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15 changes: 6 additions & 9 deletions rst_files/amss2.rst
Original file line number Diff line number Diff line change
Expand Up @@ -61,26 +61,23 @@ This lecture studies a special AMSS model in which

- the one-period gross interest rate :math:`R_t(s^t)` on risk-free debt converges to a time-invariant function of the Markov state :math:`s_t`


* For a **particular** :math:`b_0 < 0` (i.e., a positive level of initial government **loans** to the private sector), the measurability constraints **never** bind



* In this special case

- the **par value** :math:`b_{t+1}(s_t) = \bar b` of government debt at time :math:`t` and Markov state :math:`s_t` is constant across time and states,
- the **par value** :math:`b_{t+1}(s_t) = \bar b` of government debt at time :math:`t` and Markov state :math:`s_t` is constant across time and states,
but :math:`\ldots`

- the **market value** :math:`\frac{\bar b}{R_t(s_t)}` of government debt at time :math:`t` varies as a time-invariant function of the Markov state :math:`s_t`
- the **market value** :math:`\frac{\bar b}{R_t(s_t)}` of government debt at time :math:`t` varies as a time-invariant function of the Markov state :math:`s_t`

- fluctuations in the interest rate make gross earnings on government debt :math:`\frac{\bar b}{R_t(s_t)}` fully insure the gross-of-gross-interest-payments government budget against fluctuations in government expenditures
- fluctuations in the interest rate make gross earnings on government debt :math:`\frac{\bar b}{R_t(s_t)}` fully insure the gross-of-gross-interest-payments government budget against fluctuations in government expenditures

- the state variable :math:`x` in a recursive representation of a Ramsey plan is a time invariant function of the Markov state for :math:`t \geq 0`
- the state variable :math:`x` in a recursive representation of a Ramsey plan is a time invariant function of the Markov state for :math:`t \geq 0`

* In this special case, the Ramsey allocation in the AMSS model agrees with that in a :cite:`LucasStokey1983` model in which
the same amount of state-contingent debt falls due in all states tomorrow

- it is a situation in which the Ramsey planner loses nothing from not being able to purchase state-contingent debt and being restricted to exchange only risk-free debt debt
- it is a situation in which the Ramsey planner loses nothing from not being able to purchase state-contingent debt and being restricted to exchange only risk-free debt debt

* This outcome emerges only when we initialize government debt at a particular :math:`b_0 < 0`

Expand Down Expand Up @@ -781,4 +778,4 @@ Now let's compute the implied mean time to get to within .01 of the limit
print(f"Time to get within .01 of limit = {ttime}")

The slow rate of convergence and the implied time of getting within one percent of the limiting value do a good job of approximating
our long simulation above
our long simulation above