## Introduction

This is an extension of an earlier lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>` about a **complete markets**

This is an extension of an earlier lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>` about a **complete markets**

model.

In contrast to that lecture, this one describes an instance of a model authored by Bisin, Clementi, and Gottardi {cite}`BCG_2018`

in which financial markets are **incomplete**.

Instead of being able to trade equities and a full set of one-period

Arrow securities as they can in {doc}`BCG_complete_mkts <BCG_complete_mkts>`, here consumers and firms trade only equity and a bond.

Arrow securities as they can in {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`, here consumers and firms trade only equity and a bond.

It is useful to watch how outcomes differ in the two settings.

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

In the complete markets economy in {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`

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

- consumers’ portfolio choices are indeterminate

- 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.

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

A `Big K, little k` analysis played an important role in the previous lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`.

A more subtle version of a `Big K, little k` features in the BCG incomplete markets environment here.

We use it to convey the heart of what BCG call a **rational conjectures** equilibrium in which conjectures are about

equilibrium pricing functions in regions of the state space that an average consumer or firm does not visit in equilibrium.

Note that the absence of complete markets means that we can compute competitive equilibrium prices and allocations by first solving

the simple planning problem that we did in {doc}`BCG_complete_mkts <BCG_complete_mkts>`.

the simple planning problem that we did in {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`.

Instead, we compute an equilibrium by solving a system of simultaneous inequalities.

### Measures of agents and firms

As in the companion lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>` that studies a complete markets version of

As in the companion lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>` that studies a complete markets version of

the model, we follow BCG in assuming that there are unit measures of

- consumers of type $i=1$

sits at the red dot in the above graph.

This contrasts sharply with the *unqualified* Modigliani-Miller theorem

descibed in the complete markets model in the lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>`.

descibed in the complete markets model in the lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`.

There the **market’s** financial structure was indeterminate.