misc_notes

This file lays out the basics of asset-pricing using traditional methods but
heterogeneity on the part of agent beliefs.

The basic two-period consumption-based asset pricing agent (eg. see Cochrane
2001, Chapter 1) solves the following problem:

    ...fill in with agent problem from elsewhere...

This is the simplest possible asset-pricing problem, described as follows:

- Time is only two periods
- Agents begin period 1 with an initial cash endowment $y_0$ and an initial asset endowment $\xi_0$.
- Agents must choose how much asset holdings $\xi$ to retain or purchase for period 2
- The definition above restricts short selling, borrowing beyond means, and consumption greater than zero
- The only income in period 2 is the dividend $d$ payoff for the asset: $\xi \cross d$
- Agents have a risk aversion $\rho$, time discount factor $\beta$, and a set of beleifs about the asset payoff in the following period.
- Agents may differ across the features noted above, but the least controversial differences are almost certainly:
    - beliefs
    - initial cash or asset endowments

The two-period model is chosen here both for simplicity and because many infinite-horizon problems collapse to the solution to the two-period problem under broad conditions.

In addition, the two-period model can also represent the behavior of a *myopic* asset-pricing agent who lives for an indefinite number of periods, but only makes plans based on a two-period horizon. This is how we will use these agents. Future extensions will allow agents to account for further horizons.

To solve the above problem for the optimal choice of $\xi$: substitute the constraints into the objective, take a first derivative and determine where the optimal point is (confirm inside solution). This gives us the familiar Euler equation:

    ...fill in ...

...which in turn is traditionally rearranged to produce the canonical asset-pricing equation:

$$
p_t = E \left[ \frac{u'(c_{t+1})}{u'(c_{t})} (d_{t+1}+p_{t+1}) \right] = E \left[ M X \right].
$$

Here $M$ is the "stochastic discount factor" and $X$ is total payoff in the next period, which combines both dividends and capital gains. (In the two-period model, the agent assumes no trade occurs in period 2 and thus the payoff is only the final dividend.)

The traditional asset pricing approach uses the above asset pricng equation, as derived from a representative agent, to find asset prices associated with consumption and dividend streams which follow a range of processe (we will not discuss this in depth here; see Cochrane 2001 for background).

Our approach will simply do the following: define N agents, with their own endowments, preferences, and beleifs, and from each obtain the demand functions which come from each solving their own optimization problem.

Then simply find the price for which all demands are satisfied, via any number of methods:

* tatannoment (market clearing)
* bilateral trade
* order book execution
* search and matching

or others. There are a number of possible pricing finding mechanisms which may be employed in this rather general setting.



The two-period model can be easily converted

This very simple setup will allow us to explore the effects of different types of heterogeneity quickly and easily, and explore the different effects that market structures may have.


## Market Setups Explored Below

There are two main variations examined below:

1. A version with difference in how assets behave:



## Extensions

- price expectations: we will need to add:
    - an additional period to the start of the model
    - a multi-dimensional expectations operator
        - this may be avoided if we impose that dividends are constant






To add prices to expectations we will need to do some




The only descision the agents make is how much of the