Consider the perpetuity equation X D = AX + B, where (A, B) and X on the right-hand side are independent. The Kesten-Grincevičius-Goldie theorem states that P{X > x} ∼ cx −κ if EA κ = 1, EA κ log + A < ∞, and E|B| κ < ∞. We assume that E|B| ν < ∞ for some ν > κ, and consider two cases (i) EA κ = 1, EA κ log + A = ∞; (ii) EA κ < 1, EA t = ∞ for all t > κ. We show that under appropriate additional assumptions on A the asymptotic P{X > x} ∼ cx −κ (x) holds, where is a nonconstant slowly varying function. We use Goldie's renewal theoretic approach.