Consider the following economy. Two dates: t = 0, 1. One risk-free asset with a certain rate of return rf > 0. One risky asset: - At t = 1, the asset yields a risky cash flow per share. Assume that is normally distributed with mean 5 and variance of. - Price at t = 0: P (endogenous). - Total number of shares outstanding (i.e., supply) = S > 0. N (identical) investors: Price takers. - Initial wealth: Yo. Each investor maximizes the expected utility of wealth at t = 1, . - CARA utility function: U(Y) = - exp(-vY), v > 0. Price takers. - Initial wealth: Yo. Each investor maximizes the expected utility of wealth at t = 1, . - CARA utility function: U(Y) = - exp(-vY), v > 0. 1 max 5X + (1 + r) (Yo - PX) - vXo. From the FOC, we get the investor's demand function X(P). (1+r) P-voX = 0 X(P) = Imposing the market clearing condition, we get P. vo?! N NX (P) = S N 3-(1+r)P vo? S P P= 1+rf I.e., the equilibrium price is the expected cash flow minus the risk premium, discounted at the risk-free rate! (See the top of p.6, lecture note 1.) 3-(1+rf)P vo? F Modify the model of Exercise 2 of Problem Set 3 as follows. Proportion a [0, 1] of the investors are type A, and the rest are type B. To purchase the risky asset, each type-A investor needs to pay a transaction cost (such as a brokerage fee) of > 0 dollars per share. That is, if she buys a shares of the asset, she needs to spend or dollars in addition to the purchase price. Type-B investors do not need to pay the transaction cost. a) Let X, denote the number of shares of the risky asset that type i {A, B} buys. Char- acterize the competitive equilibrium (determine XA, XB and P). b) Show that OXB/00 > 0. Type-B investors do not pay the transaction cost, and therefore do not care about the level of p at all; then why would they increase investment XB when someone else's costo increases? Explain. Consider the following economy. Two dates: t = 0, 1. One risk-free asset with a certain rate of return rf > 0. One risky asset: - At t = 1, the asset yields a risky cash flow per share. Assume that is normally distributed with mean 5 and variance of. - Price at t = 0: P (endogenous). - Total number of shares outstanding (i.e., supply) = S > 0. N (identical) investors: Price takers. - Initial wealth: Yo. Each investor maximizes the expected utility of wealth at t = 1, . - CARA utility function: U(Y) = - exp(-vY), v > 0. Price takers. - Initial wealth: Yo. Each investor maximizes the expected utility of wealth at t = 1, . - CARA utility function: U(Y) = - exp(-vY), v > 0. 1 max 5X + (1 + r) (Yo - PX) - vXo. From the FOC, we get the investor's demand function X(P). (1+r) P-voX = 0 X(P) = Imposing the market clearing condition, we get P. vo?! N NX (P) = S N 3-(1+r)P vo? S P P= 1+rf I.e., the equilibrium price is the expected cash flow minus the risk premium, discounted at the risk-free rate! (See the top of p.6, lecture note 1.) 3-(1+rf)P vo? F Modify the model of Exercise 2 of Problem Set 3 as follows. Proportion a [0, 1] of the investors are type A, and the rest are type B. To purchase the risky asset, each type-A investor needs to pay a transaction cost (such as a brokerage fee) of > 0 dollars per share. That is, if she buys a shares of the asset, she needs to spend or dollars in addition to the purchase price. Type-B investors do not need to pay the transaction cost. a) Let X, denote the number of shares of the risky asset that type i {A, B} buys. Char- acterize the competitive equilibrium (determine XA, XB and P). b) Show that OXB/00 > 0. Type-B investors do not pay the transaction cost, and therefore do not care about the level of p at all; then why would they increase investment XB when someone else's costo increases? Explain
