# 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? – PhDing Oct 9 '16 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. – Alecos Papadopoulos Oct 9 '16 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. – PhDing Oct 9 '16 at 10:46