Baseline result by Grossman and Shiller (1982) is that the excess return of an asset is given by the product of two terms: a consumption-weighted harmonic average of the agents’ relative risk aversion coefficients and the expected product of aggregate consumption growth and the excess return. This can be shown here for a small time interval. For any asset, the optimum allocation for an agent implies the following necessary condition
$$ u'(c^i_t) = E_t[u'(c^i_{t+\delta})R^j_{t+\delta}] \implies E_t[u'(c^i_{t+\delta})(R_{t+\delta} - R^f_{t+\delta})] = 0 $$
Taking a first order expansion we have
$$ E_t[(u'(c_t^i) + u''(c_t^i)(c_{t+\delta}^i- c_t^i))(R_{t+\delta} - R^f_{t+\delta})] \approx 0 $$
which implies after rearranging
$$ -\frac{u'(c_t^i)}{u''(c_t^{i})}E_t[R_{t+\delta} - R^f_{t+\delta}] = E_t[(c^i_{t+\delta} - c_t^i)(R_{t+\delta} - R^f_{t+\delta})] $$
By integrating with respect to $i$ we can find an expression in terms of aggregate consumption $C_t \equiv \int_i c_t^i di$
$$ \begin{align*}E_t[R_{t+\delta} - R^f_{t+\delta}] &= \left(-\int_i \frac{u'(c_t^i)}{u''(c_t^i)}\right)^{-1}\times E_t[(C_{t+\delta} - C_t)(R_{t+\delta} - R^f_t)]\\
&= \left(-\int_i \frac{u'(c_t^i)}{u''(c_t^i)c_t^i}\frac{c_t^i}{C_t}\right)^{-1}\times E_t\left[\frac{C_{t+\delta} - C_t}{C_t}(R_{t+\delta} - R^f_t)\right]
\end{align*} $$
This result states that heterogeneity only enters through the weights in the harmonic average of relative risk aversion. If for whatever reason there is heterogeneity in consumption growth, it doesn’t matter for risk premia. Specifically, income risk that cause idiosyncratic consumption fluctuations due to imperfect risk sharing are not relevant. The risk premium with or without uninsurable idiosyncratic risks will be the same as long as all agents have the same risk aversion.
This is an “irrelevance” result. Therefore, to understand when heterogeneity matters, it’s important to use this result as a benchmark where some assumption is then broken.
Continuous time model. There is arrival and death of agents. Agents are born and endowed from birth with “trees.” Each moment, a mass of $\pi$ agents are born and face a hazard rate of $\pi$ that they will die each moment. Therefore, the mass of agents born at $s$ that have survived till $t$ is given by $\pi e^{-\pi(t-s)}$. At any given time $t$, the total population of agents is given by