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 + P .

1. a) What is the price of the stock as a function of parameters N, M and ?

2. b) Suppose N = M = 50 and = 20, are the rational traders long or short and by how

   much?

3. c) Suppose N = M = 50, = 20, and there are 100 shares outstanding, again. Also, suppose the 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.