Let us consider a simple stock market with the following characteristics. A large amount of stock is traded. In the market, there are three typical groups of traders with different strategies: fundamentalists, chartists, and noise traders. Traders can invest either in money or in stock. Since the model is designed to describe stock price movements over short periods, such as one day, the dividend from stock and the interest rate for the risk-free asset will be omitted for simplicity. Traders are myopic and bent on maximizing utility. Their utility depends on the price change they expect, and on their excess demand for stock rather than simply their demand. Their excess demand is derived from utility maximization.
Let Ytf be the amount of money that a fundamentalist holds at a time t and Xtf be the number of shares purchased by a fundamentalist at time t. Let pt be the price per share at time t. The fundamentalist’s budget constrain is given by
Ytf + ptXtf = Yt-1f + ptXt-1f —– (1)
ytf + ptxtf = 0 —– (2)
ytf = Ytf – Yt-1f
denotes the fundamentalist’s excess demand for money, and
xtf = Xtf – Xt-1f
his excess demand for stock. Suppose that the fundamentalist’s preferences are represented by the utility function,
u = α(ytf + pt+1fxtf + βxtf – (1 + βxtf) log (1 + βxtf) —– (3)
where pt+1f denotes the fundamentalist’s expectation in period t about the price in the following period t + 1. The parameters α and β are assumed to be positive. Inserting (2) into (3) the fundamentalist’s utility maximization problem becomes:
maxxtf u = α(pt+1f – pt)xtf βxtf – (1 + βxtf) log (1 + βxtf) —– (4)
The utility function u satisfies the standard properties: u′ (|xtf|) > 0, u′′(|xtf|) < 0 ∀ |xf|t ≤ |xf*|, where |xf*| denotes the absolute value of xf producing a maximum utility. Thus, the utility function is strictly concave. It depends on the price change expected by fundamentalists (pt+1f – pt) as well as fundamentalist’s excess demand for stock xtf. The first part α(pt+1f – pt)xtf implies that a rise in the expected price change increases his utility. The remaining part expresses his attitude toward risk. Even if the expected price change is positive, he does not want to invest his total wealth in the stock, and vice versa. In this sense, fundamentalists are risk averse. β is the parameter that sets the lower limitation on excess demand. All excess demand for stock derived from the utility maximization is limited to -1/β. When the expected price change (pt+1f – pt) is positive, the maximum value of the utility function is also positive. This means that fundamentalists try to buy stock. By analogy, when the expected price change (pt+1f – pt) is negative, the maximum value of the utility function is negative, which means that they try to sell. The utility maximization problem (4) is solved for the fundamentalist’s excess demand,
xtf = 1/β(exp(α(pt+1f – pt)/β) – 1) —– (5)
Excess demand increases as the expected price change (pt+1f – pt) increases. It should be noticed that the optimal value of excess supply is limited to -1/β, while the optimal value of excess demand is not restricted. Since there is little loss of generality in fixing the parameter β at unity, below, we will assume β to be constant and equal to 1. Then let us think of the fundamentalist’s expectation formation. We assume that he form his price expectation according to a simple adaptive scheme:
pt+1f = pt + ν(p* – pt) —– (6)
We see from Equation (6) that fundamentalists believe that the price moves towards the fundamental price p* by factor ν. To sum up fundamentalists’ behavior: if the price pt is below their expected price, they will try to buy stock, because they consider the stock to be undervalued. On the contrary, if the price is above the expected value, they will try to sell, because they consider the stock to be overvalued.