Understanding patterns and competitions of short‐ and long‐term rental markets: Evidence from London

In this article, we compare short‐term rental (STR) and long‐term rental (LTR) price patterns in London using one of the most popular STR platforms, Airbnb, and the LTR platform, Zoopla property website. This research aims to enhance our understanding of both LTR and STR price patterns; as well as STR dynamics specifically, using predictive modeling to analyze how the patterns might evolve. We used the coefficient of variation and correlation analysis to examine the rental price patterns of both short‐ and long‐term markets. Then we developed a rent‐based gravity model to predict STR price pattern that is sensitive to the changes in visits to tourist destinations. Based on our analysis, we concluded that: (1) STR prices tend to be higher overall with an indication of higher volatility (less stability) compared to LTR; (2) there is statistical evidence supporting the arguments that STR and LTR markets are indeed in competition; and (3) the proposed gravity model provides a robust prediction of the STR pattern with a characteristic that higher‐priced short‐term properties are found to be geographically concentrated in the core city areas and those surrounding residential areas with easy access to popular tourist attractions.

to enhance our understanding of both LTR and STR price patterns; as well as STR dynamics specifically, using predictive modeling to analyze how the patterns might evolve.
We used the coefficient of variation and correlation analysis to examine the rental price patterns of both short-and long-term markets. Then we developed a rent-based gravity model to predict STR price pattern that is sensitive to the changes in visits to tourist destinations. Based on our analysis, we concluded that: (1) STR prices tend to be higher overall with an indication of higher volatility (less stability) compared to LTR; (2) there is statistical evidence supporting the arguments that STR and LTR markets are indeed in competition; and (3) the proposed gravity model provides a robust prediction of the STR pattern with a characteristic that higher-priced short-term properties are found to be geographically concentrated in the core city areas and those surrounding residential areas with easy access to popular tourist attractions.

R E S E A R C H A R T I C L E
Understanding patterns and competitions of short-and long-term rental markets: Evidence from London

| INTRODUCTION
In recent decades, there has been a rapid development of short-term rental (STR) platforms, including Airbnb. As a digital platform, Airbnb offers short-term accommodations by providing means to connect hosts, those with entire apartments, private or shared rooms to rent, and guests, travelers who would like to rent a place. Airbnb is an example of a phenomenon that has been emerging for the last decades, known as the online platform economy. This refers to the open-access approach of utilizing assets and services (Richardson, 2015;Rifkin, 2001), mostly for-profit (Belk, 2014), and generally mediated by Internet technology (McLaren & Agyeman, 2015;Schor, 2014). Airbnb as a hospitality online platform offers two main benefits to those who participate: (1) economic incentives for hosts from renting rooms or properties; and (2) choices of short-term accommodation with extra amenities often not available at traditional accommodation such as hotels for guests.
As a topic of contemporary interest, various research has focused on examining the implications of Airbnb. This includes the impact on traditional accommodation sectors where Airbnb are argued to have a profound impact on hotel revenues (Zervas et al., 2017) as well as the long-term housing market (Ayouba et al., 2020;Barron et al., 2017;Shabrina et al., 2022;Shokoohyar et al., 2020). Shokoohyar et al. (2020) found that certain neighborhood characteristics contribute to a higher rate of return when properties are being rented as STR. Ayouba et al. (2020) pointed out that Airbnb rentals put outward pressure on rents in several French cities and Shabrina et al. (2022) presented an analysis concluding that Airbnb misuse in London (in which entire home property listings do not follow the local regulations) can be associated with disruptions toward the traditional long-term housing market. Various studies emerge within the same themes, highlighting the challenges from the emergence of Airbnb, especially in relation to the housing market (Benitez-Aurioles et al., 2020) and examining the competing supply between long-term rental (LTR; for residents) and STR (for visitors) (Barron et al., 2017).
This article contributes to providing insights into the relationship between STR and LTR price patterns in London, and the dynamics of STR specifically through the use of predictive modeling. We propose several research questions (RQs) as follows: 1. RQ1: Are there any differences in the rental price patterns of STR and LTR in London? 2. RQ2: Is there any overlap between the two rental markets, indicating competition between STR and LTR? 3. RQ3: Can we predict STR price patterns in London using the gravity model to help understand its dynamics?
In our study, we use the assumption that STR are marketed to tourists while LTR are targeted at residents who live and work in London. Using both exploratory analysis and predictive modeling, we provide insights into STR and LTR simultaneously. For the latter analysis, we develop a gravity-based model using location attributes such as proximity to tourist locations based on the ease of accessing those locations which is an important factor for successful Airbnb listings (Tussyadiah & Zach, 2017).
The article is organized as follows. In Section 2, we explore the theoretical contexts and previous studies related the STR and LTR. In Section 3, we introduce the data used for the analysis and prediction including the method used for building the doubly constrained gravity model. Section 4 presents the results of the exploratory analysis and the predictive model along with a critical discussion of the implications of our analysis. Lastly, we conclude by synthesizing the theoretical and practical implications of our analysis and presenting the limitation of the study. We have also included a methodological annex in the appendix to provide the details of our gravity model.

| STR AND THE DISPLACEMENT OF HOUSING FOR LONG-TERM RESIDENTS
The rapid emergence of STR globally has sparked a heated debate with the Airbnb platform placed at the center of these discussions. In the last years, various studies have analyzed the spatial distribution of Airbnb locations (Eugenio-Martin et al., 2019;Gutiérrez et al., 2017;Sans & Quaglieri, 2016). Airbnb has been mentioned to have a profound impact on the communities where the properties are located (Gutiérrez et al., 2017), associated with housing disruption (Barron et al., 2017;Shabrina et al., 2021) as well as disruption to other competing tourist accommodation (Zervas et al., 2017).
It is challenging to separate the discussion between STR and LTR as they come from the same supply, residential housing. Traditionally, tourists are heavily concentrated in areas where hotels are located. This could be in central areas (monocentric) or agglomerated around certain geographical points such as tourist destinations, airports, hospitals, etc. (Egan & Nield, 2000;Yang et al., 2014). However, Airbnb locations are more dispersed and geographically uneven as the listings are not just concentrated in prime urban cores, but also in other residential areas especially those in close proximity to restaurants and other amenities (Coles et al., 2017;Shabrina et al., 2021). These listings are located in highly desirable residential areas that otherwise would be available for long-term tenants (Lee, 2016;Shabrina et al., 2021;Wachsmuth & Weisler, 2017).
It has been pointed out further that Airbnb has been creating pressures, especially on housing and neighborhood in urban centers where long-term housing provision is already stressed (Cocola-Gant, 2020). Highly accessible neighborhoods, like those in the city center, are more desirable as Airbnb locations since Airbnb offers a high revenue premium (Deboosere et al., 2019). A study by Wachsmuth and Weisler (2017) shows that Airbnb allows a STR gap. Unlike the central argument proposed by Clark (1995) stating that there is a long life cycle explaining the property rent gap, Airbnb has made it possible to capitalize on the rental gap without major investment or redevelopment, only by the simple but controversial step of removing the existing tenant (Wachsmuth & Weisler, 2017). This is especially prominent in cities with high tourism activities such as Barcelona, where the consequence of Airbnb toward house prices is apparent (Garcia-López et al., 2020). STR platforms such as Airbnb have created flexibility for property owners, making it easy to follow the existing demands, adjust prices, and avoid lease-related laws; thus, STR are less likely to return to the long-term residential markets (Cocola-Gant, 2020). This could create extra challenges, especially in cities that rely heavily on rental property markets such as those in the United Kingdom and other European cities.

| DATA AND METHODS
In this study, we use various datasets to support our exploratory data analysis of STR and LTR and build our predictive model of short-term rents using a doubly constraints gravity-spatial interaction model.

| Data
We use four main datasets including Airbnb data for STR, Zoopla property data for LTR, number of visits to tourist attractions, and travel time data. The details of the datasets are shown in Table 1. Both Airbnb and Zoopla data have been aggregated into lower layer super output areas (LSOAs), a geographic hierarchy of smaller area statistic in England with average population of approximately 1500 people.

| Methods
To answer the proposed RQs, we conducted a series of statistical investigations and applied predictive modeling to examine the relationship between STR and LTR (RQ1 and RQ2) and enhance our understanding of STR price specifically (RQ3). We use the coefficient of variation (CV) and correlation analysis to examine the patterns of STR and LTR prices. We also use the gravity-spatial interaction model to calculate the predicted rental price pattern of Airbnb. The complete methodological flow used in this article is presented in Figure 1.
Through this research, we apply an exploration of the gravity model in tourism studies describing the interaction flow (or tourism flow) between two regions, areas, or locations. The interaction is influenced by the push factors associated with the flow leaving region i for tourism reasons (outflows) and the pull factors associated with the flow going to region j for tourism reasons (inflows) (Patuelli et al., 2013). In this article, we expand this basic concept and propose a novel form of gravity model for tourism analysis, by deriving the rental prices for given origin areas.
We have investigated relevant research in building the model specification. Tussyadiah and Zach (2017) suggest that proximity to points of interest and the characteristics of the neighborhood are two of the most important contributing factors to a successful Airbnb location. This is supported by the finding by Volgger et al. (2018) stating that Airbnb and non-Airbnb guests have a similar preference in terms of staying in areas with good access to well-known and iconic touristic highlights. This is strengthened by a study based on the sentiment analysis of Airbnb T A B L E 1 Data sources and variables used in the study F I G U R E 1 Flowchart of the methodology used reviews, which found that one of the key attributes of an Airbnb experience is location, followed by amenities and hosts (Cheng & Jin, 2019).
Thus, the main elements in our doubly constrained gravity model are: (1) the constraint on destination trip ends (i.e., the number of visitors to a given attraction); (2) the cost of deterrence function f(c ij ), which is the travel time data; and (3) a constraint on origin trip ends, which is the historic Airbnb supply. The model shows the relation of different locations by examining the degree of spatial interaction reflecting Tobler's First Law of Geography which states that 'everything is related to everything else but near things are more related than distant things' (Tobler, 1970). The measure of spatial separation is given by the deterrence function. Flows can be modeled where T ij is the flow from origin i to destination j, K is a scaling constant, O i is attraction at the origins i, D j is attraction at the destinations j, and c ij is the deterrence factor, following the equation: c ij represents the cost, which can be a generalized cost or weighted sum of distance d ij and travel time t ij costs. The negative exponential cost function A e −βc ij arises from the maximizing of entropy with a constraint on mean trip cost. A detailed description of the gravity model method used in this article is provided in the Methodological Annex.

| Exploration of STR and LTR price pattern
Using Zoopla property website data, we examine areas that have been experiencing major rental price changes in London. Figure 2 visualizes the percent rent changes in London LSOAs based on Zoopla data from 2010 to 2017 showcasing an East-West section view of the difference between the rental prices in the two time points. We observe sharp spikes appearing around East and Central London with some variations in the South and West parts of London near Heathrow Airport. The high and increasing rental value in central London further emphasizes London's monocentric trend, where the employment centers, retails, and many urban activities are concentrated in the core city areas. On the other hand, East and South London are areas that have been experiencing fast developments of new residential developments that might contribute to the rapid rental price changes.
Just like the British housing system, housing in London is dominated by private ownership, many of which are rented out to private tenants. According to official government data, the median monthly rental price in London is £1425 which is 190% higher than the median monthly rental prices of England (£755) between October 2020 and F I G U R E 2 3D visualization of percent rent changes in London 2010-2017. Left: East-West snapshot of rental price changes in London showcasing higher variations in Central and Eastern part of London. Right: Bird-eye snapshot of rental prices changes in London. The color represents the percentage changes according to its intensity from yellow (low changes) to red (high percentage rental price changes) September 2021 (ONS, 2021). This price difference is also apparent locally as the median rental price of Inner London areas is £1600 compared to £1300 in Outer London areas (ONS, 2021). Considering the LTR market trajectory especially with the stark housing price difference in London compared with the remainder of the UK, it is useful to investigate how the rapid influx of STR might also contribute to the overall LTR price changes. Table 2 shows the descriptive statistics for the Airbnb and Zoopla datasets in London representing the supply of STR and LTR, respectively. Our first exploration is aspatial, meaning that we examine all the available Airbnb and Zoopla listings regardless of their location. For our initial analysis, we only include STR data with at least one review as a proxy of active listings. Airbnb has grown exponentially between 2015 and 2017 while properties being marketed on the Zoopla property website are quite steady. The annual Zoopla data sample is up to around seven times larger than active Airbnb listings as the LTR market is generally larger overall.
We can see that the mean Airbnb prices in London are consistently much higher compared with the mean Zoopla price per bedroom. In 2017, the mean price per bedroom of Airbnb is 246% higher when compared with the Zoopla mean prices. For example, based on a recent market study in the United States by AlltheRooms website (https://www. alltherooms.com) that analyses STR markets, the average occupancy rate of Airbnb is around 48%. Despite Airbnb occupancy might be lower than those for long-term properties, Airbnb often has a higher rental value in total. To create a comparable measurement, we also calculate the CV for the STR and LTR prices from 2015 to 2017. CV is also known as the coefficient of dispersion and is often used as a measure of volatility (Brown, 1998). Based on the CV calculation, we found that Airbnb consistently yields a higher CV of up to 0.6 indicating that prices disperse from the mean. An explanation would be that Airbnb prices can be easily adjusted following the increase and decrease in demand, seasonality, and even personal preferences. Airbnb and other known STR markets are using dynamic pricing for the purpose of achieving demand equilibrium with high-demand hosts using larger price variations as a strategy to optimize their revenue (Gibbs et al., 2018;Shokoohyar et al., 2020).  (Almeida, 2021). Further investigation is needed to examine the implication of highly gentrified places that are populated by some of the highest priced Airbnb supply. Previous studies have highlighted that Airbnb thrives in central and touristic areas (Gutiérrez et al., 2017) which is apparent from our maps. This aligns with Gibbs et al.'s (2018) finding that physical locations matter greatly and yield high revenue. Thames. These West London areas were historically planned as residential areas and have now developed into upmarket areas with one of the most highly priced rentals.
We further investigate whether the annual rental changes of STR can be associated with the changes in the LTR market. The analysis is done at the borough level as the Zoopla data are quite sparse at the LSOA level (we do not have enough sample data for each LSOA even though LTR are obviously present), thus aggregation is needed. previous studies that have indicated Airbnb could be associated with a rental price increase for long-term residents (Barron et al., 2017;Lee, 2016;Shabrina et al., 2022;Todd et al., 2021) and even contributed to Airbnb-induced gentrification (Wachsmuth & Weisler, 2017).
We have explored the difference and overlaps in patterns of STR and LTR and provided an indication of how these two markets are in competition. For the next section, we shift the focus to modeling the STR price pattern.

| Predicting Airbnb price pattern using doubly constrained gravity-spatial interaction model
In this section, we focus on the STR price pattern and create a model to predict the Airbnb price pattern in 2019 based on the historical Airbnb attributes in 2018, the number of annual visits to tourist attractions in 2019 and easiness to reach those destinations. We implement the gravity model which provides a reasonable computation on where Airbnb locations are desirable based on the supply of Airbnb data in relation to access to various touristic destinations. Through the model, we have created a tool for predicting future Airbnb price patterns that consider the destinations' data. We use Greater London as our study area.
The gravity model simulates spatial interaction which refers to any movement as a result of human processes over space (Haynes et al., 1984). The idea is that the spatial interaction between places increases with improved accessibility to destinations, stressing the importance of accessibility as a key concept within spatial interaction models (Condeço-Melhorado et al., 2014). Bhat et al. (2000) emphasized accessibility as the ease of pursuing any kind of activities in any given location using a desired mode of travel. This type of concept uses the location perspective, in terms of how land use and transport influence access to destinations using several combinations of transport modes (Geurs & Van Wee, 2004), where locations are then assessed based on the number of activities that can be reached and weighted using travel time to destinations (Bertolini et al., 2005).
We construct our accessibility measure using the gravity-spatial interaction model. The model consists of three important components: (1) the origins (locations of Airbnb rentals in the previous year aggregated into LSOA level); (2) destinations (the points of interest that can be reached); and (3) the cost of traveling from origins to destinations given by the spatial separation. Underlying our spatial interaction model is the premise that Airbnb rental prices can be predicted based on the historical Airbnb data and external factors with assumptions that Airbnb will locate in places with high accessibility to tourist attractions (as measured by the number of visits), and those locations relate to well-sought-after neighborhoods, as indicated by their rents. It is worth noting that we filter our Airbnb data to only include those that have received at least 10 reviews to account for the listings that have been booked repeatedly.
We incorporate the number of visits to tourist attractions in London's central areas as the pull factor of our model. Attractions in urban tourism are often used to boost the image and economy of a city and provide destinations for tourists (Ashworth & Page, 2011;Judd & Fainstein, 1999;Law et al., 1993;Papadimitriou et al., 2015). Another important element in our model is the use of travel time as the cost function. An assumption in spatial interaction is that ceteris paribus, travel flow declines with the difficulty of travel. Our cost function is expressed as A e −βt ij , as in Equation 1 but with time t ij replacing cost c ij . We then derive the centroids of the 4835 LSOAs in London as our origin points and calculate the travel time to the 88 tourist destinations as our destination points using the Open Trip Planner (OTP) tool. OTP uses contraction mapping and the A* heuristic algorithm (Russell & Norvig, 2003) to find the optimal path and provide travel times between two geographical points based on the user's desired criteria (Hillsman & Barbeau, 2011). Travel times provide a more accurate representation of deterrence function compared to distance as it captures the actual time spent to travel between places as some destinations might be further apart in distance but can be reached relatively quickly using public transport.
The Furness (1965) iteration is used to iterate the base matrix, that is, A e −βc ij to the given row and column totals. For any given value of β, we solve the model iteratively and we calibrate the model by choosing the value of β which gives a mean trip cost sufficiently close to the mean. This value of β is chosen as that value that minimizes the J-divergence between the distributions (Rohde, 2016). The β values with the lowest J for our model is β = 0.008. Based on the model calibration, we derive the predicted rents of Airbnb in London. Figure 6a  year. Based on Figure 6, we can see that the prediction presents a good depiction of the observed rental pattern with higher rental prices spanning across the south-western to north-eastern axis.

F I G U R E 5 Annual number of visits to main tourists attractions in London 2019
The Airbnb gravity approach models the Airbnb market that computes accessibility of areas desirable for Airbnb locations. These areas are locations where tourists can reach a set of popular touristic destinations using public transport services. This computation is translated into rents, with the assumption that desirable Airbnb locations would have higher price premiums. As the model is computed in an aggregate form, the model does not take into account that with Airbnb's supply becoming saturated, listings will eventually compete with each other, and prices might go down due to the competing supply to reach equilibrium. This seems to be the case as shown in Figure 6c computed by subtracting the observed Airbnb mean rental price 2019 from the predicted rental price 2019. Examining the model residual is useful to investigate areas where the model is under-and over-performing (represented as blues and reds respectively). The model seems to be over-performing in some areas, especially in the East, Northeast, and some areas in South London. There is a cluster of areas where the model is over-performing meaning that the observed rents are smaller than the predicted rents.
This seems to be particularly apparent in areas with high listing counts such as those in East London. Further investigation is needed to examine this.
Considering the volatility of the Airbnb market, we have shown that it is possible to predict changes to the system such as those due to the closure of tourist destinations during the pandemic as well as development although this is beyond the scope of our article. The gravity model can be used to identify the impact of changes in accessibility on rents and as such is a useful policy tool for impact analysis if the possible driver of the impact is specified either in network changes or in trip ends (or their proxies).

F I G U R E 6
The result of doubly constrained gravity-spatial interaction model

| CONCLUSION
To summarize, our study provides insights into the competition and patterns of STR and LTR. Examining the supply of both Airbnb and Zoopla data over the years can help our understanding of the overlaps between the two markets.
This study has practical and theoretical implications that contribute to the discourse on STR and their association with long-term housing.

| Theoretical implications
This study has used a series of quantitative approaches to investigate STR and LTR patterns using statistical analysis and gravity-spatial interaction modeling. We have incorporated a total of more than 200,000 Airbnb and around 500,000 Zoopla data points to create a robust analysis for examining the STR and LTR rental patterns and this can contribute to the understanding of the spatial patterns of these markets. We also propose a model that can take into account the dynamics of STR in relation to visits to tourist destinations. By incorporating the rent-based gravity model, we expand the use of the gravity model in tourism studies that will be useful to capture a dynamically changing tourism market.

| Practical implications
The result of this study can assist planners and policymakers to understand the competition from STR in relation to LTR. We have used a historical dataset of Airbnb supply from 2015 to 2017 in our analysis and found that Airbnb price tends to be higher and more volatile compared with LTR. By closely looking at the patterns of the two markets, policymakers can use a data-driven approach to be able to keep an eye on specific areas in which Airbnb premiums are high and many properties have been transformed into STR. As the correlation analysis shows that STR and LTR prices are highly correlated, with some of the highest priced long-term properties located in areas populated with Airbnb, further exploration is needed at the local level to account for the competing needs between properties for residents and those available for tourists. Regarding the model, it can be used as a tool to simulate future aggregate Airbnb price patterns that can take into account various possible changes in the tourism sector.

| Limitations and merits
It is important to note that the results of this study should be approached with consideration of the study's limitations.
First, most of the analysis is done at an aggregate level, including the borough and LSOA levels. We need to acknowledge that analysis at an aggregate level might suffer from the Modifiable Areal Unit Problem (see Wong (2004) for details on MAUP). Despite this limitation, this study has showcased analysis using different spatial scales, including at point (the statistical analysis and CV) and aggregated levels. The approach of using a different spatial scales is necessary to provide effective analysis considering the nature of the data, such as Zoopla data that are not distributed equally.
Second, for both Airbnb and Zoopla data, we have used listed price data which can be used to showcase the prices for STR and LTR that are available and visible for renters. As they are not transaction prices, we are analyzing supply data instead of demand data. For the gravity model, we have not included competition between Airbnb listings themselves, by assuming that popular Airbnb locations would be utilized by tourists regardless of what is being offered. We acknowledge that the model might suffer from biases as we have not incorporated other possible elements that might influence the price of Airbnb listings, such as competing listings and the intrinsic elements of the property itself including the amenities provided, building quality, host responsiveness, etc., as the model restricts this toward the location aspect of Airbnb. However, the model is useful in providing a way to simulate price patterns that is sensitive to changes in destinations as external factors repeatedly mentioned as being an important determinant for Airbnb (Deboosere et al., 2019;Tussyadiah & Zach, 2017).
Third, our analysis does not take into account the existing policies that might restrict Airbnb locations. The deregulation of short-term lettings in London (Ferreri & Sanyal, 2018) still provides some restrictions for hosts, while other granular level restrictions such as building associations, etc., have also made an effort to control where Airbnb can be located. This might somewhat affect STR rental distribution, perhaps even more than the LTR market. For further studies, these various factors as mentioned above should be considered.
In summary, our study contributes to providing rigorous geographical and quantitative analysis of the possible economic disparities between STR and LTR prices. We have provided an investigation of STR and LTR in one of the most heated housing markets in the world where the growth of Airbnb supply is very fast and might put pressure on the long-term markets. We also presented evidence of overlaps between STR and LTR markets which could indicate competition between them. Lastly, we provide the first predictive analysis of Airbnb prices using a novel form of gravity-spatial interaction model, expanding the application of the traditional gravity model to tourism-related studies.

ACKNOWLEDGMENT
We would like to thank the journal editors and anonymous reviewers for their valuable comments and suggestions.
We would also like to thank Prof. Michael Batty and Dr. Elsa Arcaute from the Centre for Advanced Spatial Analysis (CASA), University College London, for their insightful feedback and continuous support, as well as Dr. Richard Milton (CASA) for providing public transport schedules in General Transit Feed Specification (GTFS) format and for contributing in the extraction of the travel time data used in this study.

DATA AVAILABILITY STATEMENT
Zoopla data is subject to third party restrictions (Glasgow Urban Big Data Centre).

A.1 | Antecedents
The gravity model we use in this research is based on the entropy maximizing derivation (Wilson, 1971). There are, however, many precursors including in transportation studies such as the work of Desart in the 1840s (Odlyzko, 2015) and migration (Ravenstein, 1889). In the 20th century, applications were developed in trade theory (Tinbergen, 1962) and retailing (Reilly, 1929), while its modern use dates from its application to retail planning (Lakshmanan & Hansen, 1965). Its current use in transport planning (Ortuzar & Willumsen, 2011) forms the basis of land-use transportation interaction models to the present day (Martinez, 2018). The gravity model has also been used in a variety of tourist analyses (Khadaroo & Seetanah, 2008;Morley et al., 2014;Patuelli et al., 2013), although its use in this context is still quite limited (Hall, 2006).

A.2 | Derivation
The derivation of the model follows that of Wilson (1970) with the difference that we use trip probabilities (p ij ) rather than trips (T ij ) and we include an extra constraint to ensure the probabilities sum to unity. This allows us to use the standard thermodynamic formalism of Gibbs (Cowan, 2005) and to include the partition function Z. We set up the Lagrangian multiplier Equation A1 which on differentiating A L with respect to p ij and setting the result equal to zero, which gives the model Equation A2.

A.3 | Estimation using mean trip cost
Knowing the origins p i and the destinations p j together with the trip costs c ij and a given value of β, the expected trips may be calculated using a Furness iteration (Furness 1965). Conventionally, the value of β is chosen to equate the modeled and observed trip costs as this corresponds to a maximum likelihood estimate. The value is found using the method of Hyman (1969). In our case, however, we recognize that β is not the only parameter in the model.

A.4 | Estimation using rent
The terms A e −λ i and A e −λ j in Equation A2 are the balancing factors which are also parameters of the model and which ensure that row and column trip totals are equal to the exogenously determined origins and destinations. The terms A − 1 β ln λ i and A − 1 β ln λ j can be identified with von Thünen location rents (Morphet, 2013). The model is estimated as above using a Furness iteration which gives the estimated trips and the balancing factors from which we calculate the estimated rents for each zone. These are then compared with the observed rents and a value of β is found which minimizes the J-divergence between observed and modeled rents (Rohde, 2016). In our model, this is done solely for origin zones that correspond to residential rents. The rents are rents per trip, which, knowing the number of origins, can be converted to total rent per zone. The zonal rents can be converted into rent rates per unit area, per household or whatever denominator is required for the study at hand.

A.5 | Accessibility
The logarithmic terms for rent have also been identified as accessibility measures (Martinez, 1995) in comparison with more conventional measures of accessibility such as by Hansen (1959) which are additive rather than multiplicative. The argument for considering them as rents, apart from their links to von Thünen, is that they have dimensions of cost, as an inspection of Equation A.2 readily shows. Consideration of rents may also be more familiar and graspable for decision-makers than are the various definitions of accessibility (Geurs & Van Wee, 2004).

A.6 | Application to London's Airbnb rent market
The model requires data on origins, destinations, and travel costs between a set of zones. Travel time is used as the measure of the trip cost and is obtained from OTP, open-source software for multi-modal route planning using a combination of the Open Street Map network and public transit schedules in the General Transit Feed Specification format (https://gtfs.org/) for multi-modal application development (Antrim et al., 2013).
Travel times using both rail and bus datasets as public transport modes were obtained from Association of Train Operating Companies. The London bus and tube data were obtained from Transport for London. Travel time data were computed for three different days, mixing weekends and weekdays. The shortest time was chosen so as to mitigate the possibility of longer travel times due to disruptions or planned engineering work on those days. Also incorporated were walking times, transit times (time spent waiting in between modes), and boarding times (number of transfers between transit modes) to increase the relevance of our travel time data. Thus, the time calculated is time spent for the whole journey from an origin to a destination. The resulting cost data matrix is of travel times to the 88 main London tourist attractions from 4835 origins based on the centroid of each London LSOA. Around 450,000 travel time records were used from which were computed 81% of the system's travel time data. The remaining travel time data is beyond the 100 min maximum threshold set for the analysis on the assumption that tourists would not make trips of greater duration.
In the absence of observed origins and destinations, we use proxies. Destinations are proxied by the annual reported visitors at each of the attractions. Origins are proxied by the number of Airbnb beds in each LSOA (Table A1). This gives us the definition of p i and p j as shown in Equations A3 and A4 below: