Life annuity plays an important role in the withdrawal phase of pensions, and it can help people deal with longevity risks effectively. This paper considers a stochastic model for a life annuity plan in continuous time, where the initial value of the individual annuity \mbox{account} is determined in advance, while the annuity payments depend on the financial situation of the account. The annuity fund is allowed to invest in a risk-free asset and a risky asset. The fund manager can adjust the annuity payment level and charge management fees to ensure the stable operation of the plan. In the objective function, we aim to maximize the utility of the weighted product of the annuity payment adjustment and the management fee, which is taken in the form of Cobb-Douglas utility. By applying dynamic programming approach, we establish the corresponding Hamilton-Jacobi-Bellman equation and derive the optimal strategies and the value function explicitly. Furthermore, the Epstein-Zin recursive utility is considered as an extension and we find the elasticity of intertemporal substitution has a positive effect on the annuity payment adjustment and the management fee. Numerical examples are presented to illustrate the sensitivity of the optimal strategies to model parameters, and the corresponding economic explanations are given, which demonstrate that this annuity plan is sustainable and can provide stable annuity payment for participants.