Ruling out the presence of noise traders.

Question 1
Classical finance does not rule out the presence of noise traders. Yet it argues that the
Efficient Markets Hypothesis continues to hold. Explain. That is, how can there be noise traders in markets but markets still produce asset prices that incorporate and reflect all available information?
Question 2
Consider the following model of a stock market. There are two groups of traders: ratio-
nal traders and optimists. There are N rational traders and each has a demand for the stock of 100 − P. Suppose that there are 100 shares of the stock outstanding. There are M
optimists and each has a demand of 100 + δ − P.
a) What is the price of the stock as a function of parameters N, M, and δ?
b) Suppose N = M = 50 and δ = 20, are the rational traders long or short and by how much?
c) Suppose N = M = 50, δ = 20, and there are 100 shares outstanding, again. Also,
suppose rational traders cannot short at all for institutional reasons. For example,
they might work for mutual funds: most mutual funds do not allow shorting. Calculate the equilibrium price and the positions of the rational traders and the optimists in this scenario.
Question 3
Consider a market where there are N rational traders. All of these traders have CARA
preferences with risk aversion parameter γ = .5. They are considering a stock that will pay a terminal dividend in the next period. The expected payoff of the dividend is $100 per share with a standard deviation of $10. Assume that the discount rate is zero. That is,
don’t worry about discounting future payoffs.
a) If there are 10 shares of the stock available, what is the price of the stock as a function of N?
b) Does the price increase or decrease as N increases? Explain the intuition of this result.
c) If N = 10, what is the price of the stock? How does the price change if γ increases to
.6? Explain the intuition for why the price moves this direction when γ increases.