Marginal utility pricing in a dividend economy ($S_t{\beta }^t{U}^{\prime }(C_t)={\mathbb{E}}_t[{\beta }^{t+1}{U}^{\prime }(C_{t+1})(S_{t+1}+d_{t+1})]$)

At an interior consumption and share-holding optimum, the Lagrange multiplier on the discrete dividend budget equation is ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. Varying the next holding by a variable measurable at date $t$ gives the displayed Euler equation. Thus the positive normalized process ${\zeta }_t={\Lambda }_t/{\Lambda }_0$ is a state-price density for traded dividend-inclusive gains. In an incomplete market, different investors can have different such densities; equality of their marginal-utility densities is an additional risk-sharing condition, not a consequence of trading one asset alone.

A positive state-price density prices a productive stock through a conditional dividend sum plus a future deflated-price term. The displayed condition removes that terminal term, yielding transversality and fundamental dividend prices. An individual consumption budget instead has terminal term $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$, so its vanishing must be imposed or proved for the particular holdings as well.

Iteration of the pricing Euler equation givesThe displayed fundamental price follows when the future dividend series is integrable and ${\mathbb{E}}_t[{\zeta }_N{S}_N]\to 0$. Without this dividend-price transversality condition, a deflated price martingale component can survive. For individual holdings, the analogous terminal term is $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$. Telescoping the discrete dividend budget equation gives the initial wealth cost of consumption minus this terminal term; its vanishing is needed for budget equality.