= Solution
Take $S_t$ to be the ex-dividend <stock> price: the date-$t$ output has already been paid when trading occurs. Receiving $\theta_td_t$ units of fruit and selling $\theta_t-\theta_{t+1}$ shares gives the <discrete dividend budget equation>
$$
\boxed{C_t=\theta_td_t+(\theta_t-\theta_{t+1})S_t,\qquad C_t+S_t\theta_{t+1}=(S_t+d_t)\theta_t.}
$$
The entering holding $\theta_t$ is chosen with date-$(t-1)$ information, while $C_t$ and the next holding $\theta_{t+1}$ may use date-$t$ information. The output is the consumption good and price numéraire; its stochastic law is exogenous to the individual's price-taking choice.
For the interior differentiable case, attach an adapted <Lagrange multiplier> $\Lambda_t$ to each <budget constraint>. On a finite horizon, with the terminal holding fixed or its continuation value included, the <optimization Lagrangian> is
$$
\mathcal L=\mathbb E\sum_t\left\{\beta^tU(C_t)+\Lambda_t\big[(S_t+d_t)\theta_t-S_t\theta_{t+1}-C_t\big]\right\}.
$$
An adapted variation in $C_t$ gives $\Lambda_t=\beta^tU'(C_t)$. An arbitrary date-$t$ measurable variation in $\theta_{t+1}$ affects two adjacent terms. Its coefficient must have zero conditional mean, so
$$
\boxed{\Lambda_tS_t=\mathbb E[\Lambda_{t+1}(S_{t+1}+d_{t+1})\mid\mathcal F_t],\qquad\Lambda_t=\beta^tU'(C_t).}
$$
These are the Euler equations of <marginal utility pricing in a dividend economy>. If the <utility function> is merely concave, a supporting slope replaces $U'$; binding consumption or trading restrictions give the corresponding inequalities rather than unconstrained equalities.
With known initial information and $\Lambda_0>0$, normalize $\zeta_t=\Lambda_t/\Lambda_0$. This is a <state-price density> for the traded asset. In particular,
$$
S_t=\mathbb E_t\left[\frac{\zeta_{t+1}}{\zeta_t}(S_{t+1}+d_{t+1})\right],\qquad
\frac{\zeta_{t+1}}{\zeta_t}=\beta\frac{U'(C_{t+1})}{U'(C_t)}.
$$
The ratio is the stochastic discount factor: payment in a state with greater marginal utility has a higher marginal value. For a priced contingent payoff $H$ at $t+1$, its price is $\mathbb E_t[(\zeta_{t+1}/\zeta_t)H]$; pricing arbitrary claims uniquely also requires <market completeness>. The deflated cumulative gains $\zeta_tS_t+\sum_{s=1}^t\zeta_sd_s$ form a <martingale> whenever the displayed terms are integrable.
Iterating the Euler equation gives
$$
\zeta_tS_t=\mathbb E_t\left[\sum_{s=t+1}^N\zeta_sd_s+\zeta_NS_N\right].
$$
Under the <dividend-price transversality condition> and the needed integrability, this becomes the fundamental price
$$
\boxed{S_t=\frac1{\zeta_t}\mathbb E_t\sum_{s>t}\zeta_sd_s.}
$$
Thus an infinite-horizon price is not automatically just a dividend sum: its terminal term must vanish. Telescoping the individual <budget constraints> similarly gives
$$
\mathbb E\sum_{t=0}^N\zeta_tC_t
=\zeta_0\theta_0(S_0+d_0)-\mathbb E[\zeta_NS_N\theta_{N+1}].
$$
A no-Ponzi lower restriction on this final term gives the consumption budget inequality; its vanishing gives equality. Together with the supporting-line inequality for a <concave> <utility function>, the Euler equations and these terminal conditions supply the usual sufficiency argument for an admissible candidate. In an <incomplete market>, each optimal agent's marginal utility can supply a different <state-price density> pricing the same traded tree.
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