Question

Consider the following model of a stock market. There are two groups of traders: rational 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 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.

Answer 1

A) N rational traders and each has a demand for the stock of 100 − P

So, total demand of rational traders= N*(100 − P)

M optimists and each has a demand of 100 + δ − P

So, total demand of optimists = M*(100 + δ− P)

Total demand = N*(100 − P) + M*(100 + δ− P) which must equal 100 shares outstanding.

Therefore, N*(100 − P) + M*(100 + δ− P) = 100

P = 100 *(N+M-1)/(N+M) + M*δ/(N+M)

B) N = M = 50 and δ = 20

Solving for P,

P = 100 *(N+M-1)/(N+M) + M*δ/(N+M)

= 99 + 10

= 109

Each rational trader is short by (100 -P) i.e. 9 shares per trader (50*9 = 450 shares in total).

C) N = M = 50 and δ = 20

As rational traders cannot short their shares, demand is only from optimists.

M*(100 + δ− P) = 100

P = 118 and the optimists are long by (100 + δ− P) i.e. 2 shares per trader (50*2 = 100 shares in total).