Suppose there is a mutual fund and each consumer buys a share in it for her endowment at t = 0. The mutual fund maximises the wealth of its shareholders when choosing IF, the investment in the long term technology. At t = 1, the mutual fund pays dividend d to each of its shareholders. At t = 1, the shares can be traded at price p°. a. Set up the mutual fund's optimisation problem and derive and interpret the first order condition. What happens when R increases and why?
There are three dates t = 0, 1, 2 and a continuum of consumers with measure one, each endowed with one unit of wealth. A fraction a of consumers is impatient, i.e. they derive utility u (c1) from consuming only at t = 1, and 1 – a are patient and derive utility u (c2) from consuming only at t = 2. The utility function is u(c) = 1 – (1/c). At t = 0 consumers don't know their type, i.e. it is unknown whether the consumer wishes to consume at date t = 1 or t = 2. At t =1 all consumers observe their own type, but cannot observe others' types (i.e. type is private information). Consumers can store their wealth across periods, with one unit stored today yielding one unit tomorrow. Alternatively, wealth can be invested in a long term technology at date t 0, which yields 0 at t = 1, but R > 1 at t = 2.
1. Suppose there is a mutual fund and each consumer buys a share in it for her endowment at t = 0. The mutual fund maximises the wealth of its shareholders when choosing IF, the investment in the long term technology. At t = 1, the mutual fund pays dividend d to each of its shareholders. At t = 1, the shares can be traded at price p°. a. Set up the mutual fund's optimisation problem and derive and interpret the first order condition. What happens when R increases and why?
b. What is the optimal consumption profile cf', c and the optimal investment IF?
c. Calculate (i.e. derive an expression for) d and p".
2. Now suppose there is no financial intermediary to handle liquidity shocks. However, at t = 1 a market for bonds opens up and agents can trade their wealth at t = 1 for wealth at t = 2. Each bond pays 1 at t = 2 and its price is pM. Calculate the consumer's optimal investment decision IM at t = 0 , the price of the bond pM, and the optimal consumption in the two states cM, cM.
3. Compare the mutual fund and bond market allocations: are c and c bigger or smaller than c and c, respectively?
4. Efficiency: a. Which allocation do you think is (Pareto) dominant? Why? Compare the prices pM and p and give an intuitive explanation. b. What should be the price in the bond market that implements the allocation cf, c if the types of the consumers is public information (i.e. observable by everyone) at t = 1 and hence securities contingent on types can be traded? Is this price higher or lower than pM? Why? Explain carefully your findings.
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