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

Now consider a modification of the constraint above: $$c_t + b_t = R^{-1} \mathbb{E}_t b_{t+1} + y_t; \forall t \geq 0$$

The new lagrangian is: $$ \mathbb{E}_0 \left\{ \sum_{t=0}^\infty \beta^t u(c_t) + \mu_{t+1} \left(R^{-1} \mathbb{E}_t b_{t+1} + y_t - c_t - b_t\right) \right\}$$

The FOC now is: $$u'(c_t) = u'(c_{t+1})$$

Which are the differences in the derivation of the FOCs in the two cases, and specifically which step implies that in the second case there is no expectation?

  • $\begingroup$ It seems like it has something to do with certianty equivalence between any two periods. In your first case the expectation of the return on future bonds isnt updated each period when compared to your second case. $\endgroup$ – EconJohn Oct 20 '19 at 7:24
  • $\begingroup$ Actually I think it is pretty much a general result, even when the third derivative is different from zero. This example should embed the idea of complete vs incomplete markets. $\endgroup$ – Luca Gi Oct 20 '19 at 13:24

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