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Copy file name to clipboardExpand all lines: lectures/BCG_incomplete_mkts.md
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## Introduction
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This is an extension of an earlier lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>` about a **complete markets**
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This is an extension of an earlier lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>` about a **complete markets**
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model.
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In contrast to that lecture, this one describes an instance of a model authored by Bisin, Clementi, and Gottardi {cite}`BCG_2018`
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in which financial markets are **incomplete**.
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Instead of being able to trade equities and a full set of one-period
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Arrow securities as they can in {doc}`BCG_complete_mkts <BCG_complete_mkts>`, here consumers and firms trade only equity and a bond.
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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.
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It is useful to watch how outcomes differ in the two settings.
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In the complete markets economy in {doc}`BCG_complete_mkts <BCG_complete_mkts>`
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In the complete markets economy in {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`
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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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- 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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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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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>`.
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A more subtle version of a `Big K, little k` features in the BCG incomplete markets environment here.
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We use it to convey the heart of what BCG call a **rational conjectures** equilibrium in which conjectures are about
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equilibrium pricing functions in regions of the state space that an average consumer or firm does not visit in equilibrium.
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Note that the absence of complete markets means that we can compute competitive equilibrium prices and allocations by first solving
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the simple planning problem that we did in {doc}`BCG_complete_mkts <BCG_complete_mkts>`.
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the simple planning problem that we did in {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`.
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Instead, we compute an equilibrium by solving a system of simultaneous inequalities.
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### Measures of agents and firms
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As in the companion lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>` that studies a complete markets version of
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As in the companion lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>` that studies a complete markets version of
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the model, we follow BCG in assuming that there are unit measures of
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- consumers of type $i=1$
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sits at the red dot in the above graph.
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This contrasts sharply with the *unqualified* Modigliani-Miller theorem
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descibed in the complete markets model in the lecture {doc}`BCG_complete_mkts <BCG_complete_mkts>`.
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descibed in the complete markets model in the lecture {doc}`Irrelevance of Capital Structure with Complete Markets <BCG_complete_mkts>`.
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There the **market’s** financial structure was indeterminate.
Copy file name to clipboardExpand all lines: lectures/classical_filtering.md
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### Implementation
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Here's the code that computes solutions to LQ control and filtering problems using the methods described here and in {doc}`lu_tricks <lu_tricks>`.
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Here's the code that computes solutions to LQ control and filtering problems using the methods described here and in {doc}`Classical Control with Linear Algebra <lu_tricks>`.
This formula is useful in solving stochastic versions of problem 1 of lecture {doc}`lu_tricks <lu_tricks>` in which the randomness emerges because $\{a_t\}$ is a stochastic
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This formula is useful in solving stochastic versions of problem 1 of lecture {doc}`Classical Control with Linear Algebra <lu_tricks>` in which the randomness emerges because $\{a_t\}$ is a stochastic
Copy file name to clipboardExpand all lines: lectures/smoothing_tax.md
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## Overview
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This lecture describes tax-smoothing models that are counterparts to consumption-smoothing models in {doc}`smoothing <smoothing>`.
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This lecture describes tax-smoothing models that are counterparts to consumption-smoothing models in {doc}`Consumption Smoothing with Complete and Incomplete Markets <smoothing>`.
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* one is in the **complete markets** tradition of Lucas and Stokey {cite}`LucasStokey1983`.
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* the other is in the **incomplete markets** tradition of Barro {cite}`Barro1979`.
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* a consumer's nonfinancial income as a government's purchases
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* a consumer's *debt* as a government's *assets*
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Thus, we can convert the consumption-smoothing models in lecture {doc}`smoothing <smoothing>` into tax-smoothing models by setting
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Thus, we can convert the consumption-smoothing models in lecture {doc}`Consumption Smoothing with Complete and Incomplete Markets <smoothing>` into tax-smoothing models by setting
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$c_t = T_t$, $y_t = G_t$, and $- b_t = a_t$, where $T_t$ is total tax
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collections, $\{G_t\}$ is an exogenous government expenditures
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process, and $a_t$ is the government's holdings of one-period risk-free bonds coming maturing at the due at the beginning of time $t$.
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import scipy.linalg as la
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```
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To exploit the isomorphism between consumption-smoothing and tax-smoothing models, we simply use code from {doc}`smoothing <smoothing>`
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To exploit the isomorphism between consumption-smoothing and tax-smoothing models, we simply use code from {doc}`Consumption Smoothing with Complete and Incomplete Markets <smoothing>`
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### Code
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In {doc}`optimal taxation in an LQ economy <lqramsey>` and {doc}`recursive optimal taxation <opt_tax_recur>`, we study **complete-markets**
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models in which the government recognizes that it can manipulate Arrow securities prices.
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Linear-quadratic versions of the Lucas-Stokey tax-smoothing model are described in {doc}`lqramsey <lqramsey>`.
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Linear-quadratic versions of the Lucas-Stokey tax-smoothing model are described in {doc}`Optimal Taxation in an LQ Economy <lqramsey>`.
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That lecture is a warm-up for the non-linear-quadratic model of tax smoothing described in {doc}`opt_tax_recur <opt_tax_recur>`.
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That lecture is a warm-up for the non-linear-quadratic model of tax smoothing described in {doc}`Optimal Taxation with State-Contingent Debt <opt_tax_recur>`.
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In both {doc}`lqramsey <lqramsey>` and {doc}`opt_tax_recur <opt_tax_recur>`, the government recognizes that its decisions affect prices.
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In both {doc}`Optimal Taxation in an LQ Economy <lqramsey>` and {doc}`Optimal Taxation with State-Contingent Debt <opt_tax_recur>`, the government recognizes that its decisions affect prices.
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In {doc}`optimal taxation with incomplete markets <amss>`, we study an **incomplete-markets** model in which the
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government also manipulates prices of government debt.
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