Market is complete in the Luca economy, with some caveats.
Market is complete (only) with respect to dividend-contingent claims
Consider the Lucas economy with one tree.
The representative investor's portfolio can be viewed as a portfolio of
dividend-contingent Arrow-Debreu securities. In particular, markets are complete with respect to dividend-contingent claims.
If $D_t$ is the exogenous dividend process and the investor has utility function $u$, the equilibrium price is characterized by the asset pricing (Euler) equation
$$
P_t = \sum_{\omega} e^{- \beta }\frac{ u'(D_{t+1})}{u'(D_t)} (\omega)) P_{t+1}(\omega) \cdot p(\omega)
$$
where $\omega$ is possible state of the world that occurs at date-$t+1$ with probability $p(\omega)$.
This decomposes the representative investor's portfolio---the Lucas tree---as a portfolio of Arrow-Debreu securities.
The date-$t$ price $Q(\omega')$ of the AD security whose date-$t+1$ payoff is $1_{\{ \omega = \omega' \}}$ is
$$
e^{- \beta }\frac{ u'(D_{t+1})}{u'(D_t)} (\omega')) \cdot p(\omega').
$$
The Lucas tree is a portfolio whose date-$t+1$ pay-off at state $\omega$ is $P_{t+1}(\omega)$. Therefore, like any portfolio in any AD economy, its date-$t$ price is sum of
$$
Q(\omega) \times P_{t+1}(\omega),
$$
over all possible states $\omega$.
It is therefore clear that market is complete with respect to $\omega$-contingent claims.
In principle, any investor in this economy has access to a full menu of AD securities, and any portfolio can be replicated by a portfolio of AD securities.
The equilibrium AD prices/stochastic discount factor is such that the optimal portfolio for the representative investor is the Lucas tree itself.
Suppose now there is a risk-free asset/saving technology.
If interest rate $r$ is given by
$$
e^{-r} = \sum_{\omega} Q(\omega),
$$
then the representative investor would not choose to lend or borrow.
This is the no-arbitrage condition for the zero coupon bond.
(If, say, $e^{-r} < \sum_{\omega} Q(\omega)$, the investor will attempt to save some of his wealth $P_t$ and only hold a fraction of the tree. There would be an excess demand of bonds.)
Therefore $r$ is the equilibrium interest rate that clears the bond market.
The asset pricing equation
\begin{align}
P_t & = \sum_{\omega} e^{- \beta + r}\frac{ u'(D_{t+1})}{u'(D_t)} (\omega)) p(\omega) \cdot e^{-r} P_{t+1}(\omega) \\
& = \sum_{\omega} \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')} \cdot e^{-r} P_{t+1}(\omega) \quad (1)
\end{align}
now says that, in a market with risk-free rate $r$ as well as AD securities,
the Lucas tree is the representative investor's optimal portfolio.
No-trade equilibrium
As is true for any representative investor model, the equilibrium is a no-trade equilibrium.
At each date-$t$, there are AD markets for $t+1$ claims.
In equilibrium, no trade occurs---the equilibrium prices in these markets are precisely the prices the ensures this market clearing condition holds on the equilibrium path.
But one can price derivatives using the stochastic discount factor/pricing kernel just the same. In the standard continuous-time formulation of the Lucas model, one can recover the Black-Scholes formula.
Non-completeness w.r.t. non-consumption risk
In the Lucas economy, only assets that are needed are those that enable the representative investor to hedge his consumption risk.
Market is not complete with respect to different states for which the dividend yield of the Lucas tree is the same.
Indeed, it has no reason to be.
In order to be induced to hold the asset (Lucas tree), the price of the asset must compensate the risk-averse representative investor for his consumption risk.
In equilibrium, he consumes the dividend. This means price, therefore the stochastic discount factor/pricing kernel, must be a function of dividend.
On the other hand, the representative investor does not care about risk unrelated to his consumption.
For example, if possible states of the world are given by $(\omega, \omega')$, and dividend $D(\omega)$ only depends on $\omega$, then market is only complete with respect to $\omega$-contingent claims.
This is referred to as "man-made uncertainty" in Ljungqvist and Sargent.
Black-Scholes formula in Lucas economy
In the standard continuous-time formulation of the Lucas model, one can recover
the Black-Scholes formula for pricing European call option in complete markets.
This is an example of market completeness with respect to dividend-contingent claims.
(In the language of mathematical finance, derivatives in complete markets are priced via the risk-neutral measure. Discounted prices are martingales under the risk-neutral measure.
This is already reflected in the discrete-time asset pricing equation $(1)$ from above:
\begin{align}
P_t & = \sum_{\omega} \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')} \cdot e^{-r} P_{t+1}(\omega).
\end{align}
The risk-neutral measure is $q(\omega) = \frac{ Q(\omega)}{\sum_{\omega'} Q(\omega')}$; it differs from the SDF by the discount factor $e^{-r}$.
The Black-Scholes formula is a continuous-time version of this.
)
Suppose the exogenous dividend process is given by
$$
\frac{d D}{D} = \mu dt + \sigma dW
$$
where $W$ is standard Brownian motion, and the investor maximizes expected utility
$$
E[\int_0^{\infty} e^{-\beta t} u(c_t) dt],
$$
over consumption stream $c_t$ adapted with respect to the filtration generated by $W$.
The asset pricing equation is
$$
P_0 = E[\int_0^{\infty} \frac{ e^{-\beta t} u'(D_t)}{u'(D_0)} D_t dt].
$$
Assume $u$ is the CRRA utility $u(c) = \frac{1}{1-\gamma} c^{1-\gamma}$.
Then $P_0$, which is just an expectation, can be computed directly:
$$
\frac{P_0}{D_0} = \frac{1}{-\beta + (1-\gamma) (\mu + \frac{1}{2} \sigma^2) + \frac{1}{2} (1-\gamma)^2 \sigma^2} \equiv \frac{1}{\delta}.
$$
So in equilibrium, price-dividend ratio $\frac{P}{D}$ is constant $\frac{1}{\delta}$ and price follows
$$
\frac{d P}{P} = \mu dt + \sigma dW.
$$
The cum-dividend return process is
$$
\frac{d P + D dt}{P} = (\mu + \delta) dt + \sigma dW. \quad (2)
$$
Suppose the equilibrium interest rate is $r$ in this economy, and time-0 price of Lucas tree is $P$. Let $E$ be the equilibrium price of an European call on the Lucas tree entered at time-$0$ maturing at time-$t$ with strike $K$. $E$ is given precisely by the standard Black-Scholes formula $C(r,P, K, t)$ for European call.
This follows from direct calculations of the pricing kernel $M = u'(D_t)$: Simply plugging in gives
$$
\frac{dM}{M} = (-\beta - \gamma \mu + \frac{1}{2}\gamma (1+\gamma) \sigma^2) dt - \gamma \sigma dW.
$$
Therefore equilibrium interest rate is
$$
r = \beta + \gamma \mu - \frac{1}{2}\gamma (1+\gamma) \sigma^2
$$
with standard observations regarding the three terms on the right-hand side---they reflect time preference, intertemporal substitution, and precautionary saving.
Therefore $\gamma \sigma = \frac{(\mu + \delta) - r}{\sigma}$
and
$$
\frac{dM}{M} = -r dt - \frac{(\mu + \delta) - r}{\sigma} dW.
$$
So the cum-dividend price process $(2)$, after discounting by $e^{-rt}$, is a martingale under risk-neutral density
$$
\frac{dL}{L} = - \frac{(\mu + \delta) - r}{\sigma} dW.
$$
This is exactly the Black-Scholes setting for pricing derivatives and the price of European call follows accordingly.
Market completeness in continuous-time
The mathematical statement that gives market completeness in continuous-time models is the Martingale Representation Theorem, which says every martingale with respect to a Brownian filtration can be represented as an Ito integral with respect to the Brownian generating that filtration.
The result is not true for general filtrations, i.e. if $W_t$ is a $(\mathcal{F_t})$-Brownian motion, it is not true in general that every $(\mathcal{F_t})$-martingale is a $dW$-integral.
This is consistent with the economic statement that market is complete only with respect to dividend-contingent claims.
In the Lucas/Black-Scholes example, if $W_t$ is a $(\mathcal{F_t})$-Brownian motion, then equilibrium price and SDF are measurable not only respect to $(\mathcal{F_t})$ but the minimal filtration generated $W_t$. In general, the minimal filtration is smaller than $(\mathcal{F_t})$.
Portfolio payoffs that can be hedged/replicated are only those that are measurable with respect to dividend (in equilibrium, price/SDF), i.e. the minimal filtration.
