"Risk-Free Consol" in Asset Pricing I: Finite State Models

[!NOTE]
Triaged from the archiving of the legacy RST repo. Originally filed by @oyamad on 2019-11-18 as lecture-python#53. QuantEcon/lecture-python was renamed to lecture-python.rst and archived on 2026-07-16; this issue was moved here because it was verified to still apply to the live lecture. Archiving context: QuantEcon/workspace-lectures#17.

Verified still present 2026-07-16. The passage is unchanged in lectures/markov_asset.md at line 675 — "The price satisfies {eq}lteeqs0 with $d_t = \zeta$, or" — which, given $d_{t+1} = g_{t+1} d_t$ (line 277), literally implies $g_{t+1} = 1$ and $m_{t+1} = \beta$. Line 681 nonetheless proceeds with $p_t = E_t[\beta g_{t+1}^{-\gamma}(\zeta + p_{t+1})]$ and no reconciling text. Every commit touching the file since has been cosmetic (style guide, figsize, logo).

What is missing. The stochastic discount factor at lucsdf2 (lines 494-498) is licensed only by $c_t = d_t$ — an equilibrium condition of the Lucas tree model — yet the consol section imports it on the strength of one sentence at line 664 ("Consider the same pure exchange representative agent economy"). @jstac acknowledged the gap in the original thread ("Thanks @oyamad, I'll make those changes soon"), but the explanation never landed: grepping the file for "zero net supply", "net supply" or "additional asset" returns nothing. The missing content is that the consol is an additional asset in zero net supply, so consumption remains the tree's dividend and $g_{t+1}$ is still the tree's dividend growth.

While we were following the lecture on Asset Pricing I: Finite State Models, we couldn't really understand the description of the model of A Risk-Free Consol: What is the source of the stochasticity here? What makes m_{t+1} not equal to beta?