Here's the reference as follows.

Life-Cycle Model PS Individual lives for 2 periods, works l, earns wl, consumes c1 in period 1, consumes c2 in period 2: young U = u(c1, l) + 8 . v(c2) Let's start with case with no taxes. Savings s = wl - C1, C2 = (1 + r)s. Capital income rs C2 Intertemporal budget: c1 + /wl 1 tr C2 Individual maximization problem: max u ( wl - + 8. v(C2) l, C2 1tr FOC wrt l: W = 0 ac1 al av FOC wrt c2: ac1 = 6 . (1+r)ac2Taxes in the Life-Cycle Model Budget with consumption tax at rate to: (1 + tc) (ci + C2