<blockquote>
<p>...the completeness of markets implies that the stochastic discount
factor is strictly positive...</p>
</blockquote>
<p>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.</p>
<p>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.)</p>
<p>For simplicity, assume agent's consumption is <span class="math-container">$\omega$</span>-contingent, where <span class="math-container">$\omega$</span> belongs to a finite set <span class="math-container">$\Omega$</span>.
Then the FOC
<span class="math-container">$$
p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right]
$$</span>
becomes
<span class="math-container">$$
p_t = \sum_{\omega \in \Omega}  \beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot x_{t+1} (\omega) \cdot P(\omega).
$$</span>
The quantity
<span class="math-container">$$
\beta \frac{u'(c_{t+1}(\omega))}{u'(c_t (\omega))} \cdot P(\omega)
$$</span>
can be viewed as the price of a claim with payoff <span class="math-container">$1_{\{ \omega' =\omega\}}(\omega')$</span> (i.e. a digital option that pays 1 if <span class="math-container">$\omega$</span> realizes and 0 otherwise).
Such claims are exactly the Arrow-Debreu securities in the market of <span class="math-container">$\omega$</span>-contingent claims.</p>
<p>(If the payoff <span class="math-container">$x_{t+1}$</span> is contingent on, say, state <span class="math-container">$(\omega, \nu)$</span>, where agent's consumption is invariant with respect to <span class="math-container">$\nu$</span>, then market is complete only with respect to <span class="math-container">$\omega$</span>-contingent claims.
Claims whose payoffs depend on <span class="math-container">$\nu$</span> cannot be replicated, i.e. market is not complete with respect to such claims.
In the FOC, the <span class="math-container">$\nu$</span>-coordinate would be integrated out.)</p>
<p>In equilibrium, the FOC becomes the asset pricing equation
<span class="math-container">$$
p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} p_{t+1}\right].
$$</span>
This means, in the market of AD securities of <span class="math-container">$\omega$</span>-contingent claims, the (representative) agent holds the portfolio that is the asset itself.</p>

CC BY-SA 4.0
2020-06-17T20:31:55.287Z
2020-06-22T22:32:28.917Z
2020-06-22T22:32:28.917Z
Michael
<p>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>
<p><span class="math-container">$$
p_t = \mathbb{E}\left[\beta \frac{u'(c_{t+1})}{u'(c_t)} x_{t+1}\right]
$$</span></p>
<p>where <span class="math-container">$p_t$</span> is the price of an asset giving <span class="math-container">$x_{t+1}$</span> in the following period.</p>

<finance><intertemporal-choice>
Proof that stochastic discount factor is positive in complete markets
Three Diag
# Proof that stochastic discount factor is positive in complete markets

(Asked by: username_0 on 2020-06-17)

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.

**Tags:** finance, intertemporal-choice

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## 1 answers

### Answer by username_1 on 2020-06-17. Score: 5

> ...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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