A dynamic investment model under time-inconsistency

In this study, we analyze a three period dynamic investment model where a principal interacts with an agent who is time-inconsistent. We use ß? preferences to capture time-inconsistency as it is widely used in the literature. Every period, the agent invests an amount and then the principal, observing the investment amount, makes an offer or not, and then the agent accepts the offer or not. For any ? and ß combination, we find the optimal investment streams for both sophisticated and naive type of time-inconsistent agent. For a sophisticated type of agent, for ßs above a threshold, we find that for intermediate values of ?, as the degree of time-inconsistency increases, the game ends in the second period rather than the first period, and for relatively high values of ?, as the degree of time-inconsistency increases, the game ends in the second period rather than the third period. When the agent is naive, however, for a range of ? values, below a certain ß value, the agent after realizing that he is time-inconsistent changes his mind about his investment decision and ends the game in the second period, even though he made an investment in the first period with which he intended to end the game in the third period. We also find that for all ? and ß values naive agent erroneously makes an overall investment that is at least as big as that of the sophisticated agent. Also, for the principal, it is better that the agent is naive rather than sophisticated for all ? and ß values.