updating sources from lecture-python-advanced (4cf6103) and tomyst (a3497b5)

mmcky · mmcky · commit 04e5dd09ce95 · 2020-10-30T09:49:47.000+11:00
diff --git a/lectures/BCG_complete_mkts.md b/lectures/BCG_complete_mkts.md
@@ -46,15 +46,15 @@ markets in one-period Arrow securities.
 
 This simplification of BCG&rsquo;s setup helps us by
 
-{cite}`Modigliani_Miller_1958`- creating a benchmark economy to compare with outcomes in BCG&rsquo;s
+`- creating a benchmark economy to compare with outcomes in BCG&rsquo;s
   incomplete markets economy
 - creating a good guess for initial values of some equilibrium objects
   to be computed in BCG&rsquo;s incomplete markets economy via an iterative
   algorithm
 - illustrating classic complete markets outcomes that include
     - indeterminacy of consumers&rsquo; portfolio choices
     - indeterminacy of firms' financial structures that underlies a
-      Modigliani-Miller theorem 
+      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
 
@@ -65,7 +65,7 @@ A Big K, little k analysis also played roles in [this quantecon lecture](https:/
 
 The economy lasts for two periods, $t=0, 1$.
 
-There are unit measure continua  of two types of consumers named $i=1,2$.
+There are two types of consumers named $i=1,2$.
 
 A scalar random variable $\epsilon$ with probability density $g(\epsilon)$ affects both
 
@@ -402,16 +402,18 @@ not now.
 
 ### Objects appearing in a competitive equilibrium
 
-We let
+Let
 
-- $a^i(\epsilon)$ be consumer $i$&rsquo;s purchases of claims
+- $a^i(\epsilon)$ be consumer $i$ &rsquo;s purchases of claims
   on time $1$ consumption in state $\epsilon$
 - $q(\epsilon)$ be a pricing kernel for one-period Arrow
   securities
 - $\theta_0^i \geq 0$ be consumer $i$&rsquo;s intial share of
   the firm, $\sum_i \theta_0^i =1$
 - $\theta^i$ be the fraction of the firm&rsquo;s shares purchased by
   consumer $i$ at time $t=0$
+- $- \bar a^i(\epsilon; \theta^i)$ be debt limits constraining consumer $i$ ' s issues of claims
+  on time $1$ consumption in state $\epsilon$
 - $V$ be the value of the representative firm
 - $\tilde V$ be the value of equity issued by the representative
   firm
@@ -471,39 +473,67 @@ As a price taker, each consumer faces a given Arrow securities pricing kernel
 $q(\epsilon)$, a given value of a firm $V$ that has chosen capital stock $k$, a price of
 equity $\tilde V$, and given prospective random dividends next period $A k^\alpha e^\epsilon$.
 
+Consumer $i$ also faces a state-by-state borrowing limit that restricts quantities of Arrow securities that he can issue.
+
+If we evaluate consumer $i$'s time $1$ budget constraint at zero consumption $c^i(\epsilon) = 0$ and solve for $-a^i(\epsilon)$
+we obtain
+
+$$
+- \bar a^i(\epsilon;\theta^i) = w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon
+$$
+
+The quantity $- \bar a^i(\epsilon;\theta^i)$ is the maximum amount that it is feasible for  consumer $i$ to repay to
+his Arrow security creditors at time $1$ in state $\epsilon$.
+
+To implement trading with one-period Arrow securities, we must impose on agent $i$ the state-by-state debt limits
+
+$$
+-a^i(\epsilon) \leq -  \bar a^i(\epsilon;\theta^i)
+$$
+
+Notice that consumer $i$'s borrowing limit depends on
+
+* his endowment at time $1$ in state $\epsilon$
+* his share $\theta^i$ of a representive firm's dividends
+
+These are the two sources of **collateral** that back the consumer's issues of Arrow securities that pay off in state $\epsilon$
+
 Consumer $i$ chooses a scalar $c_0^i$ and a function
 $c_1^i(\epsilon)$ to maximize
 
 $$
 u(c_0^i) + \beta \int u(c_1^i(\epsilon)) g (\epsilon) d \epsilon
 $$
 
-subject to time $0$ and time $1$ budget constraints
+subject to his state-by-state debt limits and the time $0$ and time $1$ budget constraints
 
 $$
 \begin{aligned}
 c_0^i & \leq w_0^i +\theta_0^i V - \int q(\epsilon) a^i(\epsilon) d \epsilon - \theta^i \tilde V \cr
-c_1^i & \leq w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon + a^i(\epsilon)
+c_1^i(\epsilon) & \leq w_1^i(\epsilon) +\theta^i A k^\alpha e^\epsilon + a^i(\epsilon)
 \end{aligned}
 $$
 
 Attach Lagrange multiplier $\lambda_0^i$ to the budget constraint
-at time $0$ and scaled Lagrange multiplier
+at time $0$,  scaled Lagrange multiplier
 $\beta \lambda_1^i(\epsilon) g(\epsilon)$ to the budget constraint
-at time $1$ and state $\epsilon$ and form the Lagrangian
+at time $1$ and state $\epsilon$, and scaled Lagrange multiplier
+$\beta \phi_1^i(\epsilon) g(\epsilon)$ to the debt limit  at time $1$ and state $\epsilon$,
+then  form the Lagrangian
 
 $$
 \begin{aligned}
 L^i & = u(c_0^i) + \beta \int u(c^i_1(\epsilon)) g(\epsilon) d \epsilon \cr
      & + \lambda_0^i [ w_0^i + \theta_0^i - \int q(\epsilon) a^i(\epsilon) d \epsilon -
           \theta^i \tilde V - c_0^i ] \cr
       & + \beta \int \lambda_1^i(\epsilon) [ w_1^i(\epsilon) + \theta^i A k^\alpha e^\epsilon
-           + a^i(\epsilon) c_1^i(\epsilon) g(\epsilon)] d \epsilon
+           + a^i(\epsilon) c_1^i(\epsilon) ] g(\epsilon) d \epsilon \cr
+      & + \beta \int \phi_1^i(\epsilon)  [ - \bar a^i(\epsilon; \theta^i) - a^i(\epsilon) ]    g(\epsilon) d \epsilon
 \end{aligned}
 $$
 
-First-order necessary conditions for an optimum with respect to
-$c_0^i, c_1^i(\epsilon)$, and $a^i(\epsilon)$ are
+Off corners, first-order necessary conditions for an optimum with respect to
+$c_0^i, c_1^i(\epsilon)$ and $a^i(\epsilon)$ are
 
 $$
 \begin{aligned}
@@ -522,7 +552,7 @@ plan to satisfy
 q(\epsilon) = \beta \left( \frac{u'(c_1^i(\epsilon))}{u'(c_0^i)} \right) g(\epsilon)
 ```
 
-To deduce a restriction on the consumer&rsquo;s choice of $\theta^i$ we
+To deduce a restriction on equilibrium prices,  we
 solve the period $1$ budget constraint to express
 $a^i(\epsilon)$ as
 
diff --git a/lectures/BCG_incomplete_mkts.md b/lectures/BCG_incomplete_mkts.md
@@ -50,18 +50,18 @@ 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>`
 
-{cite}`Modigliani_Miller_1958`- there is a unique  stochastic discount factor that prices all assets
+`- there is a unique  stochastic discount factor that prices all assets
 - consumers&rsquo; portfolio choices are indeterminate
-- firms' financial structures are indeterminate, so the model embodies an instance of a Modigliani-Miller irrelevance theorem 
+- 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
 
-{cite}`Modigliani_Miller_1958`- 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&rsquo; portfolio choices are determinate
 - while **individual** firms' financial structures are indeterminate, thus conforming to part of a Modigliani-Miller theorem,
-  , 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>`.
 
diff --git a/lectures/amss2.md b/lectures/amss2.md
@@ -75,7 +75,7 @@ Nonzero Lagrange multipliers on those constraints make the Ramsey allocation for
 
 This lecture studies a special  AMSS model in which
 
-{cite}`LucasStokey1983`* The exogenous state variable $s_t$ is governed by  a finite-state Markov chain.
+`* The exogenous state variable $s_t$ is governed by  a finite-state Markov chain.
 * With an arbitrary budget-feasible initial level of government debt, the measurability  constraints
     - bind for many periods, but $\ldots$.
     - eventually, they stop binding evermore, so $\ldots$.
@@ -92,7 +92,7 @@ This lecture studies a special  AMSS model in which
     - the **market value** $\frac{\bar b}{R_t(s_t)}$ of government debt at time $t$  varies as a time-invariant function of the Markov state $s_t$.
     - fluctuations in the interest rate make gross earnings on government debt $\frac{\bar b}{R_t(s_t)}$ fully insure the gross-of-gross-interest-payments government budget against fluctuations in government expenditures.
     - the state variable $x$ in a recursive representation of a Ramsey plan is a time-invariant function of the Markov state for $t \geq 0$.
-* In this special case, the Ramsey allocation in the AMSS model agrees with that in a  model in which
+* 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.
 * This outcome emerges only when we initialize government debt at a particular $b_0 < 0$.
diff --git a/lectures/amss3.md b/lectures/amss3.md
@@ -964,9 +964,9 @@ ergodic distribution.
 To  construct the BEGS  approximation to ${\mathcal B}^*$, we just follow steps set forth on pages 648 - 650 of section III.D of
 {cite}`BEGS1`
 
-{cite}`BEGS1`- notation in BEGS might be confusing at first sight, so
+`- notation in BEGS might be confusing at first sight, so
   it is important to stare and digest before computing
-- there are also some sign errors in the  text that we'll want
+- there are also some sign errors in the {cite}`BEGS1 text that we'll want
   to correct
 
 Here is a step-by-step description of the {cite}`BEGS1` approximation procedure.
diff --git a/lectures/arellano.md b/lectures/arellano.md
@@ -321,7 +321,7 @@ After that we'll perform some additional simulations.
 
 We use a slightly modified version of the algorithm recommended by Arellano.
 
-{cite}`arellano2008default`* The appendix to  recommends value function iteration until
+`* The appendix to {cite}`arellano2008default recommends value function iteration until
   convergence, updating the price, and then repeating.
 * Instead, we update the bond price at every value function iteration step.
 
diff --git a/lectures/arma.md b/lectures/arma.md
@@ -74,19 +74,19 @@ The famous *Fourier transform* and its inverse are used to map between the two r
 For supplementary reading, see
 
 ```{only} html
-{cite}`Ljungqvist2012`{cite}`Sargent1987`<a href=/_static/lecture_specific/arma/time_series_book.pdf download></a>{cite}`Shiryaev1995`{cite}`CryerChan2008`* , chapter 2
-* , chapter 11
+``<a href=/_static/lecture_specific/arma/time_series_book.pdf download></a>``* {cite}`Ljungqvist2012, chapter 2
+* {cite}`Sargent1987, chapter 11
 * John Cochrane's notes on time series analysis, chapter 8
-* , chapter 6
-* , all
+* {cite}`Shiryaev1995, chapter 6
+* {cite}`CryerChan2008, all
 ```
 
 ```{only} latex
-{cite}`Ljungqvist2012`{cite}`Sargent1987`{cite}`Shiryaev1995`{cite}`CryerChan2008`* , chapter 2
-* , chapter 11
+````* {cite}`Ljungqvist2012, chapter 2
+* {cite}`Sargent1987, chapter 11
 * John Cochrane's [notes on time series analysis](https://lectures.quantecon.org/_downloads/time_series_book.pdf), chapter 8
-* , chapter 6
-* , all
+* {cite}`Shiryaev1995, chapter 6
+* {cite}`CryerChan2008, all
 ```
 
 Let's start with some imports:
diff --git a/lectures/black_litterman.md b/lectures/black_litterman.md
@@ -530,7 +530,7 @@ particular density value.
 The curves defined by equation {eq}`ellipse` can be
 labeled as iso-likelihood ellipses
 
-{cite}`leamer1978specification`> **Remark:** More generally there is a class of density functions
+`> **Remark:** More generally there is a class of density functions
 > that possesses this feature, i.e.
 > 
 > $$
@@ -540,7 +540,7 @@ labeled as iso-likelihood ellipses
 > $$
 > 
 > This property is called **spherical symmetry** (see p 81. in Leamer
-> (1978) ).
+> (1978) {cite}`leamer1978specification).
 
 In our specific example, we can use the pair
 $(\bar d_1, \bar d_2)$ as being two "likelihood" values for which
diff --git a/lectures/chang_ramsey.md b/lectures/chang_ramsey.md
@@ -40,11 +40,11 @@ analyzed **competitive equilibria** and a best competitive equilibrium called a
 
 He did this by
 
-{cite}`APS1990`* characterizing a competitive equilibrium recursively in a way also employed
+`* characterizing a competitive equilibrium recursively in a way also employed
   in the {doc}`dynamic Stackelberg problems <dyn_stack>` and {doc}`Calvo model <calvo>` lectures
   to pose Stackelberg problems in linear economies, and then
 * appropriately adapting an argument of Abreu, Pearce, and Stachetti
-   to describe key features of the  set of competitive equilibria
+  {cite}`APS1990 to describe key features of the  set of competitive equilibria
 
 Roberto Chang {cite}`chang1998credible`  chose a model of  Calvo {cite}`Calvo1978`
 as a simple structure that conveys ideas that apply more broadly.
diff --git a/lectures/discrete_dp.md b/lectures/discrete_dp.md
@@ -98,7 +98,12 @@ seamlessly if you are using [Anaconda](https://www.anaconda.com/download/) as [s
 
 For background reading on dynamic programming and additional applications, see, for example,
 
-{cite}`Ljungqvist2012`{cite}`HernandezLermaLasserre1996`{cite}`puterman2005`{cite}`StokeyLucas1989`{cite}`Rust1996`{cite}`MirandaFackler2002`* , section 3.5
+``````* {cite}`Ljungqvist2012
+* {cite}`HernandezLermaLasserre1996, section 3.5
+* {cite}`puterman2005
+* {cite}`StokeyLucas1989
+* {cite}`Rust1996
+* {cite}`MirandaFackler2002
 * [EDTC](http://johnstachurski.net/edtc.html), chapter 5
 
 (discrete_dps)=
@@ -149,7 +154,7 @@ In particular, a stationary Markov policy is a map $\sigma$ from states to actio
 
 It is known that, for any arbitrary policy, there exists a stationary Markov policy that dominates it at least weakly.
 
-{cite}`puterman2005`* See section 5.5 of  for discussion and proofs.
+`* See section 5.5 of {cite}`puterman2005 for discussion and proofs.
 
 In what follows, stationary Markov policies are referred to simply as policies.
 
diff --git a/lectures/robustness.md b/lectures/robustness.md
@@ -156,7 +156,8 @@ If you want to understand more about why one serious quantitative researcher is
 
 Our discussion in this lecture is based on
 
-{cite}`HansenSargent2000`{cite}`HansenSargent2008`
+``* {cite}`HansenSargent2000
+* {cite}`HansenSargent2008
 
 ## The Model
 
diff --git a/lectures/smoothing.md b/lectures/smoothing.md
@@ -40,8 +40,8 @@ tags: [hide-output]
 
 This lecture describes two types of consumption-smoothing models.
 
-{cite}`Hall1978`* one is in the **complete markets** tradition of Kenneth Arrow <https://en.wikipedia.org/wiki/Kenneth_Arrow>
-* the other is in the **incomplete markets** tradition  of Hall 
+`* one is in the **complete markets** tradition of Kenneth Arrow <https://en.wikipedia.org/wiki/Kenneth_Arrow>
+* the other is in the **incomplete markets** tradition  of Hall {cite}`Hall1978
 
 *Complete markets* allow a consumer  to buy or sell claims contingent on all possible states of the world.
 
diff --git a/lectures/smoothing_tax.md b/lectures/smoothing_tax.md
@@ -40,8 +40,8 @@ tags: [hide-output]
 
 This lecture describes tax-smoothing models that are counterparts to consumption-smoothing models in {doc}`smoothing <smoothing>`.
 
-{cite}`LucasStokey1983`{cite}`Barro1979`* one is in the **complete markets** tradition of Lucas and Stokey .
-* the other is in the **incomplete markets** tradition  of  Barro .
+``* one is in the **complete markets** tradition of Lucas and Stokey {cite}`LucasStokey1983.
+* the other is in the **incomplete markets** tradition  of  Barro {cite}`Barro1979.
 
 *Complete markets* allow a  government to buy or sell claims contingent on all possible Markov states.
 
diff --git a/lectures/stationary_densities.md b/lectures/stationary_densities.md
@@ -673,8 +673,8 @@ For one such set of conditions see theorem 8.2.14 of [EDTC](http://johnstachursk
 
 In addition
 
-{cite}`StokeyLucas1989`{cite}`LasotaMackey1994`{cite}`MeynTweedie2009`*   contains a classic (but slightly outdated) treatment of these topics.
-* From the mathematical literature,   and  give outstanding in-depth treatments.
+```* {cite}`StokeyLucas1989  contains a classic (but slightly outdated) treatment of these topics.
+* From the mathematical literature, {cite}`LasotaMackey1994  and {cite}`MeynTweedie2009 give outstanding in-depth treatments.
 * Section 8.1.2 of [EDTC](http://johnstachurski.net/edtc.html) provides detailed intuition, and section 8.3 gives additional references.
 * [EDTC](http://johnstachurski.net/edtc.html), section 11.3.4
   provides a specific treatment for the growth model we considered in this
