‎In this paper we propose a utility function and obtain the Bayese stimate and the optimum sample size under this utility function‎. ‎This utility function is designed especially to obtain the Bayes estimate when the posterior follows a gamma distribution‎. ‎We consider a Normal with known mean‎, ‎a Pareto‎, ‎an Exponential and a Poisson distribution for an optimum sample size under the proposed utility function‎, ‎so that minimizes the cost of sampling‎. ‎In this process‎, ‎we use Lindley cost function in order to minimize the cost‎. ‎Here‎, ‎because of the complicated form of computation‎, ‎we are unable to solve it analytically and use the mumerical methids to get the optimum sample size.