The correct answer is: Only assumption I is implicit.
The statement says that equality of income throughout a community is the essential condition for maximizing the total utility which the total income available could confer on the members of that community. This means that if everyone in the community had the same amount of income, the total amount of happiness or satisfaction in the community would be as high as possible.
Assumption I says that if extra income were taken from the rich and given to the poor, the total utility experienced by the community would increase. This is consistent with the statement, because if everyone had the same amount of income, the total utility would be as high as possible.
Assumption II says that equal pay for equal work. This is not necessarily consistent with the statement, because it is possible for everyone to have the same amount of income even if people are not paid equally for their work. For example, if everyone in the community had a guaranteed income, regardless of how much they worked, then everyone would have the same amount of income, but people would not necessarily be paid equally for their work.
Therefore, only assumption I is implicit in the statement.