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Dynamic Problem of a Consumer / Investor

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10. Consider the Dynamic Problem of a Consumer/Investor:
....
s.t. at = ? ...
where ct is consumption at time t, at is assets at the beginning of period t; and the gross interest rate R > 1 are given.
(a) Write down the above sequence problem as a functional equation problem. Which variable is the state?
(b) Assuming the problem has a solution, solve the functional equation from (a) for the value function, v and policy function, g. Hint: use the "guess and verify method"!
(c) Do you need to assume anything about the relationship between R and B for the problem to have a solution?
(d) If RB = 1 what is the time path of consumption implied by the optimal policy function'? What is the intuition for this result?

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Question 1
First of all, it will be easier to plug the constraint inside the utility function, so that the maximization will be easier to carry out. We have to maximize:

Subject to

Isolating ct from the constraint, we get:

So the maximization problem becomes simply:

with no constraints (as the constraint is already plugged in)

Now, this problem can be written as a functional equation problem in the following way:

In this case, V is the value function, a represents the asset in the current time period and a' represents the assets in the following time period.

Clearly, the state variable is a (the value of the consumer's assets in the current period). This variable fully describes the state of the consumer in any period.

Question 2
We'll guess that the value function V(a) has the form:

where F is some constant that we must find. Following with this guess, the value ...

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