* In equation {eq}`rnapex`, the stochastic discount factor $m_{t+1} = \beta$, a constant.

* In equation {eq}`rnapex`, the covariance term ${\rm cov}_t (m_{t+1}, d_{t+1}+ p_{t+1})$ is zero because $m_{t+1} = \beta$.

* In equation {eq}`rnapex`, ${\mathbb E}_t m_{t+1}$ can be interpreted as the reciprocal of the one-period risk-free gross interest rate.

* When $m_{t+1}$ is covaries more negatively with the payout $p_{t+1} + d_{t+1}$, the price of the asset is lower.

* When $m_{t+1}$ covaries more negatively with the payout $p_{t+1} + d_{t+1}$, the price of the asset is lower.

Equation {eq}`lteeqs102` asserts that the covariance of the stochastic discount factor with the one period payout $d_{t+1} + p_{t+1}$ is an important determinant of the price $p_t$.

Let's start with a version of the celebrated asset pricing model of Robert E. Lucas, Jr. {cite}`Lucas1978`.

As in {cite}`Lucas1978`, suppose that the stochastic discount factor takes the form

Lucas considered an abstract pure exchange economy with these features:

* a single non-storable consumption good

* a Markov process that governs the total amount of the consumption good available each period

* a single *tree* that each period yields *fruit* that equals the total amount of consumption available to the economy

* a competitive market in *shares* in the tree that entitles their owners to corresponding shares of the *dividend* stream, i.e., the *fruit* stream, yielded by the tree

* a representative consumer who in a competitive equilibrium

* consumes the economy's entire endowment each period

* owns 100 percent of the shares in the tree

As in {cite}`Lucas1978`, we suppose that the stochastic discount factor takes the form

```{math}

:label: lucsdf

Assume the existence of an endowment that follows growth process {eq}`mass_fmce`.

The asset being priced is a claim on the endowment process.

The asset being priced is a claim on the endowment process, i.e., the *Lucas tree* described above.

Following {cite}`Lucas1978`, suppose further that in equilibrium, consumption

is equal to the endowment, so that $d_t = c_t$ for all $t$.

Following {cite}`Lucas1978`, we suppose that in equilibrium the representative consumer's consumption equals the aggregate endowment, so that $d_t = c_t$ for all $t$.

For utility, we'll assume the **constant relative risk aversion** (CRRA)

specification

J(x, y) := g(y)^{1-\gamma} P(x, y)

$$

then we can rewrite equation {eq}`eq:neweqn101} in vector form as

then we can rewrite equation {eq}`eq:neweqn101` in vector form as

$$

v = \beta J ({\mathbb 1} + v )

This is because, with a positively correlated state process, higher states indicate higher future consumption growth.

With the stochastic discount factor {eq}`lucsdf2`, higher growth decreases the

discount factor, lowering the weight placed on future returns.

discount factor, lowering the weight placed on future dividends.

#### Special Cases