  This paper diverges from previous literature by considering the utility
maximization problem in the context of investors having the freedom to actively
acquire additional information to mitigate estimation risk. We derive
closed-form value functions using CARA and CRRA utility functions and establish
a criterion for valuing extra information through certainty equivalence, while
also formulating its associated acquisition cost. By strategically employing
variational methods, we explore the optimal acquisition of information, taking
into account the trade-off between its value and cost. Our findings indicate
that acquiring earlier information holds greater worth in eliminating
estimation risk and achieving higher utility. Furthermore, we observe that
investors with lower risk aversion are more inclined to pursue information
acquisition.
