In the optimal intertemporal problem with infinite horizon, transversality condition means that the present value of captial or state variable should be zero at the infinity while no-Ponzi game condition keeps the agent away from over-borrowing. What is the difference between these two conditions? Are they mathematically equivalent with merely different economic interpretation? Thanks!

The No-Ponzi condition is an external constraint on the model. You forbid your agent from running Ponzi-schemes, i.e. acquiring infinite debt that is never repaid. Usually, it’s something along the lines of \lim_{t\to \infty} \beta^t \lambda_t B_{t+1} \geq0

The transversality condition is different. It’s an optimality condition stating that it’s not optimal to start accumulating assets and never consume them. But with respect to optimality, you would still want to run a Ponzi-scheme if allowed to run one, i.e. \lim_{t\to \infty} \beta^t \lambda_t B_{t+1} \leq0.

Together you get the usual

\lim_{t\to \infty} \beta^t \lambda_t B_{t+1} =0

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Thanks alot professor! You have completely solved my problem!!