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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 p². 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 c, cand the optimal investment IF? c. Calculate (i.e. derive an expression for) d and p.

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 p². 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 c, cand the optimal investment IF? c. Calculate (i.e. derive an expression for) d and p.

Essentials Of Investments
Essentials Of Investments
11th Edition
ISBN:9781260013924
Author:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Publisher:Bodie, Zvi, Kane, Alex, MARCUS, Alan J.
Chapter1: Investments: Background And Issues
Section: Chapter Questions
Problem 1PS
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The following question is based on the model of Diamond and Dybvig (1983). 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. %3D
Transcribed Image Text:The following question is based on the model of Diamond and Dybvig (1983). 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. %3D
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".
Transcribed Image Text: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".
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