# First order condition of the sequence problem

Consider the standard permanent income model: $$\max_{\{c_t\}_{t=0}^\infty, \{b_{t+1}\}_{t=0}^\infty} \mathbb{E}_0 \left\{ \sum_{t=0}^\infty \beta^t u(c_t)\right\}$$ s.t. $$c_t + b_t = R^{-1} b_{t+1} + y_t; \forall t \geq 0$$ Assume also that $$R^{-1} = \beta$$, $$u$$ is the usual well-behaved'' function, and all the assumption of the dominated convergence theorem are satisfied. The lagrange multipliers of the problem are $$\{\mu_{t+1}\}_{t=0}^\infty$$ and the lagrangian is: $$\mathbb{E}_0 \left\{ \sum_{t=0}^\infty \beta^t u(c_t) + \mu_{t+1} \left(R^{-1} b_{t+1} + y_t - c_t - b_t\right) \right\}$$

The usual FOC is: $$u'(c_t) = \mathbb{E}_t u'(c_{t+1})$$

How did we move from $$\mathbb{E}_0$$ in the original problem to $$\mathbb{E}_t$$ in the FOC?

• What is the textbook you describe? – Bertrand Oct 20 '19 at 18:25
• It is taken from Ljungqvist and Sargent's book. It is also in the QuantEcon website, page 734 python.quantecon.org/_downloads/pdf/… – Luca Gi Oct 20 '19 at 20:22
• My intuition is that we are looking for a solution $\{c_t\}_{t=0}^\infty$ of the form $c: Z^t \rightarrow \mathcal{C}$ and rational expectations imply bayesian updating. But I cannot find good references on this detail. – Luca Gi Oct 20 '19 at 20:33