Suppose there is a representative market maker with constant absolute risk aversion , and competition forces the

Suppose there is a representative market maker with constant absolute risk aversion α, and competition forces the bid and ask to the prices that make the market maker indifferent about trade. Suppose there is no information in market orders, which are of a unit size. Assume that the future value of a unit of the asset is normally distributed with mean μ and variance σ2.

(a) Let θ denote the number of shares owned by the market maker before trade. Compute the ask and bid prices. Show that, even though the bid and ask depend on θ, the bid-ask spread does not.

(b) Assume buys and sells are equally likely. Consider all possible sequences of three transactions. Compute the transaction price for each transaction in each sequence, and show that the transaction price changes are positively serially correlated.