There are four questions in the following pictures. Could you help me to find the solutions about them.

Q1 John and Peter are two representative consumers/investors who maximize the utility of consumption. John's utility of consumption is characterized as

ln(x) + 2ln(y)

while Peter puts more weight on the current consumption level and has a utility function of 2ln(x) + ln(y).

John has a wealth of ($10, $20) thousand, while Peter has a wealth of ($20, $15) thousand now and next year, respectively.

(a) What are the optimal consumption plans forJohn and Peter,respectively,if the interest rate is 5% per annum? (b) If John and Peter are the only investors/consumers, what is the equilibrium interest rate?

(c) Further to part (b), how much do they borrow or lend to each other?

Q2 Both Bob and Ben have a current consumption credit of $1500. However, their future consumption credits will differ depending on the economic outcome which consists of two states, "good" and "bad", with probability 0.6 and 0.4, respectively. Bob and Ben will have $2000 and $1000 if the good state occurs, and they will have $1000 and $2000 if the bad state occurs instead, respectively. Bob prefers a smooth consumption with utility function of (x-current consumption, y-future consumption) UBob(x, y) = ln(x) + ln( y), while Ben is a little impatient with a utility function of

UBen(x, y) = 1.25ln(x) + ln( y).

There are two nancial securities, NORISK and PUT, available in the market for them to make tradeoff decisions with their current and future consumption credits.

The payoffs for the two securities are given below:

1 0

P1 P2

1 1

(a) Whataretheequilibrium prices, p1 and p2, ofNORISKand PUT,respectively?

(b) Calculate the current and future optimal consumptions for each of the two investors?

(c) What is the implied risk free rate?

(d) How much do they have to hold in each of the two securities, NORISK and PUT, to facilitate their future consumptions?

Q3:Suppose the future economy is described by two states. There are three securities traded in the market. Is there any arbitrage opportunity?

Security A: Security B: Security C:

1 2 2

1.2 1.5 1

1 1 0

Q4 You are given the following information:

Security State one State two Security price

A $12 $20 15

B $24 $10 16

(a) What are the prices of the pure securities?

(b) What is the initial price of a third security C which will pay $6 in state one and $10 in state two?

(c) What is the risk free rate?

(d) What are the risk neutral probabilities? Explain how you would price the security C in part (b) under the risk neutral probabilities.