Thanks for your reply. I am seeking a numerical solution that I can plot as a function of time t. The argument v is the date of birth. To the best of my knowledge, there is no way one can get rid of the argument v. An attempt at doing so will be to define v as t-g, where g denotes the age. Even doing this does not reduce the number of arguments, as g replaces v.
Thus, l(t,v), which is the labor supply at time t of an agent born at time v, cannot be transformed into l(t). L(t) is the aggregation of l(t,v) for all the agents born between v=t-T and t.
Eq 9 is an exogenous perturbation I am adding to the system to see the way it reacts when the value of x changes over time. If there is a way to declare x(t) as an exogenous variable, it will be fine. Doing this will reduce the system to a set of 8 equations in 8 unknowns, a(t,v), c(t,v), l(t,v), L(t), K(t), r(t), w(y), Y(t). The number of equations being equal to the number of unknowns, there is a numerical solution to the system, given the values I have assigned to the parameters alpha, delta, rho, sigma, and x(t). Regards