This simplification of BCG’s setup helps us by

`- creating a benchmark economy to compare with outcomes in BCG’s

- creating a benchmark economy to compare with outcomes in BCG’s

incomplete markets economy

- creating a good guess for initial values of some equilibrium objects

to be computed in BCG’s incomplete markets economy via an iterative

algorithm

- illustrating classic complete markets outcomes that include

- indeterminacy of consumers’ portfolio choices

- indeterminacy of firms' financial structures that underlies a

Modigliani-Miller theorem {cite}`Modigliani_Miller_1958

Modigliani-Miller theorem {cite}`Modigliani_Miller_1958`

- introducing `Big K, little k` issues in a simple context that will

recur in the BCG incomplete markets environment

In the complete markets economy in {doc}`BCG_complete_mkts <BCG_complete_mkts>`

`- there is a unique stochastic discount factor that prices all assets

- there is a unique stochastic discount factor that prices all assets

- consumers’ portfolio choices are indeterminate

- firms' financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem {cite}`Modigliani_Miller_1958

- firms' financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem {cite}`Modigliani_Miller_1958`

- the aggregate of all firms' financial structures are indeterminate, a consequence of there being redundant assets

In the incomplete markets economy studied here

`- there is a not a unique equilibrium stochastic discount factor

- there is a not a unique equilibrium stochastic discount factor

- different stochastic discount factors price different assets

- consumers’ portfolio choices are determinate

- while **individual** firms' financial structures are indeterminate, thus conforming to part of a Modigliani-Miller theorem,

{cite}`Modigliani_Miller_1958, the **aggregate** of all firms' financial structures **is** determinate.

{cite}`Modigliani_Miller_1958`, the **aggregate** of all firms' financial structures **is** determinate.

A `Big K, little k` analysis played an important role in the previous lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>`.

The code below adds some functions that generate plots for instances of the `AMF_LSS_VAR` {ref}`class <amf_lss>`.

```{code-cell} python3

---

tags: [output_scroll]

---

def plot_given_paths(amf, T, ypath, mpath, spath, tpath,

mbounds, sbounds, horline=0, show_trend=True):