One of the main problems in Korea's public assistance program, the NBLS (National Basic Livelihood Security), is that the loophole of welfare system is continuously growing. Living wage program is the largest sub-program of the NBLS, and the most important determinant of amount of living wage for each beneficiary is the level of reported income. Therefore, accurate and effective income detection is essential in improving policy effects and furthermore reducing the leakage of wage expenditure as beneficiaries always have an incentive to underreport their income. Since most of them do not pay income tax, the welfare authority should exert an independent effort to effectively detect their income. Considering that living wage is a special kind of income tax of which marginal tax rate is -1, one can apply a classical theory of tax evasion to understand illegal or excessive receipt of living wage caused by income underreporting. Utilizing a classical theory given by Alingham and Sandmo (1972), this paper provides a theoretical analysis of the optimal income reporting of the beneficiary. Then an optimization problem is constructed from the government's viewpoint to derive optimal income detecting device (auditing). This paper proves that cut-off discriminated auditing outperforms random auditing and cut-off auditing which implies if the government assigns a positive audit probability to every reported income less than a certain level and the probability is inversely proportional to the level of reported income, it can minimize underreporting and then gradually reduce the leakage of wage expenditure.