Log-normality assumption in consumption based asset pricing Consider a very basic discrete time representative consumer maximization problem with CRRA utility. There exist a risky asset with time $t$ price $p_t$ that pays time $t+1$ dividend $d_{t+1}$ , and a riskless asset with price $p_t^f$ that pays a constant payoff 1 at $t+1$. We assume that the dividends are a sequence of random variables that follow a Markov process. Assume further that the consumer has no other income streams (i.e. $y_t = 0 \ \forall t$). At time t consumer invests amount $\pi_t$ in the risky asset and amount $\pi_t^0$ in the riskless asset. Therefore, the maximization problem can be stated as

\begin{align*} 
&amp; \underset{\{ c_t, \pi \}_0^\infty}{\text{max}} \ \ E_0 \sum_{t=0}^\infty \ \beta^t \ \frac{c_t^{1-\gamma} -1}{1-\gamma} \\ 
\\
\
s.t  \ \ \ \ &amp; c_t + \pi_t p_t + \pi_t^0 p_t^0 = (d_t+p_t) \pi_{t-1} + \pi_{t-1}^0  \\
&amp; c_t \geq 0
\end{align*}

Say we want to find the equilibrium riskless rate and expected equity premium. In order to close the model, it is often assumed assumed (see e.g. Claus Munk's book Financial Asset Pricing Theory chapter 8.3) that the log-consumption growth and log-risky gross returns are jointly normally distributed. I.e

\begin{align*} 
&amp; ln \ \Big(\dfrac{c_{t+1}}{c_t} \Big) \equiv \bar{g}_{t+1} \sim N(\mu_g, \sigma_g^2) \\ 
&amp; ln R_{t+1} \equiv \bar{r}_{t+1} \sim N(\mu_r, \sigma_r^2) \ , \\
\end{align*}

where gross returns are defined as 
$$R_{t+1} \equiv \frac{p_{t+1} + d_{t+1}}{p_t} \ .$$

What I don't completely understand is where do thelog-normal distribution assumptions "come from". I know that since this is a representative agent economy, consumption of the agent must equal the aggregate dividend in the economy.  But since we assumed that there is no income, $y_t = 0 \ \forall t$, the only exogenous dividend process in the economy is $d_t$ and therefore it should have the same distribution as the consumption growth. However, my impression is that when we say the risky rate has log-normal distribution this actually means the dividend process, since it is the 'random part' in the definition of returns (price $p_{t+1}$ is not exogenous but determined inside the model). To me it seems now that we have made two different assumptions about the same endowment process $d_t$. Where does the assumption for consumption come from or what does it stand for? How would the situation change if the consumer had some income stream $y_t &gt; 0$?
The typical two-period Lagrangian is

$$\Lambda = \beta^t\cdot \Big(\frac{c_t^{1-\gamma} -1}{1-\gamma} + \lambda_t\cdot \big[(d_t+p_t) \pi_{t-1} + \pi_{t-1}^0- c_t - \pi_t p_t - \pi_t^0 p_t^0\big]\Big) \\
+ \beta^{t+1}\cdot \Big(\frac{c_{t+1}^{1-\gamma} -1}{1-\gamma} + \lambda_{t+1}\cdot \big[(d_{t+1}+p_{t+1}) \pi_{t} + \pi_{t}^0- c_{t+1} - \pi_{t+1} p_{t+1} - \pi_{t+1}^0 p_{t+1}^0\big]\Big)$$ 

The first order conditions with respect to $c_t, \pi_t$ are

$$c_t^{-\gamma} = \lambda_t \implies ... \gamma\ln \frac {c_{t+1}}{c_t} = \ln \frac {\lambda_{t}}{\lambda_{t+1}} \tag{1}$$

$$-\beta^t\lambda_tp_t + \beta^{t+1}\lambda_{t+1}(d_{t+1}+p_{t+1})=0 \implies  \frac {\lambda_{t}}{\lambda_{t+1}} = \beta \frac{p_{t+1} + d_{t+1}}{p_t} \tag{2}$$

and so, using also the definition of the gross return,

$$\ln \frac {\lambda_{t}}{\lambda_{t+1}} = \ln \beta + \ln R_{t+1} \tag{3}$$

Combining $(1)$ and $(3)$ we get

$$\ln \frac {c_{t+1}}{c_t} = \frac 1 {\gamma}\ln \beta + \frac 1 {\gamma}\ln R_{t+1} \tag{4}$$

So we see that at the optimal path, consumption growth is a direct affine function of the log-risk returns. This among other things implies that their correlation coefficient is equal to unity.

The normal distribution is closed under affine transformations (alternatively, under scaling and shifting), so if we assume that log-risky returns are normally distributed, then consumption growth is also normally distributed (with different mean and variance of course).

Note that although in general, the joint normality assumption is an additional one to be made when two normal random variables are not-independent, here, the fact that the one is an affine function of the other guarantees joint normality. By Cramer's condition for bivariate normality, it must be the case that all linear combinations of two normal random variables have a univariate normal distribution. In our case we have (generic notation) the random vavriable $Y$ and the random variable $X = a+bY$. Consider

$$\delta_1X + \delta_2 Y = \delta_1(a+bY) + \delta_2 Y = \delta_1a + (\delta_1b+\delta_2)Y$$

So for any $(\delta_1, \delta_2)$ (except the zero vector which is excluded a priori), $\delta_1X + \delta_2 Y$ follows a normal distribution if $Y$ does. So it is sufficient to assume that log-risk returns follow a normal distribution to obtain joint normality also.