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I am a bit confused about when to consider a separate optimization problem and when to combine different problems into a single optimization problem.

Consider an individual who derives utility from consumption:$$\sum_{t=1}^{T}\beta^{t}u(c_{t})$$ and is given a stochastic endowment of money every period, that is $y_{t}$ . Now, if there are no savings, I understand given that $u'>0$

that optimality dictates that $c_{t}=y_{t}$ . If there were savings, however, we would write down a Lagrangean, and use FONCs to get Euler equations. My question is as follows: when can optimize for each period separately? Will it be when the periods are linked in some way? If the arguments we wish to maximize over are independent of each other (across periods), can we consider separate maximization problems?

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    $\begingroup$ You can maximize utilities period by period if there is NO linkage between periods, e.g. through saving, habit formation, capital accumulation, etc. On the other hand, in the presence of intertemporal trade-offs, e.g. consuming today vs consuming tomorrow, then you'd have to consider maximizing the (discounted) sum of utilities over time. $\endgroup$
    – Herr K.
    Nov 29, 2015 at 19:23

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