Solving Euler Equation for standard DSGE Models

Question

I am studying a rather standard DSGE Model with a standard Utility function $U = f(c_t, n_t, M_t)$ subject to a budget constraint. Now, tosolve the intertemporal optimization problem I have, or everyone has, to deal with an expectation term; often similar to $E_t \left[\lambda_{t+1}\right]$, where $\lambda_t$ is the standard Lagrangian multiplier. Is there an anlytical trick to solve this? It seems to me the that usually the Expectation is just erased (without a justification) or studied using MLE estimators or Kalman Filters and surrogates. Thanks to whoever can enlighten me on the issue. Cheers

Edit 1: $\max_{c_t, n_t, i_t, M_t, K_t, B_t}{\mathbb{E}_t}\left[ \sum_{t=0}^{\infty} \beta^t \left( \log{(c_t^i)} - \gamma_i(n_t^i)^{2}+ \log \left(\frac{B_t}{P_t} \right) \right) \right]$ under the constraint: $\frac{M_t}{P_t}+ c_t + i_t + \frac{B_t}{P_t(1+r_t)} \leq u_t n_t + \frac{Q_t}{P_t}K_t + \frac{B_{t-1}}{(1+\pi_t)P_{t-1}} +\frac{M_{t-1}}{(1+\pi_t)P_{t-1}}$ The issue is wrt to a term which comes up in both the maximisation wrt to Bonds and Money: $\beta {\mathbb{E}}_{t}\left[\frac{\lambda_{t+1}}{(1+\pi_{t+1})}\right]$ - how could I solve it? (with theother terms it would lead $ c_{t}^{-1}=\beta {\mathbb{E}}_{t}\left[\frac{c_{t+1}^{-1}}{(1+\pi_{t+1})}\right]$

Answer 1

It really depends on the problem. As to why it would disappear, sometimes you can express the multiplier in terms of variables at the present or past, as in $\lambda_{t+1}=\lambda_{(c_t,n_t,M_t)}$ , which means it's an expectation over values that are known at that time. Therefore, you have $E_t[\lambda_{t+1}]=E_t[\lambda_{(c_t,n_t,M_t)}]=\lambda_{(c_t,n_t,M_t)}=\lambda_{t+1}$

But, as said above, it depends on the problem.