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‎lectures/BCG_complete_mkts.md‎

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@@ -46,15 +46,15 @@ markets in one-period Arrow securities.
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This simplification of BCG’s setup helps us by
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`- creating a benchmark economy to compare with outcomes in BCG’s
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- creating a benchmark economy to compare with outcomes in BCG’s
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incomplete markets economy
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- creating a good guess for initial values of some equilibrium objects
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to be computed in BCG’s incomplete markets economy via an iterative
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algorithm
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- illustrating classic complete markets outcomes that include
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- indeterminacy of consumers’ portfolio choices
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- indeterminacy of firms' financial structures that underlies a
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Modigliani-Miller theorem {cite}`Modigliani_Miller_1958
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Modigliani-Miller theorem {cite}`Modigliani_Miller_1958`
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- introducing `Big K, little k` issues in a simple context that will
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recur in the BCG incomplete markets environment
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‎lectures/BCG_incomplete_mkts.md‎

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In the complete markets economy in {doc}`BCG_complete_mkts <BCG_complete_mkts>`
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`- there is a unique stochastic discount factor that prices all assets
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- there is a unique stochastic discount factor that prices all assets
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- consumers’ portfolio choices are indeterminate
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- firms' financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem {cite}`Modigliani_Miller_1958
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- firms' financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem {cite}`Modigliani_Miller_1958`
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- the aggregate of all firms' financial structures are indeterminate, a consequence of there being redundant assets
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In the incomplete markets economy studied here
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`- there is a not a unique equilibrium stochastic discount factor
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- there is a not a unique equilibrium stochastic discount factor
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- different stochastic discount factors price different assets
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- consumers’ portfolio choices are determinate
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- while **individual** firms' financial structures are indeterminate, thus conforming to part of a Modigliani-Miller theorem,
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{cite}`Modigliani_Miller_1958, the **aggregate** of all firms' financial structures **is** determinate.
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{cite}`Modigliani_Miller_1958`, the **aggregate** of all firms' financial structures **is** determinate.
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A `Big K, little k` analysis played an important role in the previous lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>`.
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‎lectures/additive_functionals.md‎

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The code below adds some functions that generate plots for instances of the `AMF_LSS_VAR` {ref}`class <amf_lss>`.
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```{code-cell} python3
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---
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tags: [output_scroll]
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---
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def plot_given_paths(amf, T, ypath, mpath, spath, tpath,
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mbounds, sbounds, horline=0, show_trend=True):
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