Q. Is Household savings depend on GDP in the cross model?

Household savings depends on Y since S_H = Y - C - NT and C and NT both rely on Y. How it depends on Y can't be conclusively be determined from this relationship as C and NT both depends positively on Y. We always presume that this dependence is positive and following illustrations explains why this assumption makes sense.

Assume that NT = t·Y where t is a constant between 0 and 1. t is the proportion of income which we pay in taxes. Following, suppose that C = c·Y_d where c is a constant between 0 and 1. c is proportion of disposable income which we use for consumption. If income Y increases by 1, NT increase by t, disposable income increases by 1 - t and C increases by c(1 - t). So S_H increases by 1 - c (1 - t) - t = (1 - c) (1 - t) > 0.

Because S = S_H + S_G + S_R and all parts on the right hand side depends positively on Y, total saving S will rely on positive Y and we write S(Y) for total savings (net total supply of savings).