General Equilibrium for Housing Liquidity in a Competitive Environment

"We develop a theoretical model of illiquidity, in which illiquid assets are being traded by two agents -- buyers and sellers. Illiquidity is defined as the expected time it takes a seller to sell his asset at the optimal price. The theoretical model is developed in the context of transactions of residential housing properties that exemplify this type of assets. Unlike markets with perfectly liquid assets, in trading illiquid assets delaying the transaction creates a positive value to both sellers and buyers, that is induced by a non-zero probability that a better deal may come up. Then, because housing markets are decentralized, home buyers are willing to accept time-related search costs as the price of finding a better match. Sellerís waiting, on the other hand, may be rewarded with a visit of the buyer who assigns the highest value to his house and is willing to pay the offered price. In our model, buyers are modeled as agents that are heterogeneous in their observed match with a house that they visit. At the same time, by their nature houses are heterogeneous and thus offer different levels of utility to different prospective buyers. A household buys a house only if the observed match is at least as high as the reservation match. During the homeownership, however, the match may get lost (for example, due to change in family size or job relocation), in which case the owner immediately puts the house up for sale, thus assuming simultaneously a role of a seller and a buyer. The objective of this paper is two-fold. The first goal is to develop a theoretical framework for the effects of competition among buyers ñ a consequence of a multiple-buyer arrival ñ on (a) equilibrium sales prices for the properties and (b) their levels of liquidity, namely time on the market. This Nash equilibrium is simultaneously determined for the two outcomes of this competitive behavior. As an illustration, if a house that is put up by seller for sale is inspected by two prospective buyers, then the probability of the house getting sold during this period is higher than in the case with only one buyer. Thus, time on the market should decline with a higher number of buyers. At the same time, the seller knows that each buyer visits two different houses each period. As a result, a buyerís match for a house exceeding his reservation match does not guarantee his buying this house. This occurs because there exists a possibility that the match observed for the second house is higher than for the first house. This in turn reduces the probability that the seller sells her house to a prospective buyer in this period, and so the time on the market is expected to increase. We hypothesize that the reduction in the probability of sale due to the possibility of a buyerís better match with a different house would be outweighed by a better chance of having her own house sold due to the presence of an additional prospective buyer. In equilibrium, in the case with two buyers this results in time on the market being shorter. In turn, the equilibrium sale price is expected to be lower with two prospective buyers visiting each house in each period. Furthermore, if the buyer arrival process is sufficiently dense, a higher number of buyers visiting same house each period is expected to result in a lower equilibrium price and time on the market. In the limit, we expect to see a perfectly liquid market. We seek to determine whether the relation between the arrival frequency and the level of liquidity is linear. The second objective is to generalize the model by extending it to a case with two classes of buyers and two classes of houses. Namely, in our model, buyers can vary in their expected tenure and can be short-term or long-term buyers and the two types of houses are (1) lower-quality houses that do not provide a match sufficiently high to appeal to a prospective buyer, and (2) higher-quality houses that do. The two types of houses should be priced differently. Then, in each period, each prospective buyer visits two houses ñ one house of each type ñ and picks the one with the highest net gain. The trade-off to be considered here can be intuitively summarized as follows: Even if a buyer observes a higher match with the first house than with the second, provided that the price exceeds substantially the price of the second house, the net gain of the first house is reduced, thus encouraging the buyerís preference to switch from the first house with a higher match to the second house with a lower match but a lower price. In our model, short-term buyersí per-period probability of buying a house is more sensitive to changes in sales prices than the corresponding probability of long-term buyers. In our model we also expect to establish the result that, in the presence of short-term and longterm buyers, there is a higher equilibrium relative proportion of short-term buyers choosing the lower-quality houses and a higher equilibrium relative proportion of long-term buyers selecting higher-quality houses as their housing choice. This happens because long-term buyers are less sensitive to changes in sales prices and can amortize a higher price paid over a longer holding horizon. This project contributes to existing, although limited, body of theoretical literature on equilibrium in the housing markets. While applied to a specific area ñ housing markets ñ we believe that this model is among the first to establish a theoretical model of complex homeownership decisions by explicitly accounting for multiple agents and expected holding horizons. Extensions of this model include generalizations to decisions between buying and renting a house, as well as to a more general topic of pricing of illiquid assets.""