Elsevier

Journal of Economic Theory

Volume 165, September 2016, Pages 643-671
Journal of Economic Theory

Transferring ownership of public housing to existing tenants: A market design approach

https://doi.org/10.1016/j.jet.2016.07.004Get rights and content

Abstract

This paper explores a housing market with an existing tenant in each house and where the existing tenants initially rent their houses. The idea is to identify equilibrium prices for the housing market given the prerequisite that a tenant can buy any house on the housing market, including the one that he is currently possessing, or continue renting the house he is currently occupying. The main contribution is the identification of an individually rational, equilibrium selecting, and group non-manipulable price mechanism in a restricted preference domain that contains almost all preference profiles. In this restricted domain, the identified mechanism is the minimum price equilibrium selecting mechanism that transfers the maximum number of ownerships to the existing tenants. We also relate the theoretical model and the main findings to the U.K. Housing Act 1980 whose main objective is to transfer ownerships of houses to existing tenants.

Introduction

Matching theory has provided fundamental tools for solving a variety of house allocation problems. Examples include procedures for allocating unoccupied houses among a set of potential tenants (Svensson, 1994), methods for reallocating houses among a group of existing tenants (Shapley and Scarf, 1974), mechanisms for determining rents on competitive housing markets (Shapley and Shubik, 1971), and rules for setting rents on housing markets with legislated rent control (Andersson and Svensson, 2014). This literature is appealing from a theoretical perspective since the investigated allocation mechanisms typically satisfy a number of desirable axioms, including, e.g., individual rationality, non-manipulability, and various efficiency notions.

The housing market considered in this paper contains a finite number of houses with an existing tenant in each house. The idea is to identify equilibrium prices for this market, given the prerequisite that a tenant can buy any house on the market, including the one that he is currently occupying, or continue renting his house. In the model, a fixed lower bound for the equilibrium prices is defined (i.e., reservation prices for the owner), and in case an existing tenant buys the particular house he is currently occupying, the tenant pays only the reservation price. The reservation price can be interpreted as a personalized or discounted price for the existing tenant as the price of that specific house for all other tenants is given by the equilibrium price which is endogenously determined by the preferences of all agents. This also means that if an existing tenant decides to “Keep the House”, the tenant can either buy it at the reservation price or continue renting it. An assignment of agents to houses and a price vector constitute an equilibrium if each agent weakly prefers his assigned consumption bundle to keeping the house and to all other houses at the given prices, and if the assignment guarantees that the maximum number of agents buy a house in case there are several assignments that are compatible with the specific equilibrium price vector. Note that the first part of the equilibrium concept can be seen as a market equilibrium condition as all agents are assigned their most preferred bundle from their consumption set at the given prices.

To solve the above described house allocation problem, it is natural to search for a price mechanism that is individually rational, equilibrium selecting, and non-manipulable. This type of mechanism guarantees that no tenant can lose from participating, that no further rationing of the houses is needed, and that the reported information is reliable. Given the interest in these three specific properties, the perhaps most natural allocation mechanism is based on a “minimum equilibrium price vector” as this type of mechanism previously has been demonstrated to satisfy these specific properties in a variety of different contexts, including, e.g., single-item auction environments (Vickrey, 1961), assignment markets (Demange and Gale, 1985, Leonard, 1983), and housing markets with rent control (Andersson and Svensson, 2014).1 Another natural price mechanism for the above described housing market is an individually rational, equilibrium selecting, and trade maximizing mechanism. The latter axiom guarantees that the maximum number of houses are transferred to the tenants in equilibrium. This is often a specific policy goal for this type of housing market as will be explained later. Here, we note that an equilibrium selecting minimum price mechanism guarantees that the number of agents who buy a house is maximal for a given minimum equilibrium price vector, whereas a trade maximizing mechanism guarantees that the number of agents who buy a house is maximal among all equilibrium price vectors.

Even if it is natural to believe that the above two mechanisms always recommend the same selection, as lower prices intuitively should increase trade, it turns out that this is generally not the case. In fact, it is not even clear what any of the above two mechanisms recommend due to the non-uniqueness of a minimum equilibrium price vector and the non-uniqueness of an allocation that maximizes trade on the full preference domain. The non-uniqueness property of the two mechanisms is somewhat unexpected, given what we know from previous literature (e.g., Demange and Gale, 1985, Leonard, 1983, Shapley and Shubik, 1971), and it is a direct consequence of the fact that agents can block trade of a house via their outside option to keep their house (see Section 4 for further discussions).

This multiplicity has the consequence that any allocation mechanism based on minimum equilibrium prices is manipulable on the full preference domain. Intuitively, this can be understood by imagining a situation when two minimum price equilibria exist for given preferences and when one agent strictly prefers one of them over the other. In this situation, it may be possible to manipulate the mechanism by misrepresenting preferences in such a way that only the strictly preferred equilibrium remains (because agents may be indifferent between several houses at the two minimum price equilibria and a certain misrepresentation guarantees the less preferred equilibrium not to remain an equilibrium after misrepresentation). Similarly, the multiplicity of an allocation that maximizes trade has the consequence that any allocation mechanism based on maximum trade is manipulable on the full preference domain. However, by considering a preference domain that contains almost all preference profiles, it turns out that a minimum equilibrium price vector is unique for each preference profile in the restricted domain, and that it is possible to base an individually rational, equilibrium selecting, and group non-manipulable allocation mechanism on this unique price vector. In addition, this mechanism turns out to be a maximum trade mechanism in the restricted domain, and it, therefore, guarantees that the maximum number of houses are transferred from the owner to the existing tenants.

To the best of our knowledge, this paper is the first to provide an individually rational, equilibrium selecting, trade maximizing, and non-manipulable allocation mechanism for a housing market with existing tenants where monetary transfers are allowed.2 The main results presented here are non-trivial extensions of similar and previously known results because most of the previous literature either considers the case with no initial endowments and where monetary transfers are allowed, or initial endowments but where monetary transfers are not allowed. Furthermore, we are also not aware of any matching model that can deal with the type of outside option considered in this paper when monetary transfers are allowed. In the following, we explain in more detail how our theoretical findings contribute to the related literature.

The idea to use a minimum equilibrium price vector as a key ingredient in an individually rational, equilibrium selecting, and non-manipulable allocation mechanism was first advocated by Vickrey (1961) in his single-unit sealed-bid second-price auction. This principle was later generalized by, e.g., Demange and Gale (1985) to the case when multiple heterogenous houses are sold.3 The significant difference between this paper and Demange and Gale (1985) is that the model in the latter paper cannot handle the case with existing tenants, and, consequently, not the case when tenants can block trade of a house through the “Keep the House” option. This outside option makes our model less “well-behaved” compared to the one in Demange and Gale (1985), e.g., the set of equilibrium price vectors generally does not have a lattice structure. In the special case when this outside option is not available, the minimum price mechanism investigated in this paper recommends the same outcome as the mechanism proposed by Demange and Gale (1985). Even though the mechanism considered here and the one in Demange and Gale (1985) are defined for different types of housing markets, they share the properties of group non-manipulability and that agents are always assigned their most preferred consumption bundle from their consumption sets at the prices determined by the mechanism.

A model with existing tenants and prices is studied by Miyagawa (2001). In his setting, each tenant owns precisely one house and is a seller as well as a buyer, so money and houses are reallocated through a price mechanism exactly as in this paper. Unlike this paper, and the ones cited in the previous paragraph, Miyagawa (2001) requires the mechanism to be “non-bossy” (Satterthwaite and Sonnenschein, 1981) which roughly means that no tenant can change the assignment for some other tenant without changing the assignment for himself. The assumption of non-bossiness has dramatic consequences on any non-manipulable allocation mechanism in this setting (see also Schummer, 2000). Namely, any individually rational, non-bossy, and non-manipulable allocation mechanism is a fixed price mechanism, and, therefore, agents need not be assigned their most preferred consumption bundle from their consumption sets. Because we do not require non-bossiness, our allocation mechanism is not a fixed price mechanism and it always selects an equilibrium outcome.

Individually rational and non-manipulable mechanisms for housing markets with existing tenants have been considered previously in the literature when monetary transfers are not allowed. Most notably Gale's Top-Trading Cycles Mechanism (Shapley and Scarf, 1974), has been further investigated by, e.g, Abdulkadiroğlu and Sönmez (1999), Ma (1994), Postlewaite and Roth (1977), Roth (1982) and Miyagawa (2002). This mechanism always selects an outcome in the core. However, the allocation mechanisms in these papers cannot generally guarantee, in similarity with Miyagawa (2001), that agents are assigned their most preferred consumption bundle from their consumption sets. Neither can they handle the case with monetary transfers.

Finally, a recent and intermediate proposal is due to Andersson and Svensson (2014) where a set of houses with no existing tenants are allocated among a set of potential tenants using a price mechanism where prices are bounded from below and from above by price ceilings imposed by the government or the local administration. In this model, they define an individually rational, equilibrium selecting and group non-manipulable allocation mechanism that, in its two limiting cases, selects the same outcomes as the mechanism in Demange and Gale (1985) and the Deferred Acceptance Algorithm (Gale and Shapley, 1962). Both in Andersson and Svensson (2014) and in this paper, agents face exogenous constraints. More precisely, in Andersson and Svensson (2014) the price ceilings are common exogenous constraints whereas the “Keep the House” options in this paper are individual exogenous constraints. The exogenous constraint in Andersson and Svensson (2014) excludes all price equilibria for some preference profiles and, consequently, a rationing mechanism must be included in their equilibrium concept. Such a rationing mechanism need not be included in the equilibrium concept considered here. Due to these differences, even though several key steps leading to the main results in Andersson and Svensson (2014) and here are similar, the proof arguments of Andersson and Svensson (2014) cannot be applied here (further details are discussed in Remark 2 and in the main text).

Public housing is a common form of housing tenure in which the property is owned by a local or central government authority. This type of tenure has traditionally referred to a situation where the central authority lets the right to occupy the units to tenants. In the last 30–35 years, however, many European countries have experienced important changes in their policies. In the United Kingdom, for example, the “Right to Buy” was implemented in the U.K. Housing Act 1980. As its name indicates, the legislation gave existing and eligible tenants the right to buy the houses they were living in.4 Here, an eligible tenant is defined as a tenant who has been living in the house for at least three years before applying for the “Right to Buy” (before May 2015, the requirement was five years).5 A tenant could, however, choose to remain in the house and pay the regulated rent. The U.K. Secretary of State for the Environment in 1979, Michael Heseltine, stated that the main motivation behind the Act was:

“... to give people what they wanted, and to reverse the trend of ever increasing dominance of the state over the life of the individual.”6

In other words, the main objective of the Act was to transfer the ownership of the houses from the local or central authority to the existing tenants. To cope with this objective, the Act also specified that tenants could buy the houses at prices significantly below the market price. In the latest update of the Act from May 2015, this “discount” may be as high as the lesser of £77,900 (£103,900 in London) or 60 percent of the market value. The market value of the house is estimated by the landlord, possibly together with a District Valuer (who will conduct an independent valuation), and should reflect the true market value of the house. As can be expected, the result of the Act was that the proportion of public housing in United Kingdom fell from 31 percent in 1979 to 17 percent in 2010.7 In fact, more than 33,000 households have taken up their “Right to Buy” since April 2012. Similar legislations, with similar effects, have been passed in several European countries, including Germany, Ireland, and Sweden, among others.

The significant difference between the U.K. Housing Act 1980, and its European equivalents, and the model considered in this paper is that we extend the situation from the case where only an existing tenant can buy the house that he is currently occupying to a situation where houses can be reallocated among all existing tenants in a pre-specified area. Note that it is impossible to immediately make this type of reallocation within the framework of the Act and at the same time transfer the ownership of the houses from the landlord to the existing tenants at the discounted prices. There are, at least, two reasons for this. First, “mutual exchange”8 is allowed in the U.K., but if a mutual exchange is conducted, the tenants must live in their new houses for at least three years before being eligible to apply for the “Right to Buy”. Second, it is not possible for two (or more) tenants to use their “Right to Buy” and then immediately reallocate the houses by selling them to each other since resale is not allowed within five years after purchasing the houses unless the discount is paid back to the landlord (in fact, if a tenant resells within 10 years after buying, the tenant has to offer first the house either to the former landlord or to another social landlord in the area).

A relevant question is of course why the extension of the current U.K. system, proposed in this paper, should be of any interest to a policy maker. As we will argue, there are, at least, three good reasons for this. First, as tenants in our framework can always choose to continue renting the house they live in or to buy it at the reservation price (one can think about the reservation price in our model as the discounted market price), exactly as in the prevailing U.K. system, but have the opportunity to buy some other house, all tenants are weakly better off in the considered model compared to the current U.K. system. Second, because all sold houses in the current U.K. system are sold at the reservation prices, but all sold houses in the considered model are sold at prices weakly higher than the reservation prices, the public authority generates a weakly higher revenue in the considered model compared to the current U.K. system. Third, the considered model captures the idea that housing needs may change over time, e.g., a family has more children or some children move out. In such cases, a tenant may want to change house and use the “Right to Buy”. These tenants are given permission to participate in our housing market, but they are not allowed to immediately participate in the prevailing U.K. system as explained in the above.

The remaining part of this paper is organized as follows. Section 2 introduces the formal model and some of basic definitions that will be used throughout the paper. The allocation mechanisms are introduced in Section 3 where also the main existence and non-manipulability results of the paper are stated. Section 4 contains some concluding remarks. All proofs are relegated to Appendix A.

Section snippets

The model and basic definitions

The agents and the houses are gathered in the finite sets A={1,,n} and H={1,,n}, respectively, where n=|A| is a natural number. Note that the number of agents and houses coincide as we assume that there is an existing tenant in each house. Agent a is the existing tenant of house h if h=a.

Each house hH has a price phR+. These prices are gathered in the price vector pR+n which is bounded from below by the reservation prices p_R+n of the owner. The reservation price of house h is p_h. The

Manipulability and non-manipulability results

As already explained in the Introduction, it is natural to let a minimum equilibrium price vector be a key ingredient in an allocation mechanism for the considered house allocation problem as this type of mechanism previously has been demonstrated to be individually rational, equilibrium selecting, and non-manipulable in various economic environments. Another natural price mechanism is based on individual rationality, equilibrium selection, and maximal trade. Here, the latter axiom guarantees

Concluding discussion

We considered a housing market with monetary transfers where tenants have the outside option to “Keep the House” reflecting two choices for existing tenants: continue renting the house or to buy it at the reservation price. It is natural to think about how this type of outside option influences our theoretical findings. If this outside option would not be available to the tenants at all, then our model reduces to the classical assignment game and the non-manipulability result in Theorem 2 can

References (26)

  • T. Andersson et al.

    Transferring ownership of public housing to existing tenants: a mechanism design approach

    (2014)
  • T. Andersson et al.

    Non-manipulable house allocation with rent control

    Econometrica

    (2014)
  • V. Crawford et al.

    Job matching with heterogeneous firms and workers

    Econometrica

    (1981)
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    We are especially grateful to two anonymous referees for their detailed comments and suggestions. We thank Alvin Roth, William Thomson, Walter Trockel, Alexander Westkamp, and Zaifu Yang for many useful and constructive comments. We would also like to thank participants at the 2015 Economic Theory and Applications Workshop in Malaga, the 2015 COST Meeting in Glasgow, the 2015 Workshop on Game Theory in Odense, the SING10 Meeting in Cracow, the 2014 Society of Social Choice and Welfare Meeting in Boston, and seminar audiences at Istanbul Bilgi University, Stockholm University, Tinbergen Institute, University of Glasgow, University of Montreal, University of Rochester, and University of York for helpful comments. All authors gratefully acknowledge financial support from the Jan Wallander and Tom Hedelius Foundation (research grant P2012-0107:1). The first author is also grateful to Ragnar Söderbergs Stiftelse (research grant E8/13) for financial support, and the second author is also grateful to the SSHRC (Canada) and the FRQSC (Québec) for financial support.

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