As machine learning (ML) becomes more prevalent in human-centric applications, there is a growing emphasis on algorithmic fairness and privacy protection. While previous research has explored these areas as separate objectives, there is a growing recognition of the complex relationship between privacy and fairness. However, previous works have primarily focused on examining the interplay between privacy and fairness through empirical investigations, with limited attention given to theoretical exploration. This study aims to bridge this gap by introducing a theoretical framework that enables a comprehensive examination of their interrelation. We shall develop and analyze an information bottleneck (IB) based information obfuscation method with local differential privacy (LDP) for fair representation learning. In contrast to many empirical studies on fairness in ML, we show that the incorporation of LDP randomizers during the encoding process can enhance the fairness of the learned representation. Our analysis will demonstrate that the disclosure of sensitive information is constrained by the privacy budget of the LDP randomizer, thereby enabling the optimization process within the IB framework to effectively suppress sensitive information while preserving the desired utility through obfuscation. Based on the proposed method, we further develop a variational representation encoding approach that simultaneously achieves fairness and LDP. Our variational encoding approach offers practical advantages. It is trained using a non-adversarial method and does not require the introduction of any variational prior. Extensive experiments will be presented to validate our theoretical results and demonstrate the ability of our proposed approach to achieve both LDP and fairness while preserving adequate utility.