# Is there an argument from first principles for the form of the no-ponzi condition?

in the ramsey model, we use the no ponzi condition $$\lim_{t\to\infty}e^{-R_t}a_t\geq 0$$

for assets $a_t$ that a household holds at time $t$.

I understand intuitively what the reasoning behind this is: the present value of the level of assets one holds in the "final period" (which goes to infinity), cannot be negative, because if it was, lenders would not be willing to borrow.

However, I have ever only encountered this argument in vague terms like that. I have never seen a rigorous argument that the condition has exactly this form (rather than some other form such as $\lim_{t \to \infty}a_t\geq 0$, or $\lim_{t \to \infty}e^{-f(t)}a_t\geq 0$ for some other function $f$).

In the Ramsey model, we have of course the equation of motion for capital, and while we can integrate this, and find something that looks like the no ponzi condition, we cannot actually derive the condition from it. It also seems to me that the no-ponzi condition is a bit ad-hoc, since we don't explicitly model lending and borrowing at all in the ramsey model.

So is there a rigorous argument from first principles, based on the assumptions of the Ramsey model, that this is the condition we need, that doesn't rely on simply vaguely stating that "otherwise the household could borrow indefinitely"?

• Although it's in discrete-time, in chapter 7 of «Economic Dynamics: Discrete Time», by Jianjun Miao, it's proven that when we assume that transversality condition, we get an optimal course of action. Nov 12 '17 at 15:41
• That is not related to my question. My question is about the no-ponzi condition, which is not the same as the transversality condition. Nov 12 '17 at 15:48
• If I remember correctly a discount factor, in the Ramsay model, would be the $e^{-f(t)}$ term inside the integral of the objective function. And usually, in these models, the discount is based on real interest rate. Hence, the $e^{-R_t}$. This condition cannot be derived from the model, I think. It's imposed by the market, i.e. it's also part of the model. What happens is that in many models, the no-ponzi condition is an instance of a transversality condition. Nov 12 '17 at 16:13