---
tags: macro, memo,public
---
$$
% My definitions
\def\ve{{\varepsilon}}
\def\dd{{\text{ d}}}
\newcommand{\dif}[2]{\frac{d #1}{d #2}} % for derivatives
\newcommand{\pd}[2]{\frac{\partial #1}{\partial #2}} % for partial derivatives
$$

# Complete Markets

## The Setting

At $t=0, 1,2, \cdots$, there is a possible state $s_t \in S$. A sequence of (specific) states is called **history**.

**Definition** We use $s^t=(s_0, s_1, \cdots, s_t)$ to refer the sequence of states from time $0$ to $t$. $\pi_t(s^t)$ is the probability of this history.

**Definition** $y_t^i(s^t)$ and $c_t^i(s^t)$ are counsumer $i$'s income and counsumption at time $t$ given the history $s^t$, respectively.

The income and consumption at every period $t$ is uncertain, which depend on $\pi_t(s^t)$. The conumer $i$'s utility is the expectated discounted cousumption of every periods among every states.

$$
U_i(c^i)=\sum_{t=0}^{\infty} \sum_{s^t} \beta^t u_i[c_t^i(s^t)] \pi_t(s^t)
\tag{8.2.1}$$

At each period, each state, there is an aggregate resource constraint:

$$
\sum_{i=0}^{I} c_t^i(s^t) \le \sum_{i=0}^{I} y_t^i(s^t) \tag{8.2.2}
$$

**Definition** An Arrow-Debreu security of time $t$ and $s^t$ is a contigent asset that gives you $1$ unit of income at time $t$ is the history $s^t$ happened.

At time $0$, consumers can trade Arrow-Debreu securities.

**Definition** $q_t^0(s^t)$ is the price of an Arrow-Debreu security of time $t$ and $s^t$ traded at time $0$.

In other words, a consumer can pay $q_t^0(s^t)$ at time $0$ and receive $1$ unit of income at time $t$ if the history $s^t$ happened. Consumers can also sell Arrow-Debreu securities to others.

Every trades happen at time $0$.

At time $0$, the consumer $i$ has the budget constraint

The reason that a consumer can sell Arrow-Debreu securities to others, i.e. make a payment in the future, is that she has some (contigent) future incomes.

In a nutshell, consumers use Arrow-Debreu securities to trade uncertainties. At time $0$, everyone knows her future consumption at time $t$ given history $s^t$.

With the price of Arrow-Debreu securities, consumer $i$ should have a budget constraint:

$$\sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)c_t^i(s^t) \le \sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)y_t^i(s^t)
\tag{8.5.1}$$

The utility maximiation problem is

$$
\max_{c_t^i(s^t)} \sum_{t=0}^{\infty} \sum_{s^t} \beta^t u_i[c_t^i(s^t)] \pi_t(s^t)\\
\text{st }\sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)c_t^i(s^t) \le \sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)y_t^i(s^t)
$$

The Lagrange function is

$$
L= \sum_{t=0}^{\infty} \sum_{s^t} \beta^t u_i[c_t^i(s^t)] \pi_t(s^t)\\ \mu_i \left(\sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)c_t^i(s^t) - \sum_{t=0}^{\infty} \sum_{s^t} q_t^0(s^t)y_t^i(s^t)\right)
$$

The first-order condition is

$$
\mu_i q_t^0(s^t) = \beta^t u_i'[c_t^i(s^t)]\pi_t(s^t)
$$