing policy on the spatial allocation of taxicabs and the aggregate number of taxi-passenger meetings. A spatial equilibrium model, in which meetings are frictionless but aggregate matching frictions can arise endogenously for certain parameter values, is calibrated using data on more than 45 million taxi rides in New York. It is shown how the set of equilibria changes for different pricing rules and different levels of aggregate market tightness, defined as the ratio of total supply to total demand. Finally, a novel data-driven algorithm for inferring unobserved demand from the data is proposed, and is applied to analyze how the relationship between demand and supply in a market with frictions compares to the frictionless equilibrium outcome. Keywords: spatial equilibrium, matching, industry dynamics, taxicabs JEL codes: ∗I would like to thank Professor Hashem Pesaran and Professor Hyungsik Roger Moon for their guidance and support. zDepartment of Economics, University of Southern California. Email: [email protected] 1 Introduction The distribution of taxicabs in big cities tends to be imbalanced, in some areas passengers have a hard time finding a taxi, in others taxis can’t easily find a passenger. In the taxi market, trades occur bi- laterally between agents, and the equilibrium outcome depends on the nature of the meeting process. For example, the matching mechanism between yellow cabs in New York and passengers, who have to hail the cab from the street, is different than for ride-hailing services such as Uber or Lyft that match passengers and cabs using an algorithm. As prices are typically fixed, the key equilibrium variables in the taxi market are meeting proba- bilities and aggregate market tightness (AMT), defined as the ratio of total supply to total demand. In this paper I first consider the spatial equilibrium model proposed by Lagos (2000), in which the taxi-passenger matching process is frictionless. Frictions in the model can arise endogenously if taxis choose to locate in such a way that there is excess supply in some locations and excess demand in others. Frictions depend on aggregate market tightness and location heterogeneity. I calibrate the model using data on over 45 million taxi rides and show that an equilibrium with frictionless matching yields excess supply of cabs in all locations, and that aggregate matches are thus determined by total demand. I then extend the model of Lagos (2000) to analyze how varying the pricing policy impacts the ratio of supply to demand in each location, as well as the aggregate number of matches. I consider both the current two-part tariff as well as a demand-dependent pricing policy. Finally, I consider a taxi-passenger matching process that involves local frictions in the sense that the number of rides originating from a neighborhood with x empty taxis looking for passengers and y customers in looking for a taxi might be smaller than minfx; yg. This type of matching process is considered by for example Frechette et al. (2015) and Buchholz (2016). Similar to Frechette et al. (2015), Buchholz (2016) and Afian et al. (2015), I propose a method of inferring unobserved taxi demand in the presence of local matching frictions. My method is based on defining a local measure of taxi availability, which can be approximated using data on taxi rides. I than compute time-specific local market tightness (tLMT), defined as the ratio of supply to demand in a specific location during a certain time interval, and compare the empirical outcomes with the results implied by the equilibrium of the frictionless matching model. 1.1 Related Literature Lagos (2000) proposes a spatial equilibrium model in which taxi drivers choose their location depending on demand and policy parameters. Matching is assumed to be frictionless, and it is shown that fric- tions arise endogenously in equilibria that correspond to certain parameter values. Lagos (2000) shows 2 that an aggregate matching function exists, and characterizes it’s properties with respect to distances between locations, the size of agent populations, as well as policy parameters. He shows that the spatial equilibrium depends on the demand transition matrix, the full set of bilateral distances, demand and policy parameters that determine fare rate and the number of medallions. Lagos (2003) uses the model of Lagos (2000) to quantify the impact of increasing taxi fares and the medallion cap on the medallion prices and the process that rules meetings between passengers and taxi- cabs in New York. I also use the model of Lagos (2000). The differences between my work and that of Lagos (2003) are as follows. Lagos (2003) assumes that all medallions are active throughout the day and uses data on 22,604 trips in 1988 to calibrate the model, characterize all equilibria, and quantify the meeting process generated by Manhattan’s market for taxicab rides. I use data from January-March 2013, which includes comprehensive information on all taxi rides during that period - around 45 million trips. Instead of assuming that all medallions are active throughout the day, I consider two different time periods - day and night - and calculate the number of active medallions from my data. Further, Lagos (2003) simplifies the map of Manhattan by dividing it into seven locations, whereas I consider forty different neighborhoods, both in and outside of Manhattan. Similar to Lagos (2003), I calibrate the frictionless spatial equilibrium model and characterize the no-frictions frontier. Further, I extend the model to include demand-driven pricing rules and calculate the equilibrium allocations for selected pricing rules. Another strand of literature considers equilibrium models of the NYC Taxi market in the presence of matching frictions. Frechette et al. (2015) propose a dynamic general equilibrium model of the NYC Taxi market, and back out unobserved demand through a matching simulation. They assume that taxi search times and passenger wait times (the latter unobserved) are an unknown function of the unob- served number of waiting passengers, searching taxis and other exogenous time-varying variables. They simulate the matching process of waiting passengers and searching taxis on a grid that represents a simplified version of Manhattan to back out the unknown function and the unobserved inputs. The authors do not consider location-specific heterogeneity. Buchholz (2016) models the driver’s choice problem using a state dependent value function. He assumes that demand for rides in each location comes from a location-specific Poisson distribution, and estimates the parameters of these distributions as well as the supply of taxis from the equilibrium of the model. My work is similar to Buchholz (2016) in the sense that I also consider location-specific supply and demand. However, rather than assuming a model for the equilibrium behavior of taxis and passengers and backing out unobserved supply and demand from the model equilibrium, I take advantage of the fact the we observe passenger pick-up and drop-off locations. I define a measure of taxi availability in each neighborhood and time period, and estimate taxi and passenger arrival rates directly from the data. This approach is similar to the one taken by Afian et al. (2015). 3 The rest of the paper is structured as follows. Section 2 describes the data. In Section 3 I present the benchmark model with frictionless matching, and analyze how the no-frictions frontier changes for different pricing policies. In Section 4 I introduce the matching process with local frictions and show estimates of local market tightness for different matching efficiency parameters. Section 5 concludes. I what follows I use lowercase letters to denote scalars, lowercase bold letters for vectors and uppercase bold letters for matrices. 2 Data I use a data set obtained by Mr. Chris Whong through a Freedom of Information Law (FOIL) request submitted to the Taxi and Limousine Commission (TLC)1. The data contains information on more than 173 million yellow cab trips in 2013. Each record contains complete information about a taxi ride. I use the following variables (a specification of all variables available in the data can be found in Appendix A). • Medallion - a unique identifier of the taxi cab. • Pickup and drop-off time and geographical coordinates Note that data on all yellow cab rides from 2009 till the present date, as well as green cab rides from mid-2013 onward and Uber rides for selected months of 2014 and 2015 is freely available in Google Big Query. However, that data does not contain information on the taxi or driver identifiers, which is crucial for inferring unmet demand. Green cabs were introduced in the summer of 2013. Hence, I restrict my analysis of the yellow cab market to the period before the summer of 2013. Specifically, I consider data from January-March, which contains information on approximately 45 million taxi rides. I restrict my analysis to this period due to the computational costs of geofencing the data. Using the geographical coordinates of the pickup and drop-off locations for each trip, I map the trip start and end points to a neighborhood. I use the neighborhood definitions of the NYC Neighborhood Tabulation Areas (NTAs)2, which are shown in Fig- ure A7 in Appendix A. Mapping pickup and dropoff coordinates to NTAs for January-March 2013 took approximately 72 hours running on an Amazon Web Services (AWS) memory-optimized PostgreSQL instance of type db.r3.8xlarge with 32 vCPUs and 244GiB Memory. 1Mr. Whong made the data set available on his website http://chriswhong.com/open-data/foil_nyc_taxi 2See https://www1.nyc.gov/site/planning/data-maps/open-data/dwn-nynta.page 4 I consider 15-minute time intervals during the time period January-March 2013.