# Deriving First Order Condition of Hansen and Singleton (1982)

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.

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 $$P_{j t} U^{\prime}\left(C_{t}\right)=\beta \mathbb{E} \left[P_{j t+1} U^{\prime}\left(C_{t+1}\right)\right]$$.
• 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 Mar 11 at 18:56