<p>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 <span class="math-container">$\lambda_{t+1}=\lambda_{(c_t,n_t,M_t)}$</span> , which means it's an expectation over values that are known at that time. Therefore, you have <span class="math-container">$E_t[\lambda_{t+1}]=E_t[\lambda_{(c_t,n_t,M_t)}]=\lambda_{(c_t,n_t,M_t)}=\lambda_{t+1}$</span></p>

<p>But, as said above, it depends on the problem.</p>

CC BY-SA 4.0
2019-11-04T17:45:15.273Z
Thanks, do you think your idea would be appliable to my issue? I edited my question with more infos.
dadelutz
CC BY-SA 4.0
2019-11-01T00:04:10.523Z
2019-11-01T00:04:10.523Z
one_teach_wonder
<p>I am studying a rather standard DSGE Model with a standard Utility function <span class="math-container">$U = f(c_t, n_t, M_t)$</span> 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 <span class="math-container">$E_t \left[\lambda_{t+1}\right]$</span>, where <span class="math-container">$\lambda_t$</span> 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</p>

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

CC BY-SA 4.0
2019-10-31T18:16:48.577Z
Can you provide more information regarding the optimization problem? Budget constraint, choice variables, et cetera
Brennan
CC BY-SA 4.0
2019-11-04T17:41:58.430Z
Hi Brennan, I edited my questions with more infos, Cheers
dadelutz
<macroeconomics><dsge><rational-expectations>
Solving Euler Equation for standard DSGE Models
dadelutz
# Solving Euler Equation for standard DSGE Models

(Asked by: username_0 on 2019-10-31)

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]$

**Tags:** macroeconomics, dsge, rational-expectations
> **Comments:**
> **username_1**: Can you provide more information regarding the optimization problem? Budget constraint, choice variables, et cetera  
> **username_0**: Hi Brennan, I edited my questions with more infos, Cheers  

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## 1 answers

### Answer by username_2 on 2019-11-01. Score: 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.

> **Comments:**
> **username_0**: Thanks, do you think your idea would be appliable to my issue? I edited my question with more infos.  

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