 # How to obtain transversality condition in DSGE

Greetings. In base RBC dynamic programming problem can be summarized as:

\max_{c_t,k_t,n_t}\mathbb{E}_0{\sum_{t=0}^{\infty}\beta^{t}u(c_t, n_t)}

s.t.

P_t(c_t+k_t-(1-\delta)k_{t-1})\le R_tk_{t-1}+W_t n_t +\Pi_tP_t

and no-Ponzi scheme condition (as here we assume k_t a completely risk free asset):

\lim_{T\to\infty}\left\{k_T\prod_{i=0}^{T}(1+\frac{R_{T+i}}{P_{T+i}}-\delta)^{-1}\right\}\ge0

(actually I’m formulating this more as an analogy to Ramsey formulation, since I think one can avoid writing the gross real interest rate and put an expectations operator on k_T similar to Galí (2008, p.8))

Where u(c_t,n_t) satisfies conditions compatible with an stable optimal path (CRRA utility), there’s no population neither technological progress, instead a cyclic component to Hicks augmenting type technology (i.e. ARMA process).

Setting up optimization with the lagrangian method:

L = \mathbb{E}_0{\sum_{t=0}^{\infty}\beta^{t}u(c_t, n_t)} - \lambda_t(P_t(c_t+k_t-(1-\delta)k_{t-1})- R_tk_{t-1}-W_t n_t -\Pi_tP_t)

FOC for c_t, n_t and k_t are the typical ones, but there’s one that is commonly obviated (since household optimal path in most cases ensures it) which is the transversality condition (and is a link of no-Ponzi scheme condition and household optimal path, to my understanding from growth courses):

\lim_{T\to\infty}\{\lambda_Tk_T\}=0

How does one arrive to the last expression given the constraints and DP problem? I understand that very rigorous treatment of this, I suspect, may be a little cumbersome or mathy, but I would appreciate just a glimpse of a reference or something on this that could be handled by a not-very-expert as me, and that could be useful for a deeper understanding of DSGE models. I have read Acemoglu’s Introduction to Modern Economic Growth chapter (6) where this is treated deeply, but I don’t get it much. (Please let me know if I miswrote something since I did it more by analogy with Ramsey, and besides that also hope that the main point is legible).

Thank you.

This condition follows from an infinite horizon limit of a finite horizon one. In a finite horizon problem, it’s not optimal to have capital left in the last period, if that capital is still valued by the agent, i.e. \lambda_Tk_T must be 0. I would recommend having a look at Chapter 3 of McCandless ABCs of RBCs.

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