There are many ways to extend the simple model. From the theoretical perspective, the first extension is to include the possibility that that and, or at least, such that the market supply function will include the term. In this case, when the speculators (can still) sell their houses, it will change the slope of the supply curve, thus further lowering the housing price.
Second, c in this paper is assumed be a parameter, although it can change over time, which is the subjective reaction of a speculator to the expected price change,. Hence, another extension of the model is to assume that the speculators have bounded rationality or naive expectation such that =. That is, the expected increase or decrease in prices from period t to period t+1 should be the same as the price increase or decrease or percentage price increase or decrease2 (or acceleration or deceleration) from period t-1 to period t. Hence, = r, where r is the reaction coefficient to the price change. In this case, we can incorporate the past prices into the demand functions without the c.
Third, we can include the law of motion that describes how changes over time. One extension, for example, is based on the adaptive or naïve expectation of the speculators. The law of motion can be. It means that the number of the speculators from period t to period t+1 will increase if, and decrease if. The coefficient shows how fast the number of speculators will change in response to the price change. However, by including such a law of motion for the number of speculators, the model will become a system of non-linear differential equations, which may be too mathematical for general readers.
4.2 An Evolutionary Game Theory Perspective
One important assumption under the evolutionary game theory is that players have bounded rationality, but can learn to be more rational. From an evolutionary game theory perspective, one extension of the model is to assume that there are two types of agents or speculators, one with rational expectation (type R) and the other one with naïve expectation (type N). There is a fixed cost that the agents have to pay to be able to have rational expectation, either through learning to be experts or paying the experts in order to obtain that knowledge. The agents with naïve expectations, the extreme form of the adaptive expectation, do not have any fixed cost.
Both types of agents can learn from each other. When one type of agent does better than the other type, say, in terms of profits, the number of one type of agents with higher profits will increase, while the number of the other type of agents will decrease. If the numbers of both types of agents follow a law of motion or replicator dynamics, it is possible to show that, as discussed, for example, by Goeree and Hommes (2000), among others, when the market price is high enough, all the agents will have only the naïve expectation. This result is similar to the situation when there are only speculators with bounded rationality that remain in the market. However, by using the approach of evolutionary game theory, the mathematics involved will become much more complicated than those presented in this paper, but this is one possible extension of the model.
4.3. An Empirical Perspective and Test
In terms of an empirical approach, the variables are usually in the natural log form. The and discussed in this paper will represents the natural log of the data. The advantage of doing so is that the coefficients estimated using OLS can be the proxies for the elasticities of the variables. In a consistent manner with our model, empirically, the supply curve in the housing market is usually (more) inelastic in the short run, and (more) elastic in the long run. And it varies across different metropolitan areas in the U.S. and countries, since different cities have different regulations and rules.
The market demand function in this paper has an intercept (ma + xa+ xc). Empirically, this intercept is a function of many variables. Because a log-linear demand function is usually assumed, the intercept becomes a sum of many other log variables. These variables, however, also vary over time, and are quite capable of generating price changes in the market. One difficulty of empirical testing is to separate the effects of these variables from the part of the market price changes that have been driven by the aggregate changes of individual expectations. And these variables can actually explain the variations in the market prices pretty well with a high R square. It is thus not surprising to see that some authors can show that there is a bubble forming in some areas while others disagree, particularly before 2003 in the US or other countries.
Another issue related to empirical testing is to that we need to determine what the long-run trend is before we can determine how much the price has deviated or increased from the long-run trend. For example, if the market price suddenly increases in one period, does that change the trend? It is not difficult to detect the formation of a bubble after a bubble has burst. But it is more difficult to detect the formation of a bubble, particularly in the early stages before it bursts. Therefore, one important extension is to integrate the empirical and theoretical studies or at least narrow the knowledge gaps between them. From the empirical and policy perspective, it is important to identify the variables that can explain the bubble in a statistically significant way, for example, through econometric analysis. In addition to the variables mentioned in the econometric studies, more efforts are needed to identify the number of speculators and the proxies of the aggregate expectation in the market. If we can determine these “indicators” of bubbles, it will be possible to better estimate how or when market price will reach the peak during the bubble as well as when the bubble will burst.
4.4. Government Intervention
The financial crisis that we experienced in recent years manifests the importance of the intervention of the government, especially, the Fed, as the lender of last resort. More importantly, instead of just cleaning up the aftermath of a financial crisis, today the Fed is moving toward a more pro-active supervising role to detect a bubble, as well as developing the policy instruments to limit the development of a bubble without damaging the economy. Therefore, how to include the intervention of the Fed or the government into the theoretical model would be another research topic to explore in the future.