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!!
Your exposition of the two is rather interesting, for despite the substantial correspondence in signification not seldom the transversality condition for bonds is bluntly spelt out as being \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}=0, rather than \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}\leq 0 - or did you mean a violation of the no Ponzi scheme condition thereby? Why merge them, in that case? Because of the agent’s craving for a Ponzi scheme concurrent to the market’s restriction thereof? Is that then, to you, a foundation of the transversality condition for bonds?
Four queries, therefrom.
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What is your intuition of the no Ponzi scheme condition \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}\geq 0? The present value of priced bonds in terms of consumption marginal utility in the anterior future cannot be negative? Why, if so, should such depict a restriction on perpetual indebtedness: why not positive and a violation thereof not negative?
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Is the no Ponzi scheme condition applicable to capital, in principle, even though it be not demanded as an asset, but simply owned? Yes, inasmuch as it be an instance of savings, perhaps?
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How would you derive \lim_{t\to\infty}\beta^{t}\lambda_{t}X_{t+1}\leq 0? Is the no Ponzi scheme pseudo-condition \lim_{t\to\infty}\beta^{t}\lambda_{t}X_{t+1}\geq 0 itself somehow derived?
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The transversality condition together with the FOCs is often related as satisfying necessity and sufficiency for the objective function’s optimisation in terms of its arguments. Is not the transversality condition merely necessary alongside the other FOCs (i.e. KKT conditions) and a convex objective function added thereto also sufficient for its optimisation in terms of its arguments?
Thank you.
To me \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}\geq 0 is better understood as a solvency supply constraint, in that transfinite private credit (over-saving) is required by the market. Its antinomy \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}\leq 0 would thus be an insolvency demand constraint, whereby transfinite private debit (under-saving) is desired by the household. The two, by antisymmetry, would yield transversality condition \lim_{t\to\infty}\beta^{t}\lambda_{t}B_{t+1}=0, which rules out transfinite indebtedness (Ponzi games). The same would apply to capital or any other store of value. Depicting the solvency supply constraint as a no Ponzi game constraint could be counterintuitive, as it were, because transfinite indebtedness seems to be effectively in place and undergone by the treasury (whence regarding its antinomy as such, if travelling that path).
I am not sure I understand your questions. The no-Ponzi condition essentially restricts your agent from accumulating infinite debt. It has nothing to do with optimal behavior by the household. Obviously, if you allow the household to borrow any asset without ever repaying that, the household would do. That’s why we forbid that in model. The argument is usually that lenders foresee this incentive and prevent such behavior.
The transversality condition instead is about optimal behavior. It naturally follows as the limit of a finite horizon optimization problem.