\(C_{1t} + \frac{C_{2t+1}}{1+r_{t+1}} = A_tw_t\)
This is the original budget constraint. This means that this change does not affect equation (2.60). Thus the fully-funded social security system has no effect on the relationship between the capital stock in successive periods.
(ii ) How, if at all, does this change affect the balanced-growth-path value of k?
Since the fully funded social security scheme does not affect the  relationship between \(k_{t+1}\) and \(k_t\), this change does not affect the balanced growth path value of k. This is so because social security pays the same rate of return as private saving and therefore individuals are indifferent as to who does the saving.