Suppose that a representative consumer has a utility function that satisfies CRRA

$U(C_t)=\frac{C_t^{1-}}{1-}$ with $>0,1$

$=ln(C_t)$ with $=1$

The agent wants to

$Ma{x}_{C_t,N}{E}_t[U(C_t)+U(C_{t+1})],in(0,1)$

subject to

$C_t=$tilde$(C{)}_t-P_{t}N$

$C_{t+1}=$tilde$(C{)}_{t+1}+(P_{t+1}+D_{t+1})N$

where

tilde$(C{)}_t$ is the consumption level before purchase of an asset.

$N$ is the number of units of the asset purchased.

$P_t$ is the price of an asset.

$D_{t+1}$ is a dividend that the asset may pay and so $(P_{t+1}+D_{t+1})$ is the future payoff.

The investor can freely buy or sell as much of the payoff at the current price.

$($i$)$ What is the first$-$order condition for an optimal consumption and portfolio choice? Explain the intuition.

$($ii$)$ Write down an expression for the consumption CAPM stochastic discount factor, $M_{t+1}.$

$($iii$)$ Write the first$-$order condition down in terms of a gross return on the asset and $M_{t+1}.$