Consider a sequential-trade market like in Glosten and Milgrom (1985). The asset valueVcan take three values,0,0.5and1. It

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Consider a sequential-trade market like in Glosten and Milgrom (1985). The asset valueVcan take three values,0,0.5and1. It takes value0.5with probability1pand the values0or1with equal probabilityp=2. If the value is0.5, in the market there are only noise traders. If, instead, the value is either0or1, then half of the traders are informed and half are noise. Noise traders buy or sell with equal probability (and never decide not to trade). Informed traders receive asymmetric binary signal with precisionq= 0.75(i.e.,Pr(st= 1|V= 1) = Pr(st= 0|V= 0) = 0.75) and maximize expected profits.

a. Suppose the probabilitypis arbitrarily close to0. What are the bid and ask prices at time1?

b. Suppose the probabilitypis arbitrarily close to0. Suppose the first two trades are buy orders. Suppose the third trader is informed and receives the signals3= 0. Will the trader buy, sell or decide not to trade? Explain your answer carefully.

c. Suppose the probabilitypis arbitrarily close to1. What are the bid and ask prices at time1?

d. Suppose the probabilitypis arbitrarily close to1. Suppose the first two trades are buy orders. Suppose the third trader is informed and receives the

signals3= 0. Will the trader buy, sell or decide not to trade? Explain your answer carefully.

e. Suppose the probabilitypis equal to2=3. Suppose in a sequence of actions, one is a no trade. What is the updated probability that the market maker attaches to the value being0.5?