Search strategies on the housing market and their implications on price dispersion

Journal of Housing Economics 26 (2014) 55–80

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Search strategies on the housing market and their implications on price dispersion Tristan-Pierre Maury a,⇑, Fabien Tripier b a b

EDHEC Business School, Lille, France University of Lille 1 – CLERSE & Cepii, France

a r t i c l e

i n f o

Article history: Received 12 April 2013 Revised 1 September 2014 Accepted 8 September 2014 Available online 19 September 2014 JEL classification: R2 R3 Keywords: Housing Search Price dispersion

a b s t r a c t When an household needs to change its home, two transactions have to be done: buy a new house and sell the preceding one. To do so, the household can either adopt a sequential search strategy or a simultaneous search strategy. In sequential strategies, it first buys (or sells) and only after tries to sell (or buy) to avoid being homeless (or holding two houses). In the simultaneous strategy, the household tries to buy and sell simultaneously on the market. This last strategy can diminish its search costs on the housing market, but exposes the household to the risk of becoming a homeless-renter or a two-houses owner. The literature generally considers only the sequential search strategy. However, we show in this article that the simultaneous strategy is (i) generally welfare improving for households, (ii) sometimes the sole equilibrium strategy, and (iii) at the origin of price dispersion on the housing market. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction The existence of price dispersion is a widely recognized fact on real estate markets. This article advances a new explanation of this phenomenon based on the search strategies of households. This explanation does not rely on any form of heterogeneity, neither in dwellings nor in households, but considers the presence on the housing market of households, which simultaneously try to buy a new home and to sell their current one. This explanation is worthwhile given the puzzling existence of price dispersion on real estate markets. Indeed, a significant part of price dispersion can be evidently attributed to the heterogeneous nature of real estate assets. Properties differ according to their structural attributes (size, building period, . . .) and their location ⇑ Corresponding author. E-mail addresses: [email protected] (T.-P. Maury), fabien. [email protected] (F. Tripier). http://dx.doi.org/10.1016/j.jhe.2014.09.001 1051-1377/Ó 2014 Elsevier Inc. All rights reserved.

(submarkets, local amenities, . . .). The large empirical hedonic price literature (see Rosen, 1974) provides evidence of the impact of each of these qualitative factors on the selling price and enables a quantitative assessment of the price gap between two almost similar dwellings differing only along one attribute or, more importantly, their location (i.e. different geographical submarkets)1. Recently, a significant strand of research has been devoted to the remaining part of real estate price dispersion: the amount of volatility that cannot be attributed to the heterogeneity of assets. Therefore, it appears that two similar dwellings (with the same attributes and close locations) can be valued differently at the same 1 For example, Gabriel et al. (1992) document how large is the regional house price dispersion in the U.S. and show that this spatial variability is linked to interregional migration of households. On the theoretical side, the role of housing supply regulation (Glaeser et al., 2005; Glaeser et al., 2006), income distribution (Gyourko et al., 2006) or differential in productivity gains across metropolitan areas (Glaeser et al., 1992, or Van Nieuwerburgh and Weill, 2010) in explaining spatial house price dispersion have been explored.

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time and that this residual heterogeneity is empirically non negligible. For example, Leung et al. (2006) trace the evolution of quality-controlled price dispersion on the Hong-Kong housing market over time and show that the amount of volatility differs from zero and is connected to macroeconomic factors.2 The question is: how does it come that two seemingly identical assets may be sold at different prices? The real estate market literature provides two typical answers relying either on unobserved (good or agent) heterogeneity, either on the liquidity dimension of housing markets. According to the first view, remaining price differentials are possibly caused by missing variables concerning good attributes (unobserved good heterogeneity) or by ex ante buyers’ or sellers’ heterogeneity. Such assumptions seem natural: some structural attributes are not observable, difficult to measure and consequently not included in hedonic estimates even if they might have an impact on selling prices. Moreover, households entering the housing market can differ across tastes, information (first-time buyers vs experienced buyers) or search costs. These factors affect their bargaining power and then the final price. For example, Read (1991) proposes a theoretical setup with search costs on the housing market in which agents have exogenously distributed preferences concerning location choice. This assumption induces equilibrium price dispersion as well as positive vacancy rates. Harding et al. (2003) adds some variables concerning the buyers’ socio-economic profile in standard hedonic house price equations. They show that transaction prices may differ according to the buyer’ age or marital status. Moreover, they build a theoretically founded proxy to evaluate the respective bargaining powers of both buyers and sellers. This proxy appears to have a significant effect on the dwelling valuation. According to the second view, the concept of market liquidity – i.e. the time a dwelling has been on the market – provides an explanation for price dispersion. Merlo and Ortalo-Magné (2004) in an empirical setup not only show that the time on market and the final sale price are correlated (which is a largely documented empirical fact in the real estate literature), they also evidence that assets with higher listed prices will encounter longer time to sell, but finally higher sale price. The time to sell is negatively linked to the listed price to sale price ratio. This suggests that ex ante identical sellers putting similar goods on the market and adopting different listing price strategies may sell at a different price. This is the liquidity assumption: price dispersion does not solely come from the heterogeneous nature of assets or the ex ante heterogeneity of agents, but also from the uncertainty in time to sale inherent in a search process. Ceteris paribus, the longer the time spent on the market, the more likely the listed (and reservation) price will be revised downward and consequently the lower the sale price. From a theoretical point of view, Fisher et al. (2003) working in a search setup, suppose that (ex ante) identical agents are affected by an exogenous source of shocks causing a continuous distribution of 2 Their work complement preceeding contributions on the housing market (see Harding et al. (2003)) or on other durable goods markets (see for example Goldberg and Verboven (2001), on the car market).

reservation prices for both buyers and sellers on (commercial) real estate markets. Even if the purpose of their contribution is to propose a liquidity-adjusted price index, this distribution is also responsible of varying liquidity and finally of a part of the dispersion in sale prices. In a recent contribution with a matching model of the housing market, Albrecht et al. (2007) suppose that ex ante identical agents currently on the market are affected by an exogenous disturbance: they may move from a relaxed to a desperate state (with high costs of being unmatched) at a Poisson rate. This generates multiple price equilibria since each desperate or relaxed seller can meet a desperate or relaxed buyer. The authors show that the variance of price is affected by the time on market. The main shortcoming with all the above mentioned literature is that price dispersion comes either from ex ante (deterministic) heterogeneity (i.e. different socio-economic profiles, tastes, search costs), either from an exogenous idiosyncratic disturbances (i.e. exogenous move from a relaxed to a desperate state). Price heterogeneity is not due to the sole endogenous functioning of the market. This contrasts with the theoretical literature on price dispersion on other markets, which has sought to provide endogenous explanation of this phenomenon. For example, Burdett and Judd (1983) propose a nonsequential search model with identical agents where price dispersion is purely endogenous. The noisy nature of the search process leads to ex post heterogeneity in agents information and consequently to price heterogeneity. Without relying on a search setup, Salop and Stiglitz (1982) prove that equilibrium price dispersion can be attained in a homogenous commodity market with ex ante identical agents and no exogenous disturbances. They show that a two price equilibrium exists in a competitive setup with no auctioneer and with costly information. More recently, on the labor market, Burdett and Mortensen (1998) provide a model where wage dispersion may exist in equilibrium with perfectly identical workers and firms. Single market wage equilibria are ruled out by strategic wage posting firms and strategic on-the-job-search workers. In this line, the main goal of this paper is to propose an original model that explains housing price dispersion without relying on ex ante agents heterogeneity, nor on any source of exogenous idiosyncratic noise. We intend to present a search model where price dispersion is purely endogenous, i.e. due to the very specific nature of the search process on the housing market. A striking feature of the housing market is the existence of agents who are simultaneously on the two sides of the market. When a household wants to change its house for family or professional reasons, two transactions have to be done: buy a new house and sell the preceding one. To do this, it has several possibilities: (i) sell its home first and go to the rental market (and possibly lately buy another house), (ii) buy a new house (and transitory own two properties) and then try to sell the old one, (iii) enter the market as both a seller and a buyer and try to achieve both transactions (if possible at the same time). The third strategy is potentially optimal since it may avoid a transient state of renter (attributes of dwellings on the rental market are generally of a lesser quality than on the owner-occupied market) and also avoid owning

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simultaneously two houses (which incurs substantial financial costs and risks – fixed-term bridge loans for example in some european countries – for indebted households). Hence, households entering the market as both a buyer and a seller may hope to get two simultaneous matches and get its new suited home without extra costs. Nevertheless, due to the uncertain nature of the search process, these agents may not escape a transient state, namely the homeless/renter state (Buyer Only agents) or the two houses owner state (Seller Only agents) and consecutive additional financial (for two-house owners) and search (they still have a house to buy or to sell) costs. Consequently, there are different kind of matches on the market. A buyer-and-seller household can match with another buyer-and-seller or a seller only household to buy a new dwelling. Similarly, it can match with another buyer-and-seller household or a buyer only household to sell the old one. All these agents get different flow values from a conclusive transaction which leads to a multiple price equilibrium, without relying on any exogenous factor. Price dispersion occurs only from the occupational status history of each household. We base our setup on Wheaton (1990) seminal search model on the housing market. In this model, only the sequential strategy (ii) is considered, i.e. households first buy their new dwelling and then try to sell the old one. Hence, only one kind of match is possible: households trying to buy their new house always meet two house owners trying to sell their old one. No price dispersion equilibrium is possible in this context. We extend Wheaton’s setup and compare equilibria where households either adopt a sequential or a simultaneous strategy when they enter the housing market. The complexity of our model prevents us to fully base our comparison on analytic properties and we use numerical analysis to characterize some of the positive and normative properties of the equilibria. The comparison highlights the importance of taking the simultaneous strategy in consideration. The first lesson drawn from our numerical analysis is that multiple equilibria (where both simultaneous and sequential strategies conduct to stable equilibria) occur for a wide range of parameter values. When the equilibria are Pareto-ranked3, the simultaneous strategy often dominates the sequential one. These results lead us to conclude that the simultaneous strategy is a credible behavior on the housing market and therefore also a credible explanation of the price dispersion induced by this strategy. We end our analysis by a description of price dispersion in the model. We prove that there are three equilibrium prices if the bargaining processes are symmetric and four otherwise, and use numerical analysis to discuss the determinants of the price dispersion amplitude. The remainder of the paper is organized as follows. The environment is presented in Section 2, the model is presented and solved in Section 3 with the sequential strategy and in Section 4 with the simultaneous strategy. The two equilibria are compared on the basis of numerical simulations in Section 5. Section 6 concludes. 3 Equilibria are ranked with respect to the steady-state welfare of a matched household, which is outside the housing market. We do not compute the transitional dynamics between steady-state equilibria.

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2. The environment The notations are fully explained in Appendix A. Time is discrete. The economy is populated by two types of households, singles and couples, which differ only in their house preferences. Let h1 denoted the mass of singles and h2 the mass of couples, with 2h the total number of households that satisfies 2h ¼ h1 þ h2 . Each household has an exogenous probability to be hit by a demographic shock and to change its family type4. Without loss of generality, we assume that the transition rate b between the two states is symmetric. The associated laws of motions are then 0

ð1Þ

0

ð2Þ

h1 ¼ h1 þ bh2  bh1 h2 ¼ h2 þ bh1  bh2

where the symbol ’ denote the next period value of the variable. The symmetry of the transition rate implies the equality of the two masses of households at the steady state: h1 ¼ h2 ¼ h. On the housing market, each type of household can live in two types of dwellings, i.e. small and large. Single (resp. couples) households are matched when they live in small (resp. large) units and mismatched otherwise. Hence, each previously matched household hit by a demographic shock will become mismatched and will have to enter the housing market to get a convenient dwelling. More precisely, this household needs to achieve two transactions: sell its current unit and buy a new (appropriate) one. To do so, mismatched households can proceed in different ways:  Enter the market as a ‘‘Buyer First’’ (BF), i.e. first search for a new dwelling. Once the household succeeded in buying a convenient new house, it then puts its old unit up for sale (and becomes a ‘‘Seller Only’’, SO). It will own two houses and bear the consequent financial costs until the sale transaction is concluded. This strategy exactly corresponds to the one described by Wheaton (1990).  Enter the market as a ‘‘Seller First’’ (SF), i.e. first put his current asset up for sale. Such a strategy is perfectly symmetric to the preceding one. Once the household succeeded in selling his asset, it then starts searching for a new one. During the spell between the two transactions, the household owns no house (or live in a rented dwelling) and becomes a ‘‘Buyer Only’’ (BO).  Enter the market as both a buyer and a seller (BS), i.e. simultaneously search for a buyer and for a new house. The potential interest of this third strategy is obvious: if the household manages to realize both transactions at the same time, then it will escape a costly state of ‘‘Seller Only’’ with two houses or of ‘‘Buyer Only’’ with no house. Nevertheless, if buying (resp. selling) transaction occurs first, then the household will encounter a ‘‘Seller Only’’ (resp. ‘‘Buyer Only’’) spell and bear the subsequent search and/or financial costs. 4 We use the same assumptions about demographic transitions than Wheaton (1990), which present the advantage to keep notations as simple as possible. Of course, a more refined setup allowing to model the transitions of all members of the household would be welcome, but would not affect the main mechanism driving our model.

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Following Wheaton (1990), we assume that when mismatched, a household can still be hit by a demographic shock. Typically, a couple with a small house, which is searching for a big house on the market, can decide to separate and then becomes matched again without concluding any transaction. In our setup, ‘‘Buyer First’’, ‘‘Seller First’’ or ‘‘Buyer and Seller’’ households will therefore be matched again after a demographic shock. On the contrary, households with either two (‘‘Seller Only’’) or no (‘‘Buyer Only’’) house and hit by a demographic shock remains unmatched. SO households will still have to sell one dwelling and BO households to buy one.

Notice that we will propose a full description of the first and third strategies only. Due to the perfect symmetry between the first two strategies, our reasoning could be easily extended to the second ‘‘Seller First’’ strategy. Fig. 2 (resp. Fig. 3) provides complete pictures of potential occupational status of a mismatched household entering the housing market as a BF (resp. a BS). The figures are completed by Fig. 1 for matched households and Figs. 4 and 5 for households becoming Buyer Only or Seller Only. Table 1 complements the description of the environment with a summary of the costs associated with the states on the housing market.

Fig. 1. The two alternative strategies in the economy. The household enters the housing market as a buyer first with the sequential strategy (left panel) and as a buyer and seller with the simultaneous strategy (rightpanel)

Fig. 2. The buyer and seller outcomes. At the end of the period, the buyer and seller can be either a buyer and seller, a matched, a seller only or a buyer only household according to its success and/or failure in buying and selling and the realization of the demographic shock.

Fig. 3. The buyer first outcomes. At the end of the period, the buyer first can be either a buyer first, a matched, or a seller only household according to its success or failure in buying and the realization of the demographic shock.

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Fig. 4. The buyer only outcomes. At the end of the period, the buyer only can be either an unmatched, a matched, of a buyer only household according to its success or failure in buying and the realization of the demographic shock.

Fig. 5. The seller only outcomes. At the end of the period, the seller only can be either a buyer first, a matched, or a seller only household only according to its success or failure in buying and the realization of the demographic shock.

Table 1 States on the housing market and associated costs. State/Costs Buyer only Seller only Buyer first Buyer seller

Cost of living in an inconvenient dwelling

Cost of living in a rented dwelling

Search cost to buy





 

In the following sections, we expose with full details the sequential strategy (the ‘‘Wheaton’’ case) and the simultaneous strategy the (‘‘BS’’ case) and establish existence conditions of both kinds of equilibria in steady-state. Due to the homogeneity assumption, each strategy can be solved independently. Indeed, all mismatched households in the model will choose the same strategy, sequential or simultaneous5. Moreover, it is straightforward to see that since there are no state variable (in particular, no aggregate source of disturbance is introduced) in the model, households have no interest in changing their strategies. For example, a mismatched household entering the market as a ‘‘Buyer First’’ at a specific period and which did not succeed in buying a new house in that period (and is not hit by a demographic shock), 5

Mixed strategies are ruled out.

 

Search cost to sell

Financial costs of holding two dwellings







will remain a BF next period rather than become a BS, since the household problem is not time-dependent. Consequently, in our setup, all households choose the same strategy and keep it until they become matched. 3. The sequential strategy equilibrium 3.1. Dynamics In a ‘‘Buyer First’’ equilibrium, each household entering the housing market first decides to buy a new dwelling corresponding to its new family structure and then to sell the old one. Hence there are only three possible states: matched (M), buyer first (BF) and seller only (SO). There is no rental market. The time sequence for unmatched households is as follows: the BF household first observes

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if it meets a seller on the housing market. This occurs with probability qb . Notice that the only type of sellers on the market is SO households, which meet BF with probability qs . Hence transactions always occurs between BF and SO households. Both buyers and sellers decide whether or not to accept the transaction and negotiate the price (in the following, we will only consider the case where transaction occurs as done by Wheaton (1990)). Let hM ; hBF and hSO respectively denote the mass of matched households, buyers and sellers (for each family type, singles or couples), with h ¼ hM þ hBF þ hSO . Let s denote the constant stock of housing of each type (small and large dwellings) in our economy. Following Wheaton (1990), we assume a permanent excess of housing, i.e. v ¼ s  h > 0. This is a necessary condition to ensure the equilibrium existence: M and BF households own exactly one house while SO households own two houses. The vacancy rate has then to be permanently positive. The mass of BF households evolves according to

parameters. The household discount factor is 0 < d < 1. The per-period utility flow depends on several factors. Let u denote the instantaneous flow of value of living in a convenient home. Moreover, if the household searches (as a buyer or a seller) on the housing market, it pays the cost jx > 0 that depends on his state x for x ¼ BF; SO. jBF combines the cost of being unmatched (and living in an unsuited home) and the search costs for a new house, while jSO comprises the search costs of a buyer as well as potential financial costs of owning two houses (capital expenditures, mortgage bridge loans for the time spell between buy date and sell date). If the household makes a transaction at price p, the utility flow uðpÞ ¼ bp is added (b > 0, this flow is positive in the case of a sell and negative in the case of a buy). Since we restrict our study to the steady state, we do not introduce time subscripts. The value function of a matched household is:

  0 hBF ¼ 1  qb ð1  bÞhBF þ bðh  hBF  hSO Þ þ qs bhSO

A matched household earns the instantaneous utility from u (and pays no additional costs) and get the discounted weighted values of staying matched with probability ð1  bÞ (i.e. no demographic shock) or becoming unmatched and entering the market as a BF with probability b. The value function of a BF is:

ð3Þ

The first term of the RHS of Eq. (3) is the mass of BF households of the last period which did not have a match   – which occurs with probability 1  qb – and did not support a change in their family type – which occurs with probability ð1  bÞ, i.e. no demographic shock – and consequently remain on the market as BF next period. The second term is the mass of matched households of the last period hit by a demographic shock – with probability b – which become unmatched and enter the market as BF. The last term is the mass of sellers which get a match and sell their old asset – with probability qs – but are immediately hit by a demographic shock and have to reenter the market as BF. The dynamics of the mass of SO households obeys 0

hSO ¼ ð1  qs ÞhSO þ qb hBF

ð4Þ

The first term of the RHS of Eq. (4) is the mass of SO households last period that did not get a match. Since SO households own two houses, they stay in that state even if they change their family types: they still have one home to sell. The second term is the mass of BF which get a match a become SO. Similarly, their status is not modified by an eventual demographic shock. The dynamics of the mass of matched households is easily deduced from Eqs. (3) and (4). To complete the presentation of the dynamic system of mass of households, we need to introduce an aggregate specification of the rate of meeting of agents on the housing market at the steady-state. We adopt the same strategy as Wheaton (1990) and treat qb as an exogenous parameter6 and deduce the value of qs ¼ qb  ðhBF =hSO Þ. 3.2. Value functions To define the value functions associated with the states on the housing market, we have to introduce additional 6

Notice that the introduction of a Cobb-Douglas matching function:  1h  h qb ¼ q hhSOB and qs ¼ q hhSOB where q and h are exogenous parameters is not straightforward and may produce multiple equilibria outcomes.

V M ¼ u þ d½bV BF þ ð1  bÞV M 

V BF ¼ u  jBF  qb bpB     þ d qb V SO þ 1  qb ðbV M þ ð1  bÞV BF Þ

ð5Þ

ð6Þ

The BF household pays the search cost jBF . With probability qb , the household matches and then buys its new house at price pB . In this case, the next period value function is dV SO since the household becomes a SO. Conversely, with   probability 1  qb , the BF household does not match and buys no house. Its next period value functionis dV M if it is hit by a demographic shock (and becomes matched) or dV BF otherwise (household stays in its BF state). The value function of a SO household is:

V SO ¼ u  jSO þ qs bpB þ d½qs ðbV BF þ ð1  bÞV M Þ þ ð1  qs ÞV SO 

ð7Þ

The SO household pays the search/financial costs jSO . With probability qs , the household matches and sell its old house at price pB . Notice that this price is the same as in Eq. (6) since the sole buyers on the market are BF households. Next period, the SO becomes matched if its family type does not change or get back on the market as a BF otherwise. With probability ð1  qs Þ, the SO household does not match and stays in this state next period. 3.3. Bargaining process Prices are the outcome of a Nash bargaining process between buyers and sellers. Each household assesses its benefit in the case of bargaining success compared with the case of bargaining failure. The unique price pB is determined according to

n o pB : arg max ½BV BF c ½SV SO 1c

ð8Þ

with BV BF ¼ V BF jqb ¼1  V BF jqb ¼0 and SV SO ¼ V SO jqs ¼1  V SO jqs ¼0 . BV BF is the value increase for the buyer in case of a

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bargaining success and SV SO is the value increase for the seller in case of a bargaining success. No equilibrium price exists if one of these two terms is negative, since no transaction could occur. In the following, we will only consider range of parameters where BV BF and SV SO are positive to keep the bargaining process consistent with the dynamic Eqs. (3) and (4). c is the parameter that governs the sharing of the value surplus. Simple manipulations lead to the following expression for the negotiated price:

d pB ¼ ð1  cÞ½V SO  ðbV M þ ð1  bÞV BF Þ b d  c½bV BF þ ð1  bÞV M  V SO  b

ð9Þ

Details of the calculations leading to Eq. (9) are given in Appendix B.1. The first term in brackets is the value surplus of a successful match for the buyer. It becomes a SO in case of transaction rather than stays either a BF with probability ð1  bÞ or a matched with probability b in case of no transaction. The second term in brackets is the equivalent value surplus of an achieved negotiation for a seller. Overall, the price pB is increasing in V SO : for large values of V SO , buyers would accept to pay a higher price to become SO and sellers would request a higher price to leave this state. Conversely, pB is decreasing in V BF and V M : the higher these value functions the lower the price buyers would be ready to pay for smaller surplus and the lower the price sellers would request for larger surplus. Finally, the price is logically decreasing with the buyer’s bargaining power in the negotiation c. 3.4. Equilibrium The steady-state equilibrium volumes of transactions and masses of households in each state (M, BF, SO) are entirely determined by the structural parameters that govern the demographic and transaction processes, namely b; s; qb and h. Parameters governing the price bargaining process (c) or utility flows (u; b; jBF ; jS ) do not influence these quantities for range of parameters where all matches lead to a transaction. Definition 1. The equilibrium price pB , the value functions fV M ; V BF ; V SO g and masses fhM ; hBF ; hSO g solve

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and uniqueness is straightforward to establish as long as V BF > V BS . The final equilibrium expression of the value function of a BF is simply

V BF ¼

jBF u þ ð1  dÞð1 þ xÞ ð1  dÞ

ð10Þ

  with x ¼ d b þ qb cð1  2bÞ =ð1  dð1  bÞÞ. As expected, V BF is decreasing with jBF . If b < 1=2 (households change their family type at an average periodicity above two periods which seems highly plausible at an annual or quarterly frequency), V BF also increases with c (the bargaining power of the buyer) and qb (the probability of a match for a buyer). Nowadays, the larger b the lesser the impact of c and qb since the gains of a conclusive match are lower if the probability of becoming matched without any transaction is higher. Finally notice that the direct impact of b on V BF is ambiguous, since it impacts positively the probability of becoming matched immediately for a BF, but, in case of a conclusive transaction, impacts negatively the value function of the new SO household (which could get back to the market just after selling). Interestingly, notice that V BF does not depend on jSO even if BF households may become SO in one period. It appears that price pB acts as a buffer against the search/ financial of SO households. Price increases as jSO decreases. But this does not impact the value of BF households since (without considering demographic shocks), they will buy and then sell a dwelling at similar price pB in the future. Proposition 1. A necessary and sufficient for the existence of the Single Equilibrium is jV > jV . See Appendix B.3 for the explicit form of jV . The above proposition is a re-expression of existence condition 4 in the equilibrium definition. It proves that we can always find a range of parameters (i.e. a sufficiently large value for jV ) ensuring the BF equilibrium existence (the system is otherwise linear). For given values of jBF and jS , the larger jV the lower the incentive for agents to deviate from the equilibrium strategy (which does not depend on jV ). 4. The simultaneous strategy equilibrium 4.1. Dynamics

1. the Nash bargaining outcome (9); 2. the definition of the value functions (5)–(7); 3. the dynamic Eqs. 3,4 considered at the steady state and the residual equation h ¼ hM þ hBF þ hSO ; 4. the following equilibrium existence condition V BF > V BS where V BS is the value function of a (unmatched) household entering the market as a Buyer/Seller rather than a Buyer First. The Nash equilibrium exists if no agent has interest in deviating from the equilibrium decisions and adopting a different strategy. Appendix B.1.1 gives the complete set of equilibrium equations. The whole system of equilibrium equations is linear subject to the inequality constraint (condition 4) given in the above definition. Consequently equilibrium existence

In the simultaneous strategy equilibrium, each mismatched household enters the housing market as a Buyer and a Seller (BS), i.e. decides to search simultaneously for a new dwelling and for a buyer for its current dwelling. If the mismatched household managed to sell its house but still not to buy the new one, it becomes a Buyer Only (BO), i.e. a homeless (or a renter) seeking to buy a new asset. Conversely, if the mismatched household has already bought its new house but still not sold the old one, it becomes a Seller Only (SO), i.e. it owns two homes and still tries to sell one on the housing market. Hence, there are four possible states in the simultaneous strategy equilibrium: matched (M), buyer/seller (BS), buyer-only (BO) and seller-only (SO). Let hM ; hBS ; hBO and hSO respectively denote the mass of matched households, BS, BO

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and SO (for each family type, singles or couples), with h ¼ hM þ hBF þ hBO þ hSO . The time sequence is as follows: the BS household observes if it has a match with a seller (either a BS or a SO) which occurs with probability qb and if it has a match with a buyer (either a BS or a BO) which occurs with probability qs . Similarly, BO – respectively SO – households observe if they have a match with a seller (either a BS or a SO) – respectively with a buyer (either a BS or a BO) – which occurs with probability qb – respectively qs . Notice that the matching probabilities are the same for BS, BO and SO since we suppose the search process is blind, i.e. agents cannot seek for a specific status of their counterpart. After observing if they have a match (or two matches for a BS) or not, both buyers and sellers decide whether or not to accept the transaction and negotiate the price. We solve the dynamics of the population under the hypothesis that each household which get a match on the housing market, accepts the transaction (whatever the type of the household it has matched). As will be later explained, this assumption will be ex post checked when solving the equilibrium. The dynamic system of masses of BS, SO and BO households is given by

8 0   > hBS ¼ b  ðh  hBS  hBO  hSO Þ þ b  qb qs hBS > >   > > > þð1  bÞ  ð1  qs Þ 1  qb hBS > <   þ b  qb hBO þ qs hSO > > 0 > > hSO ¼ ð1  qs ÞhSO þ qb ð1  qs ÞhBS > >     > 0 : hBO ¼ 1  qb hBO þ qs 1  qb hBS

ð11Þ

The next period mass of BS households (first equation) depends on: the mass of matched households (hM ) who experience a demographic shock (first term), the mass of BS households which succeed in buying and selling during the period (with probability qb qs ) but are afterward hit by the demographic shock and have to return on the housing market (second term), the mass BS households which neither sell nor buy a dwelling during the period (with prob  ability ð1  qs Þ 1  qb ) and experience no demographic shock (third term), the sum of the masses of BO and of SO which succeed in buying and selling, respectively, but experience a demographic shock (last term). The dynamic equations for masses of SO and BO households (second and third equations respectively) are simpler than for the BS households because their objectives do not change after a demographic shock. The next period mass of SO (resp. BO) is the sum of last period SO (resp. BO) with no match and last period BS households which bought a new home but did not sell the preceding one (resp. sold but did not buy). 4.2. Matches and prices Table 2 summarizes the type of matches in the economy. The four states in the housing market give rise to potentially four different transaction prices. This price dispersion occurs without relying on any source of exogenous disturbance: all mismatched households entered the market as perfectly identical BS. Their status may have changed over

Table 2 The type of matches with the simultaneous search strategy. Type of matches

The buyer

The seller

The price

1 2 3 4

Buyer Buyer Buyer Buyer

Buyer and seller Seller only Seller only Buyer and seller

p1 p2 p3 p4

and seller only and seller only

time (they become matched, SO or BO), but this is due to the sole functioning of the market. As we said, the only disturbance (i.e. the demographic shock) only forces agents to enter the market but does not create heterogeneity in the housing market per se. In this setup, we limit ourselves to consider only one price outcome for each kind of match. This means that a match between two BS households for example always conduct to a conclusive transaction at unique price p1 whatever their situation on the housing market regarding their other possible match. When a BS acting as a buyer meet a BS acting as a seller, we suppose that whether the first one has a match to sell its asset or not (if yes, with a BS or a BO household) and the second one a match to buy or not (if yes with a BS or a SO) has no influence on their bargaining. This ‘‘limited information’’ assumption keeps the number of equilibrium prices limited to four since agents engaged in a bargaining only use the information regarding this sole negotiation. We leave open for future research the ‘‘full information’’ case where agents starting a bargaining process know if they have a match besides and the status of the other household they met7. We also leave open for future research the case of asymmetric information. The Nash solution (8) can be regarded as the outcome of strategic bargaining models with alternating offers and full information as demonstrated by Binmore et al. (1986). However, it seems sensible to assume that the status of the seller (either BS or SO) and the buyer (either BS or BO) may not be revealed to the other party. For example, if it happens that p4 > p1 , it might create incentives for a BO to hide its status, because the buying price for a BS is lower. Consequently, the price p1 would not be observed. Nevertheless, the seller is aware about the heterogeneity of buyers (as well as their incentives to hide the less advantageous status) and should also use this information strategically as explained by Osborne and Rubinstein (1990). The crucial point is that considering asymmetric information would not reduce the multiplicity of prices per se.8 In a model with alternating offers, the uninformed agent interprets the moves of the informed agent to extract private information. Therefore the outcome of the bargaining process depends on the true status of the informed agent and, according to the distribution of the population, pooling equilibria with different prices may occur as in Rubinstein (1985) and Osborne and Rubinstein (1990, chapter 5). 7 The full information hypothesis could induce some agents to reject a transaction because they did not find a counterpart on the ‘‘other side’’ of the market. Such a setup would be linked to the ‘‘housing chains’’ literature (see Rosenthal (1997)) where Agent A cannot buy the property of agent B if the latter cannot buy its new house to a third agent, etc. 8 Indeed, incomplete information is per sea source of multiple equilibria (see Fudenberg and Tirole (1983)).

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4.3. Value functions We define the value functions associated with each state on the housing market: V M for a matched household, V BS for a Buyer/Seller household, V BO for a Buyer Only and V SO for a Seller Only. Compared to the preceding subsection, the whole set of parameters remains unchanged. We introduce two additional variables hb ¼ hBS =ðhBS þ hBO Þ and hs ¼ hBS = ðhBS þ hSO Þ which are the share of BS households among the mass of all buyers (respectively sellers). These ratio are taken as the probabilities for a seller and a buyer, if matched, of being matched with a BS household. The value function of matched households is close to its ‘‘sequential strategy’’ equilibrium counterpart

V M ¼ u þ d½bV BS þ ð1  bÞV M 

ð12Þ

except that when the current home becomes unsuited, the household will enter the market as a BS. The system defining the value functions of BS, BO and SO households is 8 V BS ¼ u  jV þ U ðp1 ; p3 ;p4 Þ > >  > > > þd qs qb ðbV BS þ ð1  bÞV M Þ þ qb ð1  qs ÞV SO > >      > > > þqs 1  qb V BO þ ð1  qs Þ 1  qb ðbV M þ ð1  bÞV BS Þ > > < s s V BO ¼ u  jBO þ qb ½h uðp4 Þ þ ð1  h Þuðp2 Þ     > > > þd qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO > > h   i > > > > V SO ¼ u  jSO þ qs hb uðp3 Þ þ 1  hb uðp2 Þ > > > : þd½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO 

ð13Þ

The per period utility flow of a BS is u, less the search cost

jV , plus the expected utility associated with the transaction prices U ðp1 ; p3 ; p4 Þ, defined below. Four future issues are feasible at the next period. If the household manages to buy and sell in the period, it either remains a BS in the case of demographic change and gets discounted value dV BS or becomes matched and gets dV M in the next period. If the household does not buy nor sell, it either remains BS in the absence of demographic change (then gets dV BS ), or becomes matched and gets dV BS . Otherwise, If the BS household buys his new home without selling his old one it gets dV SO and if it sells its old home without buying, it gets dV BO , the expected value of being BO. The per period utility flow of a BO household is different, since it incurs the search cost of buying for households without houses (i.e., jBO ). With probability qb , the household matches and then buys its new house at the price p4 (if it is matched with a BS – with probability hs ) or p2 (if it is matched with a SO – with probability ð1  hs Þ). In this case, the next period value of function is dV M (no change in family type) and dV BS (if a change occurs). If the household does not buy house during the period, it remains a BO (value dV SO ) in the next period whatever the realization of the demographic shock. The presentation of the value function of a SO household is similar, except that it incurs the search cost of selling for households which own two houses (i.e., jSO ). If it matches, it sells its house at price p3 (match with a BS) or at price p2 (match with a BO) and get the next period value dV M (no demographic change) or dV BS (demographic change). If the household does not sell its home during the period, it remains a SO at the next period whatever the realization of the demographic shock.

According to the definition of V BS in system (13), the utility function associated with prices is then defined as follows h   U ðp1 ; p3 ;p4 Þ ¼ qs qb hb hs uð0Þ þ hs 1  hb uðp4  p1 Þ   i þð1  hs Þhb uðp1  p3 Þ þ 1  hb ð1  hs Þuðp4  p3 Þ   i  h þ qs 1  qb hb uðp1 Þ þ 1  hb uðp4 Þ

þ qb ð1  qs Þ½hs uðp1 Þ þ ð1  hs Þuðp3 Þ   þ 1  qb ð1  qs Þuð0Þ

ð14Þ

If the household succeeds in matching twice (as a buyer and a seller, with probability qs qb ) four matches are feasible with different payoffs. If it matches twice with a BS (with probability hb hs ), the prices of its sale and its purchase are the same and the household gets   uðp1  p1 Þ ¼ uð0Þ. With probability hs 1  hb , the seller is a BS (the agent pays p1 ) and the buyer a BO (the agent earns p4 ). With probability ð1  hs Þhb , the seller is a SO (the agent pays p3 ) and the buyer is a BS (the agent earns p ). If it matches with a BO and a SO, with probability 1  1  hb ð1  hs Þ, the agent pays p3 and earns p4 . If the household succeeds only in selling (with probability   qs 1  qb ), it earns p1 in the case of a match with a BS household (with probability hb ) and p4 in the case of a   match with a BO household (with probability 1  hb ). If the household succeeds only in buying (with probability qb ð1  qs Þ), it pays p1 in the case of a match with a BS household (with probability hs ) and p3 in the case of a match with a SO household (with probability ð1  hs Þ).   Finally, with probability 1  qb ð1  qs Þ the household does not buy nor sell. 4.4. Bargaining process Prices are the outcome of a Nash bargaining process between the two households. Each household assesses its benefits in the case of bargaining success compared with the case of bargaining failure. Since this benefit varies with the household’s state, we have to define and to solve all the feasible matches on the housing market. As detailed in Table 2, the system of price is the outcome of

n o 8 c 1c > p ¼ arg max ½ B V  ½ S V  > BS BS BS BS 1 > > > n o > > > < p2 ¼ arg max ½BSO V BO c ½S BO V SO 1c n o > > p3 ¼ arg max ½BSO V BS c ½S BS V SO 1c > > > > n o > > : p ¼ arg max ½BBS V BO c ½S BO V BS 1c 4

ð15Þ

h i where BBS V X ¼ V X jqb ¼1;hs ¼1  V X jqb ¼0 is the increase in value for a household in state X to buy to a BS and h i BSO V X ¼ V X jqb ¼1;hs ¼0  V X jqb ¼0 is the increase in value to buy S BS V Y

h

to

X ¼ BS or BO. Similarly, i ¼ V Y jqs ¼1;hb ¼1  V Y jqs ¼0 is the increase in value for a

household

a

SO,

in

with

state

Y

to

sell

to

a

BS

and

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h i S BO V Y ¼ V Y jqs ¼1;hb ¼0  V Y jqs ¼0 is the increase in value to sell to a BO, with Y ¼ BS or SO. The equilibrium housing price system solution of the Nash bargaining processes satisfies (all details are in appendix C.2):

Finally, we use the reverse argument for price p4 . The positive effect of V SO is stronger on p1 than on p4 and the negative effect of V BO is smaller on p1 than on p4 . This explains the gap between p1 and p4 in equilibrium. It now clearly appears that price dispersion in our setup is

8 bp1 ¼ ð1  cÞd½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO  qs V BO  ð1  qs ÞðbV M þ ð1  bÞV BS Þ > >       > > > cd qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO  qb V SO  1  qb ðbV M þ ð1  bÞV BS Þ > > > > > bp2 ¼ ð1  2cÞd½bV BS þ ð1  bÞV M  þ d½cV SO  ð1  cÞV BO  > < bp3 ¼ ð1  cÞd½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO  qs V BO  ð1  qs ÞðbV M þ ð1  bÞV BS Þ > > > cd½bV BS þ ð1  bÞV M  V SO  > > > > > bp4 ¼ ð1  cÞd½bV BS þ ð1  bÞV M  V BO  > > >     : cd qb ðbV BS þ ð1  bÞV M  V SO Þ  1  qb ðbV M þ ð1  bÞV BS  V BO Þ Obviously, p1 (first equation) increases with the surplus in value of the buyer (first term in brackets) and decreases with the surplus in value of the seller (second term in brackets). Consequently, p1 is increasing with V SO and decreasing with V BO : the higher V SO , the higher the gain of a buy for a BS

ð16Þ

directly related to the probability of reaching the states BO and SO. Consequently, we expect p1 to lie between p3 and p4 . 4.5. Equilibrium

s

(since it could become a SO with probability ð1  q Þ) and the lower the gain of a sell for a BS (since it escapes a SO state). Similarly, the higher V BO , the lower the gain of a buy for a BS (it can no longer become a BO) and the higher the gain of a sell for a BS (it can become a BO with probabil  ity 1  qb ). The effects of V BS and V M on p1 are ambiguous

Lemma 1. The demographic structure and the transaction process imply the equality of ðiÞ the matching probabilities of buyers and sellers: qb ¼ qs ¼ q, and of ðiiÞ the proportions of buyer–seller households in the masses of buyers and of sellers: hb ¼ hs ¼ h, where h is a function of the structural parameters q and b.

and depend on the bargaining power of each agent. They have a positive effect on surplus of buyers and sellers because a conclusive match increases the probability of reaching these states. This impact is lowered by the fact that BS households could become matched (or stay BS) without transacting (and then lose this opportunity when concluding a transaction). As will be seen below, these effects of V BS and V M drop out when c ¼ 1=2 (symmetric Nash bargaining) and price p1 solely depends on V SO and V BO . A similar argument shows that p2 increases with V SO and decreases with V BO . The direct impact of V BS and V M on gains in value is positive for both BO and SO since a conclusive match increases their probability of reaching these states. Consequently the total effect depends on the respective bargaining power of the BO and the SO: positive for c < 1=2 (higher seller bargaining power) and negative for c > 1=2. Once again their impact is null in case of a symmetric Nash bargaining. p3 differs from p1 only through the second term in brackets (surplus in value of a match when selling): the difference is that the SO is ensured to become a BS or matched if it succeeds in selling, but is not concerned with these two states if it does not sell (which is not the case for a BS). The surplus of a SO decreases with V SO because it will leave this state in case of a conclusive match. In contrast with a BS, the SO is not directly concerned with V BO because it cannot reach this state in one period. Consequently, the positive effect of V SO is stronger on p3 than on p1 and the negative effect of V BO is smaller on p3 than on p1 . Such a differential explains the gap between p1 and p3 in equilibrium and generates price dispersion.

Proof. See appendix C.1.

h

The equilibrium quantity of transactions and the mass of households are entirely determined by the structural parameters that govern the demographic and transaction processes, namely q; b, and h. The equilibrium prices do not influence these quantities. This property of our model proceeds from the assumption that all matches lead to a transaction and that home supply is fixed. Such a property would be unsatisfactory if we were concerned with the size of the housing market, rather than price dispersion. We leave open for future works the issue of the influence of prices dispersion on the equilibrium quantities on the housing market. Definition 2. The steady state equilibrium price set fpi g4i¼1 and value functions fV SO ; V BO ; V BS g solve 1. 2. 3. 4.

the four Nash bargaining outcomes (16); the definition of the value functions (12,13); the definition of q and h introduced in Lemma 1; all value surpluses from a match BBS V BS ; S BS V BS ; BSO V BO ; S BO V SO ; BSO V BS ; S BS V SO ; BBS V BO , and S BO V BS are positive.  5. the existence condition: max V BF ; V SF < V BS . The equilibrium exists if no agent has interest to deviate from the equilibrium decisions. For a matched household hit by a demographic shock, to deviate means enter on the housing market as a buyer first (and gets V BF ) or a seller first (and gets V SF ) rather than as a buyer–seller. Appendix C.3 gives the complete set of equilibrium equations.

T.-P. Maury, F. Tripier / Journal of Housing Economics 26 (2014) 55–80

The point 4 of the equilibrium’s definition guarantees that all negotiations conduct to a transaction. If one of the value surplus terms is negative, at least one of the four type of transactions never occurs which reduces the number of equilibrium prices. Consequently, the dynamics of masses of households would be modified and so the proportion of BS households among buyers (hb ) or sellers (hs ). In the next section, the positivity of value gains will be numerically checked. The point 5 of the equilibrium’s Definition 2 guarantees that all matched households hit by a demographic shock have no interest to enter the housing market with a different strategy: they should enter as a buyer–seller for the simultaneous equilibrium to exist. The appendix C.4 gives  the complete expression of the payoffs V BF ; V SF , which computation is quite complex since it requires to solve the Nash Bargaining process for matches with a single agent deviating from the equilibrium strategy. Due to the almost-linear nature of the model, a condition of existence of the equilibrium can be easily established. Proposition 2. Let g ¼ fp1 ; p2 ; p3 ; p4 ; V SO ; V BO ; V BS ; V M g be the vector of all endogenous variables defined in the steadystate equilibrium. Let the linear system defined by the four Nash bargaining Eqs. (16)and the four value functions (12,13) be rewritten as follows Ag ¼ b where A is a ð8  8Þ matrix and b a ð8  1Þ vector. It is straightforward to prove that the positivity of the eight value surpluses (condition 4 of equilibrium definition) and the two existence conditions (condition 5) can be reexpressed as follows C g > d with C a ð10  8Þ matrix and d a ð10  8Þ vector. Hence, if A is a full rank matrix, a necessary and sufficient condition for existence and unicity of equilibrium is C:A1 b > d Proof. A; b; C and d are given in Appendix.

h

As in the Sequential Equilibrium case, the steady state equilibrium is linear as long as conditions 4 and 5 are verified. Even though the conditions of existence and unicity of the simultaneous equilibrium are easy to establish, the way structural parameters enter A; b; C and d is highly non linear and we will have to proceed to numerical experiments. 4.6. Price dispersion We first characterize the price dispersion for symmetric bargaining power (c ¼ 1=2) and then for the asymmetric case (c – 1=2Þ:. Proposition 3. For symmetric Nash bargaining programs, the equilibrium prices satisfy

2b 2b jBO  jSO p ¼ p ¼ V SO  V BO ¼ d 1 d 2 1d

ð17Þ

2b p ¼ ð1  qÞðV BS þ V M Þ  qV BO þ ð2  qÞV SO d 3

ð18Þ

2b p ¼ ð1  qÞðV BS þ V M Þ þ qðV SO þ V BO Þ  2V BO d 4

ð19Þ

65

Corollary 1. For symmetric Nash bargaining programs, there are three distinct prices on the housing market. The price p1 is equal to p2 and to the mean of prices p3 and p4 :

p1 ¼ p2 ¼ ðp3 þ p4 Þ=2

ð20Þ

Prices p1 and p2 are linear and increasing functions of the differential in value functions for a SO household and for a BO household. If these value functions are equal, the equilibrium price is null. In this case, the buyer and the seller have the same expected value for the next period and then the same threat point in the bargaining process. Prices are positive if the value of being SO exceeds the value of BO, i.e. jBO  jSO > 0. The link between these value functions and the prices proceeds from a compensation mechanism between the utility drawn from price and the utility associated with the expected change of situation on the housing market. The household accepts the transaction if the payoff is high enough. Ceteris paribus, a damaging in future expected situation for the seller has to be compensated by a higher price in the transaction to ensure his acceptation. To analyze the determinants of these prices, we first comment the match composed of SO and BO. The two households face the same probability of being BS or matched at the next period. With identical power in the bargaining process (c ¼ 1=2), the associated effects of value functions (V BS and V M ) cancel and the price depends only on the difference between V SO and V BO . The higher the value of being seller only, the higher the negotiated price. Reciprocally, the lower the value of being buyer only, the higher the negotiated price. The negotiated price ensures the consent of agents to buy or to sell. If the value of being a SO is high, a high price is needed to obtain the seller’s consent. If the value of being a BO is low, the buyer is ready to accept a high price. The issue of the bargaining process for this match is finally rather intuitive. Another interesting result of our model is that the price is the same for a match composed of two BS households than for match composed of one BO and one SO. As for the match of type 1, the equality of the bargaining power of agents cancels out the effects of value functions associated with the states of BS and matched. The two agents are in a symmetric position for these states. They face the same probability of being BS or matched in the future. However, they are in asymmetric position with regards to the state of SO and BO. The BS which sells its house avoids the risk of being with two houses on the housing market (that is a SO) for the next period, but exposes more heavily to the risk of being a homeless (that is a BO). On the contrary, the BS which buys its house avoid the risk of being a homeless, but exposes himself to the risk of being a two houses owner. Consequently, a rise in the value of being BO (or a fall in the value of SO) increases the utility associated with the expected change of situation for the seller which then accepts a lower price. Price p3 is increasing with the value function of SO and decreasing with the value functions of BS, BO, and matched. For the SO household, selling its house solves its problem on the housing market. At the end of the period, it is matched with a convenient home (but is exposed to demographic risk). Utilities V BS and V M are discounted

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by bd and ð1  bÞd, respectively. The BS household is in a more complex situation. First, its probability to solve its housing problem during this period is lower than for a SO (or a BO) household. It depends on the probability q of selling its house. Second, if it does not sell its house, buying the house prevents from benefiting from a reversal of demographic situation that would solve its housing problem without buying nor selling. Utilities V BS and V M are discounted by ½bq  ð1  qÞð1  bÞd ¼ ½b  ð1  qÞd < bd and ½ð1  bÞq  ð1  qÞbd ¼ ðq  bÞd < ð1  bÞd, respectively. Consequently, an increase in V BS or V M rises the gains of a conclusive match for the SO more heavily than the gains of the BS. The seller accepts to sell at a lower price as asked by the buyer. By selling, the SO household looses value V SO whereas buying the house exposes the BS to the risk of being SO (in case it does not sell during the period). An increase in V SO diminishes the payoff for the SO and increases the payoff for the BS, which implies a rise in the price p3 . Finally, the utility V BO impacts only the payoff of the BS. An increase in V BO increases the payoff for the BS. Since the BS is acting a buyer, this lowers the sale price p3 . A similar argument applies for price p4 . Corollary 2. For symmetric Nash bargaining programs, the amount of price dispersion, i.e. p4  p1 , on the housing market is

p4  p1 ¼

d ð1  qÞ½ðV BS þ V M Þ  ðV SO þ V BO Þ 2b

ð21Þ

The RHS of Eq. (21) is the difference in (future) value gains to buy for a BO compared to a BS. If transaction occurs, the BO is ensured not to become a SO next period (whereas such a risk is supported by a BS seeking to buy) and is ensured to leave its current BO state (while this risk is anyway null for a BS). Hence, the lower V BO or V SO , the larger the gains of a match for a BO compared to those of a BS. The reverse is true from a BS point of view. Moreover, the lower V BS or V M , the larger the gains of a match for a BS compared to those of a BO, since the latter encounters a higher probability being in these states next period than the former if a (buying) transaction occurs. When V SO þ V BO is low (or V BS þ V M is large) the gains of conclusive match with a BS are higher for a BO than for a BS. Hence, the BO is ready to accept a high price (p4 ) to ensure the transaction occurs and p4  p1 is strictly positive. To end the study of price dispersion, it is worth mentioning that for asymmetric Nash Bargaining a fourth price appears in the economy as shown in the following proposition. Proposition 4. For asymmetric Nash bargaining programs, the equilibrium prices satisfy

p2  p1 ¼

dð1  2cÞ ð1  qÞ½ðV BS þ V M Þ  ðV SO þ V BO Þ b

ð22Þ

Interestingly the sign of p2  p1 depends on the sign of ðV BS þ V M Þ  ðV SO þ V BO Þ which also determines the amount of price dispersion ðp4  p1 Þ in the symmetric case. When V SO þ V BO is low (V BS þ V M is large) the gains of a conclusive

match are higher for a BO than for a BS (acting as a buyer) as already explained. But similarly, these gains are also higher for a SO than for a BS (acting as a seller), since the former encounters a higher probability becoming a BS or a matched household if the transaction is achieved. Consequently, if the respective bargaining powers of BO and SO households, which determine price p2 , are the same (i.e. c ¼ 1=2 ), both surplus in value compared to BS-BS transactions cancel out and p2 ¼ p1 . Evidently, for large (resp. low) values of bargaining power of buyers i.e. when c is above (resp. below) 1/2, the BO household can extract a larger (resp. lower) part of this gain in value than a SO household and price decreases p2 < p1 . The opposite argument applies when ðV SO þ V BO Þ is high and above ðV BS þ V M Þ.

5. Numerical analysis We proceed to numerical experiments to determine the range of parameters for which the sequential strategy equilibrium, the simultaneous strategy equilibrium or both of them exist. When multiple equilibria occur, they are ranked with respect to the steady-state welfare of matched households. Then, we discuss the existence and the amount of price dispersion. It is worth mentioning that the results of this numerical analysis are dependent on the assumption of exogenous moves. In our setup, all households accept the transaction when matched, which occurs with exogenous probability q, because we assume they have to move when hit by a demographic shock. If moves were endogenous, households may follow alternative search strategies that would lead to other dynamics of vacancy and time to sale. 5.1. Calibration We suppose time is annual. In this numerical analysis, some parameters will not be allowed to vary: we impose d ¼ 0:95 which suggests a 5% annual discount rate. Without loss of generality, the population size h is equal to one. The utility flow of being matched and of being paid for a home are scale parameters: we impose u ¼ 1 and b ¼ 1. Moreover, we try to reduce the disparities between the two equilibria by assuming similar matching probabilities. In the simultaneous strategy equilibrium, we impose q ¼ 0:7 (this value will be allowed to vary) which means that the average duration before getting matched is 1=0:7 ¼ 1:43 years. In the sequential strategy equilibrium, we adjust the vacancy rate v =h such as qs ¼ qb ¼ q ¼ 0:7. Finally we impose b ¼ 0:10. The probability of a change in family type is 10% in a year. The other parameters, which correspond to the states’ costs, will be allowed to vary: the costs of being a BS (jBS ), a BF (jBF ), a SO (jSO ) and a BO (jBO ). 5.2. Comparing equilibria In equilibrium Definitions 1 and 2, we determine the respective existence conditions of both kinds of equilibria. They exist if no agent has interest in refusing to conclude a transaction, nor to deviate from equilibrium decisions of

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14

Search cost of seller only κ

SO

12

10

8

6

4

2

− − − − − − − − − − − − − − − − −

5

− − − − − − − − − − − − − − − − − −

x x

− − − − − − − − − − − − − − − − −

x x x x x x x x x x x x x − − − − − x x x x x x x x x x x x − − − − x x x x x x x x x x x x x x x x x x x x x x x − x x x − − x x − − − x

− − − − − − − − − − −

− − − − − − − − − − −

6

− − − − − − − − − − −

− − − − − − − − − − −

− − − − − − − − − − −

− − − − − − − − − − −

7

o o x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − − x x x x x x x x x x x x x x x x x x x x x x x x x x − − − − − x x x x x x x x x x x x x x x x x x x x x x x x − − − − x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x − x x x − − x x − − − x

− − − −

− − − −

− − − −

− − − −

8

− − − −

− − − −

− − − −

o o o o o o o o o o o o o o o o o o o o o o x o o o x x o o x x x o x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x − x x x − − x x − − − x

o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x x

9 10 11 Search cost of buyer only κBO

o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x x

12

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x x

13

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x x

14

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x x

o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o x x x x x

15

Fig. 6. Existence of Equilibria and Pareto Ranking. The symbol () is for cases where the simultaneous strategy is the unique equilibrium. For multiple equilibria, symbol (x) means the welfare is higher with the simultaneous strategy, whereas higher is with the sequential strategy for symbol (o).

other agents. In line with Wheaton (1990) results, the sequential strategy equilibrium is characterized by a unique price, while the simultaneous strategy equilibrium is characterized by potential price dispersion. Different cases are possible for each set of parameters:  No equilibrium exists. No price.  The sequential strategy equilibrium exists, but the simultaneous strategy equilibrium does not exist. Price is unique.  The sequential strategy equilibrium does not exist, but the simultaneous strategy equilibrium exists. Multiplicity of prices occurs.  Multiple equilibria exist. Price dispersion is a possible outcome. In such a case, steady state equilibria are Pareto ranked according to the value of a matched household (V M ) in both economies. Matched households experience the same probability b of a change in their family type and must choose between the sequential strategy (enter as a BF) and the simultaneous strategy (enter as a BS) if such event occurs. If their discounted flows of utility is higher in the second case, it suggests that the simultaneous strategy may shorten the unmatched spell and that price dispersion has an indirect positive impact on welfare at steady-state.9

9 The comparison of V M in both equilibria should only be seen as a proxy of a Pareto ranking procedure, since we directly compare steady state equilibria and leave out the transient dynamics before joining the steady state.

5.3. Existence and ranking of equilibria Figs. 6 and 7 provide a comparison of the equilibria with sequential or simultaneous strategy for span of values of jBS ; jBO and jSO . In this experiment, we impose jBF ¼ jBS , i.e. we assume that the costs of simultaneously searching for a buyer and a seller are not fundamentally greater than those of just searching for a seller. A crucial point in our numerical analysis is that the results qualitatively depend on the relative positions of jBS ; jBO and jSO . Recall that jBS summarizes the costs of being unmatched (to live in a non convenient home) and the costs of searching for both a seller and a buyer, jBO is the cost of being homeless (or renter) and searching for a new home, jSO is the cost of searching for a buyer plus the potential financial costs. Hence we can reasonably suppose that jBO > max fjBS ; jSO g. Being a homeless is the worse state in our economy10. Without loss of generality, we arbitrarily choose a reference value for jBO (we impose11 jBO ¼ 8). The relative position of jBS and jSO is more ambiguous. The SO household lives in a convenient home (which is not the case for a BS), but bears financial costs that could potentially outweigh the search costs of a BS. As a reference we impose jBS ¼ 5 and jSO ¼ 2 but we will allow cases where jSO > jBS .

10 This is consistent with our focus on the ownership market, without an explicit modeling of the rental market. 11 We need to impose such a large value for jBO to guarantee that prices will remain positive in all our numerical experiments, see Proposition 2.

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12

Search cost of buyer seller and buyer first κ

BS

= κBF

14

10

8

6

4

2

− − − − −

− − − − −

− − − − − −

x o o o

o o o o

− − − − − − x x − − − − − − − − x x x x − − − x x x x x x − − x x x x x x x x x x x o

2

x x x o

x x o o

x x o o

x x o o

x o o o

x o o o

x o o o

4

− − − − − − x x x x x o o o o

− − − − − − − x x x x x o o o o

− − − − − − − x x x x x o o o o o

− − − − − − − − x x x x x o o o o o

− − − − − − − − x x x x x x o o o o o

− − − − − − − − −

− − − − − − − − − −

− − − − − − − − − − −

x x x x x o o o o o o

x x x x x o o o o o o

x x x x x o o o o o o

− − − − − − − − − − −

− − − − − − − − − − − −

x x x x o o o o o o o

x x x x o o o o o o o

− − − − − − − − − − − − −

− − − − − − − − − − − − − −

x x x o o o o o o o o

x x x o o o o o o o o

− − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − −

x x o o o o o o o o o

x x o o o o o o o o o

− − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − −

x o o o o o o o o o

x o o o o o o o o o

o o o o o o o o o o

− − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − −

o o o o o o o o o o

o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − −

− − − − − − − − − − − − − x − x x x − x x x x x − − x x x x x x x x

x o o o o o o o o o o

x o o o o o o o o o

− − − − − − − − − − − − − −

− − − − − − − − − − − − − x x x x x x x x x x − − x x x x x x x x x x x x

x x x o o o o o o o o

x x o o o o o o o o o

− − − − − − − − − − − −

− − − − − − − − − − − x x x x x x x x x x x x x x − − x x x x x x x x x x x x x x x x

x x x x o o o o o o o

x x x o o o o o o o o

− − − − − − x x x x x x x x x x x x x x x x x x − x x x x x x x x x x x x x x x x x x x x − − x x x x x x x x x x x x x x x x x x x x o o o x x x x x x o o o o o

6

o o o o o o o o o o

o o o o o o o o o o

o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − x x x x x x x x x x o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − x x x x x x x x x x o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − −

x x x x x x x x x x o o o o o o o o o o o

x x x x x x x x x o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − − − −

x x x x x x x x x o o o o o o o o o o o o

x x x x x x x x o o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − − − − − −

x x x x x x x x o o o o o o o o o o o o o

x x x x x x x x o o o o o o o o o o o o o

x x x x x x x o o o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − − − − − − − −

x x x x x x x o o o o o o o o o o o o o o

x x x x x x x o o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − − − −

− − − − − − − − − − − − − − − − − − − − − − x x − x x x x x − − x x x x x x x

x x x x x x x x x o o o o o o o o o o o o

x x x x x x x x x o o o o o o o o o o o o

− − − − − − − − − − − − − − − − − − − − − x x x x x x x x x − − x x x x x x x x x x x x x x x x x x x x x o o o o o o o o o o o

l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

8 10 Search cost of buyer only κBO

12

14

Fig. 7. Existence of Equilibria and Pareto Ranking. The symbol () is for cases where the simultaneous strategy is the unique equilibrium and (|) for cases where the sequential strategy is the unique equilibrium. For multiple equilibria, symbol (x) means that the welfare is higher with the simultaneous strategy, whereas higher with the sequential strategy for symbol (o).

Overall, the figures show that for large spans of values of jBS ; jBO and jSO , the simultaneous strategy equilibrium robustly exists, which is not the case for the sequential strategy equilibrium. Fig. 6 provides an analysis of the existence of the two equilibria for different values of jBO (> jBS ) and jS0 (< jB0 ). In this figure, jBS is set to its reference value (i.e.; jBS ¼ 5). For low values of the cost of being homeless (i.e. jBO close to jBS ) the sequential strategy equilibrium does not exist. Therefore, since the simultaneous equilibrium still exists even for very large values of jB (more than three times the search costs of a BS household): the larger jB , the more frequent the occurrence of multiple equilibria. This result is natural since an unmatched buyer adopting the sequential strategy is ensured not to experience a spell of BO in the future. On the contrary, a BS household incurs the risk of becoming a BO and bearing costs jBO if it sells its home but simultaneously does not find a seller. Consequently, large values of jB prevent households from deviating from the sequential strategy. When multiple equilibria exist, the simultaneous strategy remain Pareto dominant over the sequential strategy (V M > V BF ) as long as jBO is not too large (for example jBO < 8:5 for the reference value jBS ¼ 5). This suggests that the main gain of being a matched household adopting simultaneous strategy instead of the sequential one – shorter expected unmatched spells – overwhelms the losses (positive probability of becoming a homeless/ renter in the BO state). In the simultaneous strategy equilibrium, the average duration of an unmatched spell

   is (in years) dsimultaneous ¼ ð3  2qÞ= b þ 2ð1  2bÞ 2q  q2 which is lower than its counterpart with sequential strategy dsequential ¼ 2=ðb þ q  2bqÞ for a large span of values of b and q. For example, for our reference values of b (b ¼ 0:10), the average time spent on the housing market with the sequential strategy for q ¼ 0:7 is more than 3 years (dsequential ¼ 3:03), but only slightly above 1 year with the simultaneous strategy12. Fig. 6 also provides an analysis of the existence of the two equilibria for different values of jSO . Results are very similar to those obtained for different span of values of jBO since, as noticed in Section 3, the value function of a BF household does not depend on jSO which impacts only the equilibrium price in the simultaneous strategy equilibrium. Consequently, the higher jSO , the lower the interest for household in deviating from the sequential strategy and the higher the probability to have multiple equilibria since the simultaneous equilibrium always exists, as for jBO . Both equilibria exist when jSO 2 ½jBS ; jBO . On the contrary, for low values of jSO , the simultaneous strategy is the only equilibrium outcome. In this case, the risk of becoming a SO when adopting the simultaneous strategy is smaller than the gains consecutive to a shorter expected unmatched spell. Finally, notice that the simultaneous strategy is Pareto dominant 12 This gap is mitigated for larger values of b: the higher b the higher the probability of becoming matched directly with a demographic shock rather than with conclusive transactions. Hence, the lower the relative interest of a simultaneous strategy compared to the sequential one.

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over the sequential one if jSO is small enough (below 5) when jBO is set at its reference value. Finally, Fig. 7 numerically confirms the robustness of the simultaneous equilibrium existence for a large span of values of fjBO ; jBS g. This figure suggests that the positive effect of jBO on the occurence of multiple equilibria lowers as jBF ð¼ jBS Þ grows. The sequential equilibrium does not exist for large span of values of jBF (even in sensible cases where jBF < jBO ). The higher jBF , the lower the welfare value of a BF in the sequential equilibrium. Consequently, households have a greater interest in deviating and adopting the simultaneous strategy since their probability to leave their current state (with probability q2 þ 2qð1  qÞ, apart from the demographic chock) is larger than with the sequential strategy (with probability q), since, in the first case, they only have to sell or buy to leave their current BS state. The existence condition of the equilibrium with simultaneous strategy is not verified only when we have simultaneously high search costs jBO and small search costs jBS . Notice that all these results appear to be robust for other plausible calibrations of parameter b (i.e. 0 < b < 0:2) and q (i.e. 0:5 < q < 0:9). In these cases, the simultaneous strategy generally exists and multiple equilibria only occurs for low values of jBF and/or large values of jBO . All these results (the stable existence of the equilibrium with simultaneous strategy and its numerically large Pareto dominance over the equilibrium with sequential strategy) confirms the robustness of price dispersion in our setup.

5.4. Price dispersion Let us now focus on the amount of price dispersion in the equilibrium with simultaneous strategy. Once again, we restrict our attention to the symmetric Nash bargaining program (c ¼ 1=2Þ. As noticed in Proposition 3, p1 ¼ p2 in the symmetric case and p1 is the mean of ½p3 ; p4 . So there are only three different prices in equilibrium. Fig. 8 provides the values of p1 ; p3 and p4 for different values of jBO ; jSO together with the rate of relative dispersion, i.e.ðp4  p1 Þ=p1 . Notice that we restrict our analysis to parameter values where the simultaneous equilibrium with price dispersion does exist (i.e. we do not consider too large values for jBO ). As the figure suggests, the higher the gap between jBO and jSO , the higher p1 (see Eq. 17). Differently said, the price of a home is directly measured by the welfare difference between owning two houses and being a homeless. Ceteris paribus, the larger jBO , the larger the relative welfare of being an owner-occupier (either in a convenient dwelling or not) rather than a BO household. Hence, as jBO grows, price p1 becomes larger. On the contrary, the larger jSO the lower this relative welfare: a BS household acting as a buyer will be ready to accept a lower price p1 to escape a transient state of BO. The degree of price dispersion appears to be impacted by the values of jBO and jSO . The difference between the price paid by a BO household and the price paid by a BS household (when they both meet a BS household acting as a seller), i.e. p4  p1 , is decreasing with respect to jBO

a. Prices

b. Price dispersion

80

5 p3

p1

p4

(p4−p1)/p1

60

4

40

3

20

2

0

1

−20

0

2 4 6 search cost of seller only κ

8

0

0

SO

8

SO

c. Prices

d. Price dispersion

80

0.02

70

0.019 0.018

60

0.017

50

0.016

40

0.015

30 20

2 4 6 search cost of seller only κ

0.014 5

6 7 8 9 search cost of buyer only κ

BO

10

0.013

5

6 7 8 9 search cost of buyer only κ

10

BO

Fig. 8. Equilibrium prices and price dispersion in the simultaneous strategy equilibrium. Panels (a) and (c) show the equilibrium prices for various values of search costs of seller only and buyer only respectively. Panels (b) and (d) give the associated rate of price dispersion on the housing market.

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and jSO . The relative price dispersion, i.e. ðp4  p1 Þ=p1 , is increasing with jSO and decreasing with jBO . This suggests in particular that the sum of BS and matched households welfare values ðV BS þ V M Þ decreases slightly faster than the sum of SO and BO households welfare values ðV SO þ V BO Þ as jBO grows, see Corollary 2. Such a small gap is due to a ‘‘discount differential’’ between these four welfare values: as jBO grows, V BO decreases at a high rate (since the BO household currently bears the search/homeless cost), V BS decreases at a lower rate (since BS households experience a positive probability to pay the cost jB in one period) and V M and V SO at an even lower rate (matched and SO households cannot become BO and pay jBO before two periods). SO households in particular encounter a reduced riskof becoming a BO in the next two periods. This latter effect proves to dominate the other one in our numerical experiment which explains the drop in price dispersion as jBO grows.The same reasoning applies for jSO .

6. Conclusion Every year many households change their residential location either for personal or professional reasons. Moving home is a tricky operation, which can be highly costly for households in case of troubles. The households can experience homeless (rental market) spells, if they sell their old dwelling before buying the new one, high financial costs, if they are stuck with two houses, or the annoyance of living in a non convenient dwelling for a long time. These features of the housing market are well-known and have already been considered in the literature, notably through the presence of matching frictions and search costs. However, few attention has been devoted to the search strategy of households. This is especially the case for the simultaneous strategy where the household tries to simultaneously buy and sell the dwellings. Indeed, the literature exemplified by Wheaton (1990) generally considers only a sequential strategy where the household first buys the new dwelling before selling the old one. The comparison of the two strategies provided in this article shows however that the simultaneous strategy is more stable than the sequential strategy and may deliver higher welfare. The numerical analysis of our theoretical model indicates that for a large range of parameters’ values, the equilibrium with simultaneous strategy generally exists but not the equilibrium with sequential strategy. In the case of multiple equilibria, each of the two strategies can be associated with the higher steady-state welfare depending on parameters values. Besides its implications on the equilibrium (existence, uniqueness, and welfare), the simultaneous strategy also provides a rational explanation for the puzzling price dispersion observed on the housing market. When households enter the housing market both as buyer and seller, the transactions are done by pairs of households which are at different stages of their housing market pathes. Since these households have different prospects on the housing market, the bargained prices are different. It is worth mentioning that this price dispersion proceeds only from the endogenous functioning

of the housing market and does not require any exogenous form of heterogeneity either on dwellings or on households. Our model has been kept sufficiently simple to provide analytical proof of the existence of price dispersion. Its simplicity may also limit its scope. First, prices and their dispersion are not linked with transaction volumes and time to sales on the housing market. Taking time to sale into account in a dynamic setup would enrich the price distribution compared to the discrete distribution of our benchmark. It is well known that households with a BS status that become SO should use bridge loans, which are available in some countries. Such loans typically have low maturities with huge penalties if the SO household is not able to sell its home before loan maturity. In this case, the loan maturity would introduce non stationarity in the setup and the heterogeneity in time to sale would impact the bargaining process, thereby creating a continuous price distribution. Second,we do not tackle the challenge of solving both the bargaining processes and the strategy’s choice while taking the existence of chains on the housing market into account. However, we think that the model we proposed in this article improves our understanding of the housing market and could be fruitfully enriched in these directions in future researches. Appendix A. Notations

Population h1 the mass of single households h2 the mass of couple households 2h the total mass of households hZ mass of households in state Z for each family type and for Z ¼ fM; BS; BF; SF; BO; SOg hb the share of buyer seller households among the mass of all buyers hs the share of buyer seller households among the mass of all sellers s the stock of dwellings for each type (small and large) v the mass of vacant dwellings for each type (small and large)

Preferences d the discount factor b the constant marginal utility jZ the per-period cost associated to the state Z for Z ¼ fBF; BO; SOg jV the per-period cost associated to the state BS c the bargaining power of buyers in the bargaining processes on prices

States M BS BF

for households who leave in the dwelling that corresponds to their family type for buyer seller households who try to simultaneously buy a new dwelling and sell the old one for buyer first households who first search for a new dwelling before selling the old one

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T.-P. Maury, F. Tripier / Journal of Housing Economics 26 (2014) 55–80

SF BO SO

for seller first households who first sell the old dwelling before buying the new one for households who own no house and search to buy a new one for households who own two houses and search to sell one of them

Value functions VZ the value function associated to the state Z.for Z ¼ fM; BS; BF; SF; BO; SOg BV Z the increase in the value function for a household in state Z who buys a dwelling SV Z the increase in the value function for a household in state Z who sells a dwelling BX V Z the increase in the value function for a household in state Z who buys a dwelling to a household in state X SX V Z the increase in the value function for a household in state Z who sells a dwelling to a household in state X V BF the value function of the household who deviates from the simultaneous strategy by entering as a buyer first V SF the value function of the household who deviates from the simultaneous strategy by entering as a seller first V BS the value function of the household who deviates from the sequential strategy by entering as a buyer seller V B0 the value function of the household who has deviated from the sequential strategy, by entering as a buyer seller, and has sold its dwelling

Transition rates b the symmetric transition rate between the two family states (single and couple) qb the matching probability with a seller for the buyers qs the matching probability with a buyer for the sellers Prices pB p1BF p2BF p3BF fpi g4i¼1

the price bargained by a buyer first and a seller only in the equilibrium with sequential strategy the price bargained by a buyer first and the deviating buyer seller in the equilibrium with sequential strategy the price bargained by a the deviating buyer seller and a seller only in the equilibrium with sequential strategy the price bargained by a the deviating buyer seller, who has already sold its house, and a seller only in the equilibrium with sequential strategy the four prices in the equilibrium with simultaneous strategy (see Table 2 for details)

SV SO ¼ V SO jqs ¼1  V SO jqs ¼0 ¼ bpB þ dðbV BF þ ð1  bÞV M Þ  dV SO

ðB:1Þ

BV BF ¼ V BF jqb ¼1  V BF jqb ¼0 ¼ bpB  dðbV M þ ð1  bÞV BF Þ þ dV SO

ðB:2Þ

Introducing the Eqs. B.1,B.2 in the sharing rule gives

c½bpB þ dðbV BF þ ð1  bÞV M Þ  dV SO  ¼ ð1  cÞ½bpB þ dV SO  dðbV M þ ð1  bÞV BF Þ

ðB:3Þ

It is therefore straightforward to deduce the expression of pB given by (9). B.1.1. The system of equations The set of endogenous variables of the equilibrium with sequential strategy fpB ; V M ; V BF ; V SO ; hBF ; hSO g solves the following system of equations

d pB ¼ ð1  cÞ½V SO  ðbV M þ ð1  bÞV BF Þ b d  c½bV BF þ ð1  bÞV M  V SO  b

ðB:4Þ

V M ¼ u þ d½bV BF þ ð1  bÞV M 

ðB:5Þ

V BF ¼ u  jBF þ qb uðpB Þ     þ d qb V SO þ 1  qb ðbV M þ ð1  bÞV BF Þ

ðB:6Þ

V SO ¼ u  jSO þ qs uðpB Þ þ d½qs ðbV BF þ ð1  bÞV M Þ þ ð1  qs ÞV SO 

ðB:7Þ

  0 hBF ¼ 1  qb ð1  bÞhBF þ bðh  hBF  hSO Þ þ qs bhSO

ðB:8Þ

0

hSO ¼ ð1  qs ÞhSO þ qb hBF

ðB:9Þ

B.1.2. The equilibrium value for V BF To get (10), the Eq. (9) is rearranged as follows

b p ¼ ð1  cÞV SO  ð1  cÞ½bV M þ ð1  bÞV BF  d B  cbV BF  cð1  bÞV M þ cV SO

ðB:10Þ

or

b p ¼ V SO  ½ð1  cÞb þ cð1  bÞV M d B  ½ð1  bÞð1  cÞ þ cbV BF

ðB:11Þ

Introducing this expression for the price pB in (6) leads to V BF ¼ u  jBF þ qb d½V SO þ ½ð1  cÞb þ cð1  bÞV M þ ½ð1  bÞð1  cÞ þ cbV BF      þd qb V SO þ 1  qb ðbV M þ ð1  bÞV BF Þ ðB:12Þ

Appendix B. The equilibrium with sequential strategy B.1. The price The bargaining process (8) leads to the standard sharing rule cSV SO ¼ ð1  cÞBV BF , where the inputs of the bargaining process are calculated using (6,7)

After the suppression of the values for V SO , we regroup the coefficients for V M and V BF to get

    V BF ¼ u  jBF þ ð1  cÞbqb þ cð1  bÞqb þ 1  qb b dV M     þ ð1  bÞð1  cÞqb þ cbqb þ 1  qb ð1  bÞ dV BF ðB:13Þ

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which can be simplified as follows

V BF



 ¼ u  jBF þ cq ð1  2bÞ þ b dV M   þ 1  b  cqb ð1  2bÞ dV BF b

ðB:14Þ

This expression for V BF depends on the value function V M . Using its definition given by (5), we get its expression in function of the value function V BF and the structural parameters

VM ¼

u db þ V BF 1  dð1  bÞ 1  dð1  bÞ

ðB:15Þ

Introducing the expression for V M given by (B.15) in (B.14) gives

   V BF ¼ u  jBF þ cqb ð1  2bÞ þ b 1ddu þ 1ddb dV BF ð1bÞ ð1bÞ   þ 1  b  cqb ð1  2bÞ dV BF ðB:16Þ and



 cqb ð1  2bÞ þ b V BF 1  bd þ 1  b  cqb ð1  2bÞ d 1  dð1  bÞ

 cqb ð1  2bÞ þ b ¼ d þ 1 u  jBF 1  dð1  bÞ ðB:17Þ Introducing the following value for the block of structural parameter x

x¼d

cqb ð1  2bÞ þ b

ðB:18Þ

1  dð1  bÞ

gives

    V BF 1  bx þ 1  b  cqb ð1  2bÞ d

types of match are feasible on the market with three associated prices. The three matches are defined in the Table B.1. To establish the condition of existence of the equilibrium given by the Proposition 1, we need to find the value of these prices to calculate the net payoff of deviation. We use the symbol  for the deviating household, (its value functions are V BS and V B0 ): 1

2

V BS ¼ u  jV þ qs bpBF  qb bpBF    þ d qs qb bV BS þ ð1  bÞV M þ qb ð1  qs ÞV SO      þ qs 1  qb V BO þ ð1  qs Þ 1  qb bV M þ ð1  bÞV BS ðB:22Þ 3

V BO ¼ u  jB  qb bpBF       þ d qb bV BS þ ð1  bÞV M þ 1  qb V BO notice that

V SO

¼ V SO .

B.2.2. The equilibrium price p1BF The price p1BF is the solution of the bargaining process

n o p1BF : arg max ½BV BF c ½SV BS 1c

ðB:24Þ

The first input of the bargaining process (B.24), namely BV BF , is given by (B.2), with p1BF instead of pB . The second input, namely SV BS , is calculated as follows

  SV BS ¼ V BS qs ¼1  V BS qs ¼0    1 ¼ bpBF þ d qb bV BS þ ð1  bÞV M  V SO     1  qb bV M þ ð1  bÞV BS  V BO

ðB:19Þ

using (B.22) for the definition of Introducing these inputs in the sharing rule cSV BS ¼ ð1  cÞBV BF solution of (B.24) gives

cbp1BF þ cd qb bV BS þ ð1  bÞV M  V SO

¼ ðx þ 1Þu  jBF

ðB:20Þ

Using once again the definition of x given by (B.18) leads to

V BF ½1  dbx  d þ x½1  dð1  bÞ ¼ ðx þ 1Þu  jBF



    1  qb bV M þ ð1  bÞV BS  V BO h i 1 ¼ ð1  cÞ bpBF  dðbV M þ ð1  bÞV BF Þ þ dV SO ðB:26Þ

and

   V BF 1  dbx  d þ d b þ cqb ð1  2bÞ

ðB:25Þ

V BS .

 

¼ ðx þ 1Þu  jBF

ðB:23Þ

and therefore, after algebra, we obtain 1

bpBF ¼ dð1  cÞ½V SO  ðbV M þ ð1  bÞV BF Þ    cdqb bV BS þ ð1  bÞV M  V SO    þ cd 1  qb bV M þ ð1  bÞV BS  V BO

ðB:27Þ

ðB:21Þ

and, after simplifications, we finally obtain (10).

B.2.3. The equilibrium price p2BF The price p2BF is the solution of the bargaining process

B.2. The equilibrium prices with a deviating household

n o p2BF : arg max ½BV BS c ½SV SO 1c

B.2.1. Definition If an household deviates from the sequential strategy and enters the market as a buyer and seller, three additional

The second input of the bargaining process (B.28), namely SV SO , is given by (B.1), with p2BF instead of pB . The first input, namely BV BS , is calculated as follows

ðB:28Þ

Table B.1 Type of match with a deviating household. Type of Match

The buyer

The seller

The price

1

A buyer first

The unique buyer and seller

p1BF

2

The unique BS

A seller only

p2BF

3

BO who has been BS first

A seller only

p3BF

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  BV BS ¼ V BS qb ¼1  V BS qb ¼0    2 ¼ bpBF þ d qs bV BS þ ð1  bÞV M  V BO    ð1  qs Þ bV M þ ð1  bÞV BS  V SO

ðB:29Þ

V BS .

using (B.22) for the definition of Introducing these inputs in the sharing rule cSV SO ¼ ð1  cÞBV BS solution of (B.28) gives

cbp2BF þ dcðbV BF þ ð1  bÞV M Þ  cdV SO    2 2 ¼ bpBF þ cbpBF þ ð1  cÞd qs bV BS þ ð1  bÞV M  V BO    ð1  qs Þ bV M þ ð1  bÞV BS  V SO

ðB:30Þ

and therefore, after algebra, we obtain

B.3.2. Transformations of prices and value functions  3 We begin with the prices piBF i¼1 , which equilibrium expressions are transformed as follows

2

bpBF ¼ ð1  cÞdqs ðbV BS þ ð1  bÞV M  V BO Þ  ð1  cÞdð1  qs ÞðbV M þ ð1  bÞV BS  V SO Þ  cd½bV BF þ ð1  bÞV M  V SO 

ðB:31Þ

B.2.4. The equilibrium price p3BF The price p3BF is the solution of the bargaining process

p3BF

n o : arg max ½BV BO c ½SV SO 1c

ðB:32Þ

The second input of the bargaining process (B.28), namely SV SO , is given by (B.1), with p3BF instead of pB . The first input, namely BV BO , is calculated as follows

BV BO

  ¼ V BO qb ¼1  V BO qb ¼0   3 ¼ bpBF þ d bV BS þ ð1  bÞV M  V BO

h

ðB:36Þ

  2 bpBF ¼ B þ ð1  cÞd ðb  ð1  qs ÞÞV BS  qs V BO

ðB:37Þ

  3 bpBF ¼ C þ ð1  cÞd bV BS  V BO

ðB:38Þ

where A; B, and C are independent from

A ¼ dð1  cÞ½V SO  ðbV M þ ð1  bÞV BF Þ      cd qb ðð1  bÞV M  V SO Þ  1  qb bV M

ðB:39Þ

 cd½bV BF þ ð1  bÞV M  V SO 

ðB:41Þ We do a similar transformation for

V BO

¼ D þ cdq

ðB:34Þ ¼

  3 bpBF ¼ ð1  cÞd bV BS þ ð1  bÞV M  V BO

ðB:40Þ

C ¼ ð1  cÞdð1  bÞV M  cd½bV BF þ ð1  bÞV M  V SO 

and therefore, after algebra, we obtain

 cd½bV BF þ ð1  bÞV M  V SO 

jV and given by

B ¼ ð1  cÞd½qs ð1  bÞV M  ð1  qs ÞðbV M  V SO Þ

i

h  i 3 ¼ ð1  cÞ bpBF þ d bV BS þ ð1  bÞV M  V BO

    1 bpBF ¼ A þ cd 1  b  qb V BS  cd 1  qb V BO

ðB:33Þ

using (B.23) for the definition of V BO . Introducing these inputs in the sharing rule cSV SO ¼ ð1  cÞBV BO solution (B.32) gives

c bp3BF þ dðbV BF þ ð1  bÞV M Þ  dV SO

it seems evident that V BS is decreasing with jV . The difficulty comes from the fact that this parameter impacts also  3 the prices piBF i¼1 and the value function of the deviating household if it becomes BO. Therefore, we start with the definition of V BS given by Eq. (B.22) and re-express it as a function of the structural parameters using the equilibrium  3 expression of piBF i¼1 , equations (B.22) (B.35), and of V BO , Eq. (B.23). Finally, notice that the value functions associated with the single strategy (V M ; V BF , and V SO ) do not depend on jV . Consequently, they are treated as exogenous in this proposition.

b

bV BS



þ d 1  cq

b



V BO ,

which satisfies

V BO

D cdqb b þ V b 1  dð1  cq Þ 1  dð1  cqb Þ BS

ðB:42Þ

after introducing (B.38) into (B.23), where D is also independent from jV and given by

ðB:35Þ

B.3. The condition of existence B.3.1. Remarks Proposition 1 gives a necessary and sufficient condition of existence of the equilibrium with sequential strategy. This proposition relies on a threshold value for the search cost of buyer and seller, jV , such that the equilibrium exists if jV > jV and does not exist otherwise. This property is intuitive. Since jV is paid only by the deviating household which tries to simultaneously buy and sell, one can always find a value for jV (eventually very high if necessary) such that deviating from the sequential strategy is not optimal. To confirm this intuition, we only need to prove that the value function of the deviate household (V BS ) is strictly decreasing with jV since the value function of the buyer first (V BF ) is not impacted by jV . By inspecting Eq. (B.22),

D ¼ u  jB  qb C þ dqb ð1  bÞV M

ðB:43Þ V BS

We are now in position to compute the value as a function of jV . Let us first introduce the prices given by (B.36) and (B.37) into (B.22), we obtain

      V BS ¼ u  jV þ qs A þ cd 1  b  qb V BS  cd 1  qb V BO     qb B þ ð1  cÞd ðb  ð1  qs ÞÞV BS  qs V BO    þ d qs qb bV BS þ ð1  bÞV M þ qb ð1  qs ÞV SO      þ qs 1  qb V BO þ ð1  qs Þ 1  qb bV M þ ð1  bÞV BS ðB:44Þ We define as follows the term E, which is independent of

jV E ¼ u þ qs A  qb B     þ d qs qb ð1  bÞV M þ qb ð1  qs ÞV SO þ ð1  qs Þ 1  qb bV M ðB:45Þ

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Introducing (B.45) into (B.44) gives       V BS ¼ E  jV þ qs cd 1  b  qb V BS  cd 1  qb V BO     qb ð1  cÞd ðb  ð1  qs ÞÞV BS  qs V BO       þd qs qb bV BS þ qs 1  qb V BO þ ð1  qs Þ 1  qb ð1  bÞV BS ðB:46Þ

or equivalently "

V BS

#   qs c 1  b  qb  qb ð1  cÞðb  ð1  qs ÞÞ ¼ E  jV þ dV BS   þqs qb b þ ð1  qs Þ 1  qb ð1  bÞ      þ qb ð1  cÞqs  qs c 1  qb þ qs 1  qb dV BO ðB:47Þ

or equivalently " V BS ¼ E  jV þ

#   ð1  qs Þ 1  qb ð1  bÞ þ qs qb b dV BS   þqs c 1  b  qb  qb ð1  cÞðb  ð1  qs ÞÞ

þqs ð1  cÞdV BO ðB:48Þ

B.4. Numerical simulation Equations are used sequentially in the following order: (B.18) for x, (10) for V BF , (B.15) for V M ; (B.55) for V SO ; (((B.39)–(B.41))-(B.43)–(B.45) for A  E; (B.51) for V BS ; (B.1) for SV SO ; and (B.2) for BV BF . Eq. (B.55) is obtained by introducing the expression (9) for the price pB into the expression (7) for the value function V SO

V SO ¼ u  jSO þ qs dðð1  cÞ½V SO  ðbV M þ ð1  bÞV BF Þ  c½bV BF þ ð1  bÞV M  V SO Þ þ d½qs ðbV BF þ ð1  bÞV M Þ þ ð1  qs ÞV SO 

ðB:54Þ

and, after algebra, we get

V SO ¼

u  jSO dqs ð1  cÞð1  2bÞ þ ðV M  V BF Þ 1d 1d

ðB:55Þ

Appendix C. The simultaneous strategy equilibrium

Second, we introduce the value function defined by (B.42) into (B.48)

C.1. The demographic structure

V BS ¼ E  jV þ

Proof. Proof of Lemma 1. We first use the dynamic system (11) to get the expression of the gap between the masses of SO and BO households for tomorrow

"

   # ð1  qs Þ 1  qb ð1  bÞ þ qs qb b þ qs c 1  b  qb

qb ð1  cÞðb  ð1  qs ÞÞ

bb D þqs ð1  cÞd 1d 1 þ 1dcdq V BS b b c q c q 1 ð Þ ð Þ

dV BS

ðB:49Þ

to obtain

  0 0 hSO  hBO ¼ ð1  qs ÞhSO þ qb ð1  qs ÞhBS  1  qb hBO    qs 1  qb hBS

s

ð1cÞq d V BS ¼ E  jV þ 1d D ð1cqb Þ 2    3 s b ð1  q Þ 1  q ð1  bÞ þ qs qb b þ qs c 1  b  qb 4 5dV  þ 1cÞqs cdqb b BS qb ð1  cÞðb  ð1  qs ÞÞ þ ð1d ð1cqb Þ ðB:50Þ

ðC:1Þ

qb ðhBS þ hBO Þ ¼ qs ðhBS þ hSO Þ

s

¼

 qs hBS þ qs qb hBS     ¼ ðhSO  hBO Þ þ qb hBS  qs hBS þ qb hBO  qs hSO Therefore, it implies the equality of purchases and sales

and finally

V BS

¼ hSO  qs hSO þ qb hBS  qb qs hBS  hBO þ qb hBO

ð1cÞq d E  jV þ 1d D ð1cqb Þ



1cÞqs cdqb b 1  d ð1  bÞð1  qs þ cðqs  qb ÞÞ þ ð1d b ð1cq Þ ðB:51Þ

since

    ð1  qs Þ 1  qb ð1  bÞ þ qs qb b þ qs c 1  b  qb  qb ð1  cÞðb  ð1  qs ÞÞ     ¼ ð1  bÞ 1  qs þ c qs  qb ðB:52Þ

hBO ¼ ð1  qÞhBS and hSO ¼ ð1  qÞhBS s

B.3.3. The condition of existence The condition of existence of the equilibrium with sequential strategy is deduced from the condition V BF > V BS . Using the Eqs. (10) and (B.51) for the value functions, we obtain jV > jV for



ð1  cÞqs d jBF u jV ¼ E þ D þ  ð1  dÞð1 þ xÞ ð1  dÞ 1  dð1  cqb Þ

   ð1  cÞqs cdqb b  1  d ð1  bÞ 1  qs þ c qs  qb þ b 1  dð1  cq Þ

ðB:53Þ with E; D and x are defined by (B.45)–(B.43)-(B.18) and do not depend on jV .

ðC:2Þ

 0 0  at the steady-state because hSO  hBO ¼ ðhSO  hBO Þ. Therefore, assuming an equal number of SO and BO 0 0 households hSO ¼ hBO is consistent with hSO ¼ hBO according to (C.1) and implies qb ¼ qs  q according to (C.2). The steady-state the masses of BO households and SO households solution of (11) are therefore

b

It directly follows that h ¼ h .

ðC:3Þ

j

C.2. The equilibrium housing price system C.2.1. The price p1 The bargaining process (15) leads to the sharing rule: cS BS V BS ¼ ð1  cÞBBS V BS , where the inputs of the bargaining process are calculated using (13,14)

BBS V BS

¼ V BS jqb ¼1;hs ¼1  V BS jqb ¼0 ¼ bp1 þ d½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO  d½qs V BO þ ð1  qs ÞðbV M þ ð1  bÞV BS Þ ðC:4Þ

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S BS V BS

¼ V BS jqs ¼1;hb ¼1  V BS jqs ¼0     ¼ bp1 þ d qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO     d qb V SO þ 1  qb ðbV M þ ð1  bÞV BS Þ ðC:5Þ

C.2.4. The price p4 The bargaining process (15) leads to the sharing rule cS BO V BS ¼ ð1  cÞBBS V BO , where the inputs of the bargaining process are calculated using (13,14)

BBS V B0

¼ V BO jqb ¼1;hs ¼1  V BO jqb ¼0

since

ðC:12Þ

¼ bp4 þ dðbV BS þ ð1  bÞV M Þ  dV BO

U ðp1 ; p3 ; p4 Þqb ¼1;hs ¼1  U ðp1 ; p3 ; p4 Þqb ¼0 h i ¼  U ðp1 ; p3 ; p4 Þqs ¼1;hb ¼1  U ðp1 ; p3 ; p4 Þqs ¼0

S BO V BS

¼ V BS jqs ¼1;hb ¼0  V BS jqs ¼0     ¼ bp4 þ d qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO     d qb V SO þ 1  qb ðbV M þ ð1  bÞV BS Þ

ðC:6Þ

¼ bp1

ðC:13Þ

Introducing Eqs. C.4,C.5 in the sharing rule gives

    cbp1 þ dc qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO      dc qb V SO þ 1  qb ðbV M þ ð1  bÞV BS Þ

Introducing Eqs. C.12,C.13 in the sharing rule gives



It is therefore straightforward to deduce the expression of p1 given by (16). C.2.2. The price p2 The bargaining process (15) leads to the sharing rule: cS BO V SO ¼ ð1  cÞBSO V BO , where the inputs of the bargaining process are calculated using (13,14)

S BO V SO

¼ V SO jqs ¼1;hb ¼0  V SO jqs ¼0 ¼ bp2 þ dðbV BS þ ð1  bÞV M Þ  dV SO

ðC:7Þ

ðC:8Þ

It is therefore straightforward to deduce the expression of p4 given by (16). C.3. Existence and unicity of simultaneous equilibrium C.3.1. The equilibrium system of equations for prices and value functions Under Lemma 1 the utility function (14) writes as follows

U ðp1 ; p3 ; p4 Þ ¼ q2 ½hð1  hÞbðp4  p1 Þ þ ð1  hÞhbðp1  p3 Þ þ ð1  hÞð1  hÞbðp4  p3 Þ þ qð1  qÞ½hbp1 þ ð1  hÞbp4  þ qð1  qÞ½hbp1  ð1  hÞbp3 

bp2 ¼ ð1  cÞ½dðbV BS þ ð1  bÞV M Þ  dV BO   cdðbV BS þ ð1  bÞV M Þ þ cdV SO It is therefore straightforward to deduce the expression of p2 given by (16). C.2.3. The price p3 The bargaining process (15) leads to the sharing rule ð1  cÞBSO V BS ¼ cS BS V SO , where the inputs of the bargaining process are calculated using (13,14)

¼ V SO jqs ¼1;hb ¼1  V SO jqs ¼0 ¼ bp3 þ dðbV BS þ ð1  bÞV M Þ  dV SO

BSO V BS

ðC:9Þ

¼ V BS jqb ¼1;hs ¼0  V BS jqb ¼0 ¼ bp3 þ d½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO  d½qs V BO þ ð1  qs ÞðbV M þ ð1  bÞV BS Þ ðC:10Þ

Introducing Eqs. C.9,C.10 in the sharing rule gives

ð1  cÞ½bp3 þ d½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO  d½qs V BO þ ð1  qs ÞðbV M þ ð1  bÞV BS Þ ¼ c½bp3 þ dðbV BS þ ð1  bÞV M Þ  dV SO 



ðC:14Þ

Introducing Eqs. C.7,C.8 in the sharing rule gives

S BS V SO



¼ ð1  cÞ½bp4 þ dðbV BS þ ð1  bÞV M Þ  dV BO 

þ ð1  qs ÞV SO   d½qs V BO þ ð1  qs ÞðbV M þ ð1  bÞV BS Þ

¼ V BO jqb ¼1;hs ¼0  V BO jqb ¼0 ¼ bp2 þ dðbV BS þ ð1  bÞV M Þ  dV BO



    d qb V SO þ 1  qb ðbV M þ ð1  bÞV BS Þ

¼ ð1  cÞ½bp1 þ d½qs ðbV BS þ ð1  bÞV M Þ

BSO V BO



c bp4 þ d qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO

ðC:11Þ

It is therefore straightforward to deduce the expression of p3 given by (16).

ðC:15Þ

and, after simplifications, is equal to U ðp1 ; p3 ; p4 Þ ¼ qð1  hÞbðp4  p3 Þ. We use this expression to re-express the systems for prices and value functions, (16) and (13) respectively, under Lemma 1 8 qð1  hÞbp3  qð1  hÞbp4  qð1  qÞdV SO  qð1  qÞdV BO þ > > h   i h i > > < 1  q2 b þ ð1  qÞ2 ð1  bÞ d V  ð1  qÞ2 b þ q2 ð1  bÞ dV ¼ u  j BS M V > > qð1  hÞbp2 þ qhbp4 þ ½1  ð1  qÞdV BO  dqbV BS  dqð1  bÞV M ¼ u  jB > > : qð1  hÞbp2  qhbp3 þ ½1  ð1  qÞdV SO  dqbV BS  dqð1  bÞV M ¼ u  jS ðC:16Þ

and 8 > > > > > > > > > > > <

b p d 1

 ½ð1  cÞ  qð1  2cÞV SO þ ½c þ ð1  2cÞqV BO

þð1  2cÞð1  b  qÞV BS þ ð1  2cÞðb  qÞV M ¼ 0 b p  cV SO þ ð1  cÞV BO  ð1  2cÞbV BS  ð1  2cÞð1  bÞV M ¼ 0 d 2

b p  ½ð1  cÞð1  qs Þ þ cV SO þ ð1  cÞqs V BO þ ½cb þ ð1  cÞð1  b  qs ÞV BS d 3 > > > > ½ð1  cÞðqs  bÞ  cð1  bÞV M ¼ 0 > > > > b > p  c qV þ ½ ð 1  c Þ þ cð1  qÞV BO þ ½cqb  ð1  cÞb  cð1  qÞð1  bÞV BS > SO 4 >d : þ½cqð1  bÞ  ð1  cÞð1  bÞ  cð1  qÞbV M ¼ 0 ðC:17Þ

b p  ð2  qÞV SO þ qV BO þ ð1  qÞV BS þ ð1  qÞV M ¼ 0 dc 3 b p  qV SO þ ð2  qÞV BO þ ðq  1ÞV BS þ ðq  1ÞV M ¼ 0 dc 4

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b b p þ p ¼ 2ðV SO  V BO Þ dc 3 dc 4 b p ¼ ðV SO  V BO Þ dc 1 where RHS terms are constant and LHS terms are endogenous variables (prices and value functions). We transform similarly the value function (12) as

8 bp1 þ dð1  qÞV SO  dqV BO þ d½qb  ð1  qÞð1  bÞV BS þ d½qð1  bÞ  ð1  qÞbV M > 0 > > > > > bp1  dqV SO þ dð1  qÞV BO þ d½qb  ð1  qÞð1  bÞV BS þ d½qð1  bÞ  ð1  qÞbV M > 0 > > > > > bp2  dV BO þ dbV BS þ dð1  bÞV M > 0 > > > < bp  dV þ dbV þ dð1  bÞV > 0 SO BS M 2 > bp3  dV SO þ dbV BS þ dð1  bÞV M > 0 > > > > > bp3 þ dð1  qÞV SO  dqV BO þ d½qb  ð1  qÞð1  bÞV BS þ d½qð1  bÞ  ð1  qÞbV M > 0 > > > > > bp4  dV BO þ dbV BS þ dð1  bÞV M > 0 > > : bp4  dqV SO þ dð1  qÞV BO þ d½qb  ð1  qÞð1  bÞV BS þ d½qð1  bÞ  ð1  qÞbV M > 0 ðC:22Þ

dbV BS þ ½1  ð1  bÞdV M ¼ u

ðC:18Þ

C.4. The equilibrium with a deviating household

fp1 ; p2 ; p3 ; p4 ; V SO ; V BO ; V BS ; V M g

ðC:19Þ

C.4.1. The deviating household chooses to buy first If a household deviates from the simultaneous strategy and enters the market as a buyer first, two additional types of match are feasible on the market with two additional associated prices. After the household succeeds in buying its house, it becomes a seller only similar to other households with two dwellings. The two matches are defined in the Table C.4.1. The value function of the deviating household choosing to buy first is

Let us give the explicit formulae of A; b; C and d in Proposition 2. The matrix A is split into two matrices h i A ¼ Ap84 AV84 with

h i A ¼ Ap84 AV84 where

3 0 0 0 0 60 0 qð1  hÞb qð1  hÞb 7 7 6 7 6 7 6 0 qbð1  hÞ 0 qbh 7 6 7 6 0 qbð1  hÞ qbh 0 7 6 7 Ap ¼ 6 7 6b 0 0 0 7 6d 7 6 b 7 60 0 0 d 7 6 7 60 b 0 0 5 4 d 2

0

0

0

BF

BF

V BF ¼ u  jV  qb hs bpBS  qb ð1  hs ÞbpSO       þ d qb V SO þ 1  qb ð1  bÞV BF þ 1  qb bV M ðC:23Þ

ðC:20Þ price pBF BS price pBF BS is

The The process

the solution of the following bargaining

b d

3 0 0 db 1  ð1  bÞd   7 6 6 qð1  qÞd 1  q2 b þ ð1  qÞ2 ð1  bÞ d qð1  qÞd dq2 ð1  bÞ  dð1  qÞ2 b 7 7 6 7 6 7 6 ð Þd dqb dq ð 1  b Þ 0 1  1  q 7 6 7 6 ½1  ð1  qÞd 0 dqb dqð1  bÞ 7 6 7 6 AV ¼ 6 ½ð1  cÞ  qð1  2cÞ 7 c þ ð 1  2 c Þq ð 1  2 c Þ ð 1  q  b Þ ð 1  2 c Þ ð b  q Þ 7 6 7 6 7 6 c 1c ð1  2cÞb ð1  2cÞð1  bÞ 7 6 s s 7 6 ½ð1  cÞð1  qÞ þ c ð 1  c Þq c b þ ð 1  c Þ ð 1  b  q Þ  ð 1  c Þ ð q  b Þ þ c ð 1  b Þ 6



7 7 6 c qb  ð 1  c Þb c q ð 1  b Þ  ð 1  c Þ ð 1  b Þ 5 4 ð1  cÞ þ cð1  qÞ cq cð1  qÞð1  bÞ cð1  qÞb 2

2

u

3

6 u  jV 7 7 6 7 6 6 u  jB 7 7 6 6uj 7 S 7 6 b¼6 7 6 0 7 7 6 6 0 7 7 6 7 6 4 0 5 0

pBF BS : arg max

n

o c BBS V BF ½S BF V BS 1c

ðC:21Þ

ðC:24Þ

where the increase in value for the seller is

S BF V BS

¼ V BS jqs ¼1  V BS jqs ¼0    BF ¼ bpBS þ d qb ðbV BS þ ð1  bÞV M Þ þ 1  qb V BO    qb V SO  1  qb ðbV M þ ð1  bÞV BS Þ ðC:25Þ BF qs bpBS

using Eq. (13) where replaces U ðp1 ; p3 ; p4 Þ. The increase in value for the buyer is C.3.2. The inequality constraints The condition (4) of the equilibrium definition states that all surpluses defined by the Eqs. (C.4,C.5)-(((C.7)–(C.10), (C.12), (C.13)) are positive

  BBS V BF ¼ V BF qb ¼1;hs ¼1  V BF qb ¼0   BF ¼ bpBS þ d V SO  ð1  bÞV BF  bV M

ðC:26Þ

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T.-P. Maury, F. Tripier / Journal of Housing Economics 26 (2014) 55–80 Table C.4.1 Type of match with a deviating household. Type of match

The buyer

The seller

The price

1

The unique buyer first

A buyer and seller

pBF BS

2

The unique buyer first

A seller only

pBF SO

using Eq. (C.23). Introducing the inputs (C.25) (C.26) in the sharing rule ð1  cÞBBS V BF ¼ cS BF V BS gives

V BF

  BF bpBS ¼ cdqb ðbV BS þ ð1  bÞV M Þ  1  qb dcV BO   þcdqb V SO þ 1  qb cdðbV M þ ð1  bÞV BS Þ

    ¼ u  jV þ qb c b  1  qb hs dV BS     þqb hs 1  qb dcV BO þ qb hs c 1  qb dV SO     þ qb c 1  2b  hs 1  qb þ b dV M   þð1  bÞ 1  qb c dV BF

þdð1  cÞV SO  dð1  cÞð1  bÞV BF  dð1  cÞbV M ðC:27Þ The price pBF SO The price pBF SO is the solution of the following bargaining process

pBF SO : arg max

n

o c BSO V BF ½S BF V SO 1c

ðC:28Þ

where the increase in value for the seller is

S BF V SO ¼ V SO jqs ¼1  V SO jqs ¼0 BF

¼ bpSO þ d½bV BS þ ð1  bÞV M  V SO  s

BF bpSO

h

b



ðC:29Þ  i bp2 .

b

using Eq. (13) where q replaces h bp3  1  h The increase in value for the buyer is

  BBS V BF ¼ V BF qb ¼1;hs ¼0  V BF qb ¼0   BF ¼ bpSO þ d V SO  ð1  bÞV BF  bV M

ðC:30Þ

using Eq. (C.23). Introducing the inputs (C.29), (C.30) in the sharing rule ð1  cÞBSO V BF ¼ cS BF V SO gives

  BF bpSO ¼ dð1  cÞ V SO  ð1  bÞV BF  bV M  dc½bV BS þ ð1  bÞV M  V SO 

ðC:31Þ

The value functionV BF To get the value of V BF , we first simplify the expressions of prices (C.27)-(C.31) as follows BF

bpBS

   ¼ c 1  qb ð1  bÞ  qb b dV BS    þ 1  qb cb  cqb ð1  bÞ  ð1  cÞb dV M      1  qb dcV BO þ cqb þ ð1  cÞ dV SO

ðC:35Þ since      qb hs c 1  qb ð1  bÞ  qb b  qb ð1  hs Þcb d        ¼ qb c b  1  qb hs d  qb hs cqb þ ð1  cÞ þ qb ð1  hs Þ  qb d       ¼ qb hs c 1  qb d  qb hs 1  qb cb  cqb ð1  bÞ    ð1  cÞb  1  qb b þ qb ð1  hs Þ½ð1  cÞb þ cð1  bÞ d  b      ¼ q c 1  2b  hs 1  qb þ b qb hs ð1  cÞð1  bÞ   þ qb ð1  hs Þð1  cÞð1  bÞ þ 1  qb ð1  bÞ d   ¼ ð 1  b Þ 1  qb c d V BF can be reformulated as



   1  ð1  bÞ 1  qb c d V BF     ¼ u  jV þ qb c b  1  qb hs dV BS  b    þ q c 1  2b  hs 1  qb þ b dV M     þ qb hs 1  qb dcV BO þ qb hs c 1  qb dV SO

ðC:36Þ

C.4.2. The deviating household chooses to sell first If a household deviates from the simultaneous strategy and enters the market as a seller first, two additional types of match are feasible on the market with two additional associated prices. After the household succeeds in selling its house, it becomes a buyer only similar to other households with no dwelling. The two matches are defined in the Table C.4.2. The value function of the deviating household choosing to sell first is

  SF SF V SF ¼ u  jV þ qs hb bpBS þ qs 1  hb bpBO  s  þ d q V BO þ ð1  qs Þð1  bÞV SF þ ð1  qs ÞbV M

ð1  cÞð1  bÞdV BF

ðC:37Þ

ðC:32Þ C.4.3. The price pSF BS The price pSF BS is

BF

bpSO ¼ dV SO  dð1  cÞð1  bÞV BF  ½ð1  cÞb þ cð1  bÞdV M  dcbV BS

ðC:33Þ

and then introduce these expressions into (C.23) V BF

    ¼ u  jV  qb hs c 1  qb ð1  bÞ  qb b  qb ð1  hs Þcb dV BS    b s b  s b s b b þq h 1  q dcV BO  q h cq þ ð1  cÞ þ q ð1  h Þ  qb dV SO ( b s b    ) b q h c q ð 1  bÞ  1  q c b þ bð 1  c Þ þ dV M   þ 1  qb b þ qb ð1  hs Þ½ð1  cÞb þ cð1  bÞ  b s   þ q h ð1  cÞð1  bÞ þ qb ð1  hs Þð1  cÞð1  bÞ þ 1  qb ð1  bÞ dV BF

ðC:34Þ and finally

the solution of the following bargaining

process

n  o c  1c pSF BS : arg max ½B BF V BS  S BS V SF

ðC:38Þ

Table C.4.2 Type of match with a deviating household. Type of match

The buyer

The seller

The price

1

A buyer seller

The unique seller first

pSF BS

2

A buyer only

The unique seller first

pSF BO

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T.-P. Maury, F. Tripier / Journal of Housing Economics 26 (2014) 55–80

where the increase in value for the buyer is

or

BBF V BS ¼ V BS jqb ¼1  V BS jqb ¼0

n     V SF ¼ u  jV þ qs hb ð1  cÞqs b  ð1  cÞ 1  qb ð1  bÞ

SF

¼ bpBS þ d½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO    ðC:39Þ  qs V BO  1  qb ðbV M þ ð1  bÞV BS Þ

  o n þ 1  hb bð1  cÞ dV BS þ qs hb ½ð1  cÞqs ð1  bÞ       ð1  cÞ 1  qb b þ cb þ qs 1  hb ½ð1  bÞð1  cÞ

SF

using Eq. (13) where qb bpBS replaces U ðp1 ; p3 ; p4 Þ. The increase in value for the seller is

S BS V SF



¼

V SF qs ¼1;hb ¼1

¼

SF bpBS

n   þ bc þ ð1  qs ÞbgdV M þ qs hb cð1  bÞ þ qs 1  hb ð1  bÞc



V BF qs ¼0

   þ d V BO  ð1  bÞV SF  bV M

n o þ ð1  qs Þð1  bÞgdV SF þ qs hb dð1  cÞð1  qs Þ V SO

ðC:40Þ

n   o  qs hb ½ð1  cÞqs þ c þ qs 1  hb  qs dV BO

using Eq. (C.37). Introducing the inputs (C.25) (C.26) in the sharing rule ð1  cÞBBF V BS ¼ cS BS V SF gives or

SF

bpBS ¼ dð1  cÞ½qs ðbV BS þ ð1  bÞV M Þ þ ð1  qs ÞV SO     qs V BO  1  qb ðbV M þ ð1  bÞV BS Þ    cd V BO  ð1  bÞV SF  bV M

V SF ¼ u  jV þ qs

ðC:41Þ

ðC:42Þ

where the increase in value for the buyer is

n   o þ qs hb cð1  bÞ þ ð1  qs Þð1  bÞ þ qs 1  hb ð1  bÞc dV SF

SF

ðC:43Þ

BF

using Eq. (13) where qb bpBS replaces ½hs bp4  ð1  hs Þbp2 . The increase in value for the seller is

S BO V SF

  ¼ V SF qs ¼1;hb ¼0  V BF qs ¼0   SF ¼ bpBO þ d V BO  ð1  bÞV SF  bV M

ðC:44Þ

using Eq. (C.37). Introducing the inputs (C.43), (C.44) in the sharing rule ð1  cÞBBF V BO ¼ cS BO V SF gives SF

bpBO ¼ ð1  cÞd½bV BS þ ð1  bÞV M  V BO     cd V BO  ð1  bÞV SF  bV M

ðC:45Þ

The value functionV SF To get the value of V BF , we first simplify the expressions of prices (C.41)-(C.45) as follows     BF bpBS ¼ ð1  cÞqs b  ð1  cÞ 1  qb ð1  bÞ dV BS     þ ð1  cÞqs ð1  bÞ  ð1  cÞ 1  qb b þ cb dV M þdð1  cÞð1  qs ÞV SO  ½ð1  cÞqs þ cdV BO þ cdð1  bÞV SF ðC:46Þ SF

bpBO ¼ bð1  cÞdV BS þ ½ð1  bÞð1  cÞ þ bcdV M  dV BO þ ð1  bÞcdV SF

n o þð1  qs Þqs hb dð1  cÞV SO þ qs hb  qs hb ½qs ð1  cÞ þ c dV BO

or h  i V SF ¼ u  jV þ qs ð1  cÞ b  hb 1  qb dV BS

BBF V BO ¼ V BO jqb ¼1  V BO jqb ¼0 ¼ bpBO þ d½bV BS þ ð1  bÞV M  V BO 

  o 1  hb bð1  cÞ  hb ð1  cÞ 1  qb  b dV BS

n h   i o   þ qs ð1  cÞ hb qs ð1  bÞ  hb 1  qb b þ 1  hb ð1  bÞ  b þ b dV M

C.4.4. The price pSF BO The price pSF BO is the solution of the following bargaining process

n  o c  1c pSF BO : arg max ½B BF V BO  S BO V SF

n

ðC:47Þ

and then introduce these expressions into (C.37)      V SF ¼ u  jV þ qs hb ð1  cÞqs b  ð1  cÞ 1  qb ð1  bÞ dV BS      s b s b þ q h ð1  cÞq ð1  bÞ  ð1  cÞ 1  q b þ cb dV M   þ qs hb dð1  cÞð1  qs ÞV SO  ½ð1  cÞqs þ cdV BO þ cdð1  bÞV SF    b þ qs 1  h bð1  cÞdV BS þ ½ð1  bÞð1  cÞ þ bcdV M  dV BO þ ð1  bÞcdV SF  s  þ d q V BO þ ð1  qs Þð1  bÞV SF þ ð1  qs ÞbV M ðC:48Þ

n o h  i þ qs ð1  cÞ 1  2b  hb ð1  qs Þ  bhb qs  qb þ b dV M þð1  bÞ½1  qs ð1  cÞdV SF n o þð1  qs Þqs hb dð1  cÞV SO þ qs hb  qs hb ½qs ð1  cÞ þ c dV BO

And finally, ½1  ð1  bÞ½1  qs ð1  cÞV SF h  i ¼ u  jV þ qs ð1  cÞ b  hb 1  qb dV BS n o h  i þ qs ð1  cÞ 1  2b  hb ð1  qs Þ  bhb qs  qb þ b dV M n o þð1  qs Þqs hb dð1  cÞV SO þ qs hb  qs hb ½qs ð1  cÞ þ c dV BO ðC:49Þ

The existence condition  The existence condition max V BF ; V SF < V BS implies, firstly, V BS  V BF > 0 where using (C.36)

u  jV 1  ð1  bÞð1  qb cÞd     qb c b  1  qb hs dV BS  1  ð1  bÞð1  qb cÞd    qb c 1  2b  hs 1  qb þ b dV M  1  ð1  bÞð1  qb cÞd   s qb h 1  qb c dV BO  1  ð1  bÞð1  qb cÞd   qb hs c 1  qb dV SO  1  ð1  bÞð1  qb cÞd

V BS  V BF ¼ V BS 

ðC:50Þ

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T.-P. Maury, F. Tripier / Journal of Housing Economics 26 (2014) 55–80

2

therefore qb hs cð1qb Þ qb hs ð1qb Þc  1ð1bÞ 1qb c d dV SO þ  1ð1bÞ 1qb c d dV BO ð Þ ð Þ   qb c½bð1qb Þhs  qb c½12bhs ð1qb Þþb þ 1  1ð1bÞ 1qb c d d V BS  1ð1bÞ 1qb c d dV M ð Þ ð Þ ujV 1ð1bÞð1qb cÞd

>

ðC:51Þ The existence condition implies, secondly, V BS  V SF > 0, where using (C.49) jV V BS  V SF ¼ V BS  1ð1bu  Þ½1qs ð1cÞ qs ð1cÞ½12bhb ð1qs Þbhb ðqs qb Þþb



1ð1bÞ½1qs ð1cÞ ð1qs Þqs hb dð1cÞ

 1ð1bÞ½1qs ð1cÞ V SO 

qs ð1cÞ½bhb ð1qb Þ 1ð1bÞ½1qs ð1cÞ

dV BS

dV M

qs hb qs hb ½qs ð1cÞþc 1ð1bÞ½1qs ð1cÞ

therefore s b

s b

s b

s

Þq h dð1cÞ h ½q ð1cÞþc  1ð1q V  q1h ðq dV BO ð1bÞ½1qs ð1cÞ SO 1bÞ½1qs ð1cÞ   b s b q ð1cÞ½bh ð1q Þ þ 1  1ð1bÞ½1qs ð1cÞ d V BS

 >

qs ð1cÞ½12bhb ð1qs Þbhb ðqs qb Þþb 1ð1bÞ½1qs ð1cÞ

ðC:53Þ

dV M

The existence condition requires to satisfy the 10 inequality constraints defined by (C.22)-(C.51)-(C.53). These constraints are reexpressed as follows in Proposition 2: C g > d where g ¼ fp1 ; p2 ; p3 ; p4 ; V SO ; V BO ; V BS ; V M g and for h i the matrices C ¼ C p104 C V104 and d (under Lemma 1)

6 6 6 6 6 6 6 6 6 6 6 6 p C ¼6 6 6 6 6 6 6 6 6 6 6 6 4

b b 0 0 0 0 0 0 0 0

2

0

0

0

3

0

0 0 0 0 ujV ½1dð1bÞð1qcÞ

ðujV Þ 1dð1bÞð1qð1cÞÞ

This appendix provides the detailed calculus associated with the Proposition 3. It is straightforward to deduce the equality of the price p1 with the price p2 for symmetric bargaining programs. Setting c ¼ 1=2 in Eqs. (16) gives

ðC:54Þ

To get an expression of these prices as a function of structural parameters and not value functions, we compute the difference between V SO and V BO using (13) and obtain

½1  ð1  qÞdðV SO  V BO Þ ¼ jB  jS þ qb½hp3 þ 2ð1  hÞp2 þ hp4  ðC:55Þ For the expression of p2 given by (C.54), we have the following expression for the difference in value functions

¼

jB  jS þ qbhðp3 þ p4 Þ 1  ð1  qÞd  qð1  hÞd jB  jS þ qbhðp3 þ p4 Þ

ðC:56Þ

1  ð1  qhÞd

and we deduce the following expression for equilibrium prices p1 and p2

p1 ¼ p2 ¼

dc b





jB  jS þ qbhðp3 þ p4 Þ 1  ð1  qhÞd

ðC:57Þ

The next step consists in the determination of the equilibrium value of ðp3 þ p4 Þ, which is obtained from Eqs. (16)

d p3 þ p4 ¼ ðV SO  V BO Þ ¼ 2  p1 b

0

dð1  qÞ dq 6 6 dq d 1  qÞ ð 6 6 6 0 d 6 6 6 6 d 0 6 6 6 dð1  qÞ dq 6 6 CV ¼ 6 d 0 6 6 6 6 0 d 6 6 6 dq dð1  qÞ 6 6 6 ð1qÞdc 6  dqhcð1qÞ  ½1dqhð1b Þð1qcÞ 6 ½1dð1bÞð1qcÞ 4 Þqhdð1cÞ dqhð1cÞð1qÞ  1dðð1q  1bÞð1qð1cÞÞ 1dð1bÞð1qð1cÞÞ

0

V SO  V BO ¼

7 0 0 0 7 7 7 b 0 0 7 7 7 b 0 0 7 7 7 0 b 0 7 7 7 7 0 b 0 7 7 7 0 0 b 7 7 7 0 0 b 7 7 7 0 0 0 7 5 0

0

7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5

b b p ¼ p ¼ V SO  V BO dc 1 dc 2

ujV 1ð1bÞ½1qs ð1cÞ

2

0

C.5. The equilibrium price dispersion

dV BO ðC:52Þ

s

6 6 6 6 6 6 6 6 6 6 d¼6 6 6 6 6 6 6 6 6 6 4

3

0

d½qb  ð1  qÞð1  bÞ d½qð1  bÞ  ð1  qÞb

3

7 d½qb  ð1  qÞð1  bÞ d½qð1  bÞ  ð1  qÞb 7 7 7 7 db dð1  bÞ 7 7 7 7 db dð1  bÞ 7 7 d½qb  ð1  qÞð1  bÞ d½qð1  bÞ  ð1  qÞb 7 7 7 7 db dð1  bÞ 7 7 7 7 db dð1  bÞ 7 7 d½qb  ð1  qÞð1  bÞ d½qð1  bÞ  ð1  qÞb 7 7 7 7 qdcfbð1qÞhg ð12bð1qÞhÞþbg 7 1  ½1d  dfq½c1d ð1bÞð1qcÞ ð1bÞð1qcÞ 7 5 dqð1cÞ½bhð1qÞ dfqð1cÞ½12bð1qÞhþbg 1  1d  ð1bÞð1qð1cÞÞ 1dð1bÞð1qð1cÞÞ

ðC:58Þ

We then conclude that the price p1 is mean of prices p3 and p4 The exact value of p1 reported in the Proposition 3 is obtained by plugging (C.58) in (C.57). Under Lemma 1, we deduce from (C.17) the values of p3 and p4 reported in the Proposition 3. References Albrecht, J., Anderson, A., Smith, E., Vroman, S., 2007. Opportunistic matching in the housing market. Int. Econ. Rev. 48 (2), 641–664. Binmore, K., Rubinstein, A., Wolinsky, A., 1986. The Nash bargaining solution in economic modelling. RAND J. Econ., 176–188. Burdett, K., Judd, K., 1983. Equilibrium price dispersion. Econometrica 51 (4), 955–969.

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