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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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:
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.
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 ...
A Dynamic Problem of a Consumer / Investor is investigated. The solution is detailed and well presented.