<p>This has nothing to do with GMM, which is a statistical method for estimating parameters. This is a macroeconomics model or say a dynamic optimization problem.</p>
<p>From FOCs for <span class="math-container">$C_t$</span>, <span class="math-container">$C_{t+1}$</span>, and <span class="math-container">$Q_{jt}$</span>, youg can get <span class="math-container">$$\mathbb{E}\frac{\beta U'(C_{t+1})}{U'(C_t)} = \frac{\lambda_{t+1}}{\lambda_t}$$</span>, and <span class="math-container">$$\lambda_t P_{jt} = \lambda_{t+1} P_{jt+1}$$</span>.</p>
<p>Then by replacing <span class="math-container">$\lambda$</span> you have <span class="math-container">$$P_{j t} U^{\prime}\left(C_{t}\right)=\beta \mathbb{E} \left[P_{j t+1} U^{\prime}\left(C_{t+1}\right)\right]$$</span>.</p>

CC BY-SA 4.0
2021-03-11T01:07:45.903Z
Thank you! How did you get the first two equations (would it be possible to add some intermediate steps)?
user2978524
CC BY-SA 4.0
2021-03-11T18:56:01.087Z
OK. I figured it out. We consider $\mathcal{L} = \sum_{t=0}^\infty \bigg\{ \mathbb{E} \beta^t U(C_t) - \lambda_t \bigg( C_t + \sum_{j=1}^J (P_{jt} Q_{jt} - P_{jt} - Q_{j, t-1} ) - W_t \bigg)   \bigg\}$ and take the derivative with respect to $C_t$, $C_{t+1}$, and $Q_{jt}$, as the answer says.
user2978524
CC BY-SA 4.0
2021-03-10T23:57:52.760Z
2021-03-10T23:57:52.760Z
Alalalalaki
<p>Hansen and Singleton (1982) considers the maximization of expected utility,</p>
<p><span class="math-container">\begin{align*}
   \max \mathbb{E} \sum_{t=0}^\infty \beta^t U(C_t)
\end{align*}</span>
with respect to the budget constraint,
<span class="math-container">\begin{align*}
C_t + \sum_{j=1}^J P_{jt} Q_{jt} = \sum_{j=1}^J P_{jt} Q_{j, t-1} + W_t,\ t= 0, 1, \cdots,
\end{align*}</span>
where <span class="math-container">$C_t$</span> is the consumption at time <span class="math-container">$t$</span>, <span class="math-container">$P_{jt}$</span> is the price of security <span class="math-container">$j$</span> at time <span class="math-container">$t$</span>, <span class="math-container">$Q_{jy}$</span> is the amount of security <span class="math-container">$j$</span> at time <span class="math-container">$t$</span>, <span class="math-container">$J$</span> is the number of securities and <span class="math-container">$W_t$</span> is the labor income at time <span class="math-container">$t$</span>.</p>
<p>Then, they show that the first order condition is
<span class="math-container">\begin{align*}
P_{jt} = \mathbb{E} \bigg[ \beta \frac{U'(C_{t+1})}{U'(C_t)} P_{j, t+1}  \bigg],\ j = 1, \cdots, J.
\end{align*}</span></p>
<p>I tried to see it in the simplest case, where <span class="math-container">$t=1$</span> and maximize <span class="math-container">$U(C_0) + \beta U(C_1)$</span> with the Lagrange multiplier, but I could not get the similar first order condition. This seems to be a canonical example of non-linear GMM.</p>

<econometrics>
Deriving First Order Condition of Hansen and Singleton (1982)
user2978524
# Deriving First Order Condition of Hansen and Singleton (1982)

(Asked by: username_0 on 2021-03-10)

Hansen and Singleton (1982) considers the maximization of expected utility,

\begin{align\*}
\max \mathbb{E} \sum\_{t=0}^\infty \beta^t U(C\_t)
\end{align\*}
with respect to the budget constraint,
\begin{align\*}
C\_t + \sum\_{j=1}^J P\_{jt} Q\_{jt} = \sum\_{j=1}^J P\_{jt} Q\_{j, t-1} + W\_t,\ t= 0, 1, \cdots,
\end{align\*}
where $C\_t$ is the consumption at time $t$, $P\_{jt}$ is the price of security $j$ at time $t$, $Q\_{jy}$ is the amount of security $j$ at time $t$, $J$ is the number of securities and $W\_t$ is the labor income at time $t$.

Then, they show that the first order condition is
\begin{align\*}
P\_{jt} = \mathbb{E} \bigg[ \beta \frac{U'(C\_{t+1})}{U'(C\_t)} P\_{j, t+1} \bigg],\ j = 1, \cdots, J.
\end{align\*}

I tried to see it in the simplest case, where $t=1$ and maximize $U(C\_0) + \beta U(C\_1)$ with the Lagrange multiplier, but I could not get the similar first order condition. This seems to be a canonical example of non-linear GMM.

**Tags:** econometrics

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## 1 answers

### Answer by username_1 on 2021-03-10. Score: 5 | Accepted answer

This has nothing to do with GMM, which is a statistical method for estimating parameters. This is a macroeconomics model or say a dynamic optimization problem.

From FOCs for $C\_t$, $C\_{t+1}$, and $Q\_{jt}$, youg can get $$\mathbb{E}\frac{\beta U'(C\_{t+1})}{U'(C\_t)} = \frac{\lambda\_{t+1}}{\lambda\_t}$$, and $$\lambda\_t P\_{jt} = \lambda\_{t+1} P\_{jt+1}$$.

Then by replacing $\lambda$ you have $$P\_{j t} U^{\prime}\left(C\_{t}\right)=\beta \mathbb{E} \left[P\_{j t+1} U^{\prime}\left(C\_{t+1}\right)\right]$$.

> **Comments:**
> **username_0**: Thank you! How did you get the first two equations (would it be possible to add some intermediate steps)?  
> **username_0**: OK. I figured it out. We consider $\mathcal{L} = \sum_{t=0}^\infty \bigg\{ \mathbb{E} \beta^t U(C_t) - \lambda_t \bigg( C_t + \sum_{j=1}^J (P_{jt} Q_{jt} - P_{jt} - Q_{j, t-1} ) - W_t \bigg)   \bigg\}$ and take the derivative with respect to $C_t$, $C_{t+1}$, and $Q_{jt}$, as the answer says.  

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