Fundamentalists – Risky Asset – Chartists

Let us consider the market for a risky asset, composed of two groups of traders having different trading strategies: fundamentalists and chartists. Fundamentalists are assumed to have a reasonable knowledge of the fundamental value of the risky asset. The fundamentalist’s strategy can be described as follows: if the price p_t is below the fundamental value p^∗, then the fundamentalist tries to buy the risky asset because it is undervalued; if the price p_t is above the fundamental value p^∗, then he tries to sell the risky asset because it is overvalued. The excess demand for the risky asset is given by:

x^f_t = exp(α(p^∗ − p_t)) − 1, α > 0 —– (1)

where α is the parameter that denotes the strength of the non-linearity of the fundamentalist excess demand function (1). The fundamentalist’s excess demand function (1) is derived in a one-period utility optimizing framework. The technical details of the derivation of (1) within a utility maximizing framework are given in Kaizoji. We can see that Equation (1) has captured the distinctive features of the fundamentalist’s strategy. While fundamentalists calculate the fundamental value, chartists estimate a trend in the price change. Chartists can be assumed to form their expectation of the price of the risky asset according to the simple adaptive scheme:

p^e_t+1 = p^e_t + µ(p_t − p^e_t), 0 < µ ≥ 1 —– (2)

where p^e_t denotes the price at period t expected by chartists, and the parameter µ is the error correction coefficient. As above, the chartist’s excess demand function is given by:

x^c_t = exp(β(p^e_t+1 − p_t)) − 1, β > 0 —– (3)

where β is the parameter that denotes the strength of the non-linearity of the fundamentalist excess demand function (1). The chartist’s excess demand function (3) means that chartists try to buy the risky asset when they anticipate that the price will rise within the next period; inversely, that they try to sell the risky asset when they expect the risky asset price to fall within the next period. Let us now consider the adjustment process of the price in the market. We assume the existence of a market-maker who mediates the trading. If the excess demand in period t is positive (negative), the market maker raises (reduces) the price for the next period t+1. Let κ be the fraction of chartists in the total number of traders. Then the process of price adjustment can bewritten as

p_t+1 − p_t = θN[(1 − κ)x^f_t + κx^c_t] —– (4)

where θ denotes the speed of the adjustment of the price, and N the total number of traders…..

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