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I am looking for a reference (with a proof possibly) to understand why the completeness of markets implies that the stochastic discount factor is strictly positive in the context of intertemporal consumption, i.e. when a agent FOC implies

$$ p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right] $$

where $p_t$ is the price of an asset giving $x_{t+1}$ in the following period.

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...the completeness of markets implies that the stochastic discount factor is strictly positive...

This statement is not quite correct. Rather, the agent's optimality condition implies that market is complete with respect to the agent's consumption-relevant states, and that the SDF must be strictly positive on those states.

The SDF is, more or less, same as Arrow-Debreu prices on consumption-relevant states, which must be strictly positive. Otherwise equilibrium would not exist. Marginal utility cannot be zero in equilibrium. (SDF is the Radon-Nikodym derivative of AD prices with respect to the physical measure. SDF is strictly positive if and only if AD prices are.)

For simplicity, assume agent's consumption is $\omega$-contingent, where $\omega$ belongs to a finite set $\Omega$. Then the FOC $$ p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right] $$ becomes $$ p_t = \sum_{\omega \in \Omega} \beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot x_{t+1} (\omega) \cdot P(\omega). $$ The quantity $$ \beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot P(\omega) $$ can be viewed as the price of a claim with payoff $1_{\{ \omega' =\omega\}}(\omega')$ (i.e. a digital option that pays 1 if $\omega$ realizes and 0 otherwise). Such claims are exactly the Arrow-Debreu securities in the market of $\omega$-contingent claims.

(If the payoff $x_{t+1}$ is contingent on, say, state $(\omega, \nu)$, where agent's consumption is invariant with respect to $\nu$, then market is complete only with respect to $\omega$-contingent claims. Claims whose payoffs depend on $\nu$ cannot be replicated, i.e. market is not complete with respect to such claims. In the FOC, the $\nu$-coordinate would be integrated out.)

In equilibrium, the FOC becomes the asset pricing equation $$ p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} p_{t+1}\right]. $$ This means, in the market of AD securities of $\omega$-contingent claims, the (representative) agent holds the portfolio that is the asset itself.

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