# No Ponzi game condition and transversality condition are the same?

Given the following non-stochastic planning problem with finite horizon, \begin{align} &\max_{\{k_{t+1}\}}\sum^T_{t=0}\beta^tU[f(k_t-k_{t+1})] \\ \text{s.t. } & 0\leq k_{t+1}\leq f(k_t)\\ & k_0 >0 \text{ (given)}. \end{align} I found that in order to make the first order conditions necessary and sufficient I have to add the so called no Ponzi game condition, i.e. \begin{gather} \lim_{T \rightarrow \infty} \frac{k_{T+1}}{R_{T+1}} \geq 0 \end{gather}

When written with the equal sign, this condition can be interpreted as the willingness of not keeping any capital at the end of life. And this is the same interpretation of the so called transversality condition.

Thus, is it right to interpret the no Ponzi game condition as a finite horizon version of the transversality condition? If not, which is the difference between them?

Is it right to interpret the no Ponzi game condition as a finite horizon version of the transversality condition?

No. The "No-Ponzi-Game" or "solvency" condition is an external constraint imposed on the individual by the market/other participants. The individual would very much like to violate it.

The Transversality condition must be satisfied in order for the individual to maximize indeed its intertemporal utility. It is an optimization condition.

So they are conceptually very different aspects of the problem.

Finally the No-ponzi-game/solvency condition is not inherently of finite horizon -it extends to the infinite horizon also.

• Thank you for the clarification. But, when should I use one or the other when dealing with Kydland-Prescott model? Commented Oct 9, 2016 at 7:22
• @Alessandro In the theoretical solution of the model, both should be satisfied. What happens (and it may be the source of some confusion) is that in most cases, a single mathematical expression expresses the satisfaction of both. Commented Oct 9, 2016 at 10:32
• Thanks, because the fact is that in our Adv. Macro course, we usually use the transversality condition as a condition to find the optimum but we never add no-Ponzi game. The only model we have added was a model as the one above, in which we get through the FOCs a second order difference equation so that we need two boundary conditions, one of them being nPg. Commented Oct 9, 2016 at 10:46
• It seems like the TVC which is an optimization condition also leads to a constraint, i.e. the individual does not save at the end of life (i.e. say kT=0). Is it true to say that the TVC ensures that savings are not positive, and the nPg condition ensures that savings are not negative? But, you can summarize both by simply imposing that last period saving=0 and you consume everything. Commented Aug 29, 2022 at 19:53
• @KwameBrown Nice way to summarize it from a technical point of view, but the point of my answer was to stress that "savings not positive" springs from the utility function and non-satiation, so it is a desirable constraint, while "savings not negative" is an undesirable constraint imposed by the environment... except perhaps if "posthumous respect" is an argument in the utility function. Commented Aug 29, 2022 at 21:15