In the context of the dynamic investment problem described, if the consultants' information was correlated, the optimal strategy would likely involve adjusting the decision-making process to account for this correlation. When consultants provide correlated information, the investor can use this information to gain a more comprehensive understanding of the state of nature. This could lead to a more refined decision-making process, where the investor considers the information provided by multiple consultants in a more integrated manner.
If the investor had a time preference for making a decision, it would impact the optimal strategy by introducing a trade-off between the value of waiting for more information and the cost of delaying the decision. The investor would need to balance the potential benefits of gathering more information against the cost of waiting, taking into account the time value of money and the potential risks associated with delaying the decision.