I am an economics student, and I have run into a question where I must apply a transversality condition in order to prove that we have a balanced growth path (all variables grow at the same constant rate).
I need to derive the infinite horizon transversality condition, and I found online that I need to use the limit of the finite horizon transversality condition as T approaches infinity as a constraint and solve the maximization again in order to get the infinite horizon transversality condition.
By this logic, I need to solve the following:
$$\underset{\{k_t\}_{t=0}^\infty}{max}\;\sum_{t=0}^\infty\beta^t\bigg(\frac{((1+(1-\delta))k_t-k_{t+1})^{1-\gamma}}{1-\gamma}\bigg)$$ $$s.t.\; \underset{T\rightarrow\infty}{lim}\;\bigg[\bigg(\frac{\beta^T}{((1+(1-\delta))k_T-k_{T+1})^\gamma}\bigg)k_{T+1}\bigg]=0$$
In order to attempt to avoid the homework problem trap, I can make it more general:
Let $F(k_T,k_{T+1})$ be the transversality condition (finite horizon), and let $U(k_t,k_{t+1})$ be the objective function. $$\implies \underset{\{k_t\}_{t=0}^\infty}{max}\;\sum_{t=0}^\infty\beta^t\bigg(U(k_t,k_{t+1})\bigg)$$ $$s.t.\; \underset{T\rightarrow\infty}{lim}\;F(k_T,k_{T+1})=0$$
My main question is: How do I solve such a problem? Do I do a Lagrangian? If that is the case, how do I evaluate the derivative of the limit? Any help would be greatly appreciated.