There are (m+x) buyers and n sellers in the housing market, with m > 0, n > 0, and. The m buyers are non-speculators, while x buyers are the speculators. There are four stages in the game. In the first stage, only mnon-speculators and n sellers enter the market. In the second stage, x speculators enter the market. In the third stage, the housing price is too high for the non-speculators to stay in the market. Only x speculators remain in the market. In the fourth stage, m speculators re-enter the market with x speculators in the market, with, i.e., if there are any speculators left. In addition, any buyer or seller in any period can exit the market by refusing to buy or sell the houses. Hence, the number of speculators, x, is not fixed, neither is m or n.
The demand function for a non-speculator in period t is a decreasing function of the market price (adjusted by inflation) in period t, i.e., = a – b, with a > 0 and b > 0. The demand function of a speculator in period t depends on the market price in period t as well as their reaction to the expected price increase from period t to period t+1, = a – b+, where is the reaction of an individual speculator i at time t to the expected future price change, or, similar to the perspective of Riddel (1999), the expectation of a real change in the housing price in the next and future periods. This also implies that = 0 for non-speculators in period t. Since the focus of this paper is on the herding behavior in the formation process of bubbles, for simplicity, we assume that = c > 0, for each speculator i in every period. In this case, c becomes an element in the trend of housing prices. It can also be interpreted as the mean of the expected price increase of speculators in the housing market. From the horizontal aggregation, the market demand curve in period t is = ma – mb+ xa – xb + xc, if x > 0, and = ma – mb if x = 0.
For the supply side, each seller of houses has the supply function = +, with,, and . The supply function in period t includes the previous period price because of the time lag to construct a new house. The assumption that h is greater than zero means that the producers are willing to produce or construct more houses available for sale in period t if the housing price in period t-1 is higher. The coefficient is usually negative because of the positive reservation price before the sellers are willing to sell the house. The coefficient is usually positive because it is the slope of the supply function. Alternatively, there are also two other perspectives for market supply functions. First, the coefficient of comes from the sellers that have rational expectation that can correctly anticipate the market price, while the coefficient of comes from the sellers that have adaptive or naïve expectation. Second, the coefficient of mainly comes from existing houses, while the coefficient of mainly comes from the newly constructed houses that are built based on the expected price, which is the simplyas a result of naïve expectation.
Generally speaking, the housing prices should increase more in the areas where housing supplies tend to be more inelastic, while the areas with more elastic housing supply should have fewer and smaller bubbles, and with smaller price increases, as discussed by Glaeser et al. (2008). For our purposes, the focus of this paper is on the speculation from the demand side. Hence, we focus on the situation when the housing supply function is most (or perfectly) inelastic. For simplicity, we also assume that,, and . Therefore, from the horizontal aggregation, the market supply function in period t is = n.
In each stage, we assume the market clears such that =. The buyers of houses in period t buy the houses that have been produced in the previously period, t – 1. The total number of houses will change only if n or h changes.
In the first stage when x = 0, market equilibrium implies that ma – mb= n, which can be simplified as. In the second stage when the speculators enter the market, market equilibrium =implies that.