[styleguide] adjust lecture titles to match the QuantEcon style guide rules around capitalisations. (#240)

* Update additive_functionals.md

* Update amss.md

* Update amss2.md

* amss and arellano

* Update arma.md

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* Update calvo.md

* update

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* Update lectures/calvo.md

Co-authored-by: Matt McKay <mmcky@users.noreply.github.com>

---------

Co-authored-by: Matt McKay <mmcky@users.noreply.github.com>

Author: longye-tian

lectures/BCG_incomplete_mkts.md

@@ -242,7 +242,7 @@ show up in differences in the two types of consumers’ demands for a
 typical firm’s bonds and equity, the only two assets that agents can now
 trade.
 
-## Asset Markets
+## Asset markets
 
 Markets are incomplete: *ex cathedra* we the model builders declare that only equities and bonds issued by representative
 firms can be traded.
@@ -548,7 +548,7 @@ $C$’s that appear in the pricing functions, then
 - $\check q = q(K,B)$ and
   $\check p = p(K,B)$.
 
-## Pseudo Code
+## Pseudo code
 
 Before displaying our Python code for computing a BCG incomplete markets equilibrium,
 we’ll sketch some pseudo code that describes its logical flow.

lectures/additive_functionals.md

@@ -77,7 +77,7 @@ import matplotlib.pyplot as plt
 from scipy.stats import norm, lognorm
 ```
 
-## A Particular Additive Functional
+## A particular additive functional
 
 {cite}`Hansen_2012_Eca` describes a general class of additive functionals.
 
@@ -123,7 +123,7 @@ initial condition for $y$.
 The nonstationary random process $\{y_t\}_{t=0}^\infty$ displays
 systematic but random *arithmetic growth*.
 
-### Linear State-Space Representation
+### Linear state-space representation
 
 A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
 
@@ -872,7 +872,7 @@ Notice tell-tale signs of these probability coverage shaded areas
 * the green one for the stationary component $s_t$ converges to a
   constant band
 
-### Associated Multiplicative Functional
+### Associated multiplicative functional
 
 Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.
 
@@ -929,7 +929,7 @@ It is interesting to how the martingale behaves as $T \rightarrow +\infty$.
 
 Let's see what happens when we set $T = 12000$ instead of $150$.
 
-### Peculiar Large Sample Property
+### Peculiar large sample property
 
 Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the  martingale component
 $\widetilde M_t$ of the multiplicative decomposition
@@ -958,7 +958,7 @@ It remains constant at unity, illustrating the first property.
 
 The purple 95 percent frequency coverage interval collapses around zero, illustrating the second property.
 
-## More About the Multiplicative Martingale
+## More about the multiplicative martingale
 
 Let's drill down and study probability distribution of the multiplicative martingale  $\{\widetilde M_t\}_{t=0}^\infty$  in
 more detail.
@@ -973,7 +973,7 @@ where $H =  [F + D(I-A)^{-1} B]$.
 
 It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.
 
-### Simulating a Multiplicative Martingale Again
+### Simulating a multiplicative martingale again
 
 Next, we want a program to simulate the likelihood ratio process $\{ \tilde{M}_t \}_{t=0}^\infty$.
 
@@ -984,7 +984,7 @@ After accomplishing this, we want to display and study histograms of $\tilde{M}_
 
 Here is code that accomplishes these tasks.
 
-### Sample Paths
+### Sample paths
 
 Let's write a program to simulate sample paths of $\{ x_t, y_{t} \}_{t=0}^{\infty}$.
 
@@ -1257,7 +1257,7 @@ These probability density functions help us understand mechanics underlying the
 * Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$
   even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
 
-### Multiplicative Martingale as Likelihood Ratio Process
+### Multiplicative martingale as likelihood ratio process
 
 [This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes**
 and **likelihood ratio processes**.

lectures/amss.md

@@ -61,7 +61,7 @@ In this lecture, we
 
 We begin with an introduction to the model.
 
-## Competitive Equilibrium with Distorting Taxes
+## Competitive equilibrium with distorting taxes
 
 Many but not all features of the economy are identical to those of {doc}`the Lucas-Stokey economy <opt_tax_recur>`.
 
@@ -118,7 +118,7 @@ AMSS allow the government to issue only one-period risk-free debt each period.
 
 Ruling out complete markets in this way is a step in the direction of making total tax collections behave more like that prescribed in Robert Barro (1979) {cite}`Barro1979` than they do in Lucas and Stokey (1983) {cite}`LucasStokey1983`.
 
-### Risk-free One-Period Debt Only
+### Risk-free one-period debt only
 
 In period $t$ and history $s^t$, let
 
@@ -244,7 +244,7 @@ b_t(s^{t-1}) =  \mathbb E_t \sum_{j=0}^\infty \beta^j
 
 Equation {eq}`TS_gov_wo4a` must hold for each $s^t$ for each $t \geq 1$.
 
-### Comparison with Lucas-Stokey Economy
+### Comparison with Lucas-Stokey economy
 
 The expression on the right side of {eq}`TS_gov_wo4a` in the Lucas-Stokey (1983) economy would  equal the present value of a continuation stream of government net-of-interest surpluses evaluated at what would be competitive equilibrium Arrow-Debreu prices at date $t$.
 
@@ -254,7 +254,7 @@ In the AMSS economy, the restriction that government debt be risk-free imposes t
 
 In a language used in the literature on incomplete markets models, it can be said that the AMSS model requires that at each $(t, s^t)$ what would be the present value of continuation government net-of-interest surpluses in the Lucas-Stokey model must belong to  the **marketable subspace** of the AMSS model.
 
-### Ramsey Problem Without State-contingent Debt
+### Ramsey problem without state-contingent debt
 
 After we have substituted the resource constraint into the utility function, we can express the Ramsey problem as being to choose an allocation that solves
 
@@ -286,7 +286,7 @@ and
 
 given $b_0(s^{-1})$.
 
-#### Lagrangian Formulation
+#### Lagrangian formulation
 
 Let $\gamma_0(s^0)$ be a non-negative Lagrange multiplier on constraint {eq}`AMSS_44`.
 
@@ -316,7 +316,7 @@ That would let us reduce the beginning-of-period indebtedness for some other his
 
 These features flow from  the fact that the government cannot use state-contingent debt and therefore cannot allocate its indebtedness  efficiently across future states.
 
-### Some Calculations
+### Some calculations
 
 It is helpful to apply two transformations to the Lagrangian.
 
@@ -408,7 +408,7 @@ tags: [collapse-20]
 To analyze the AMSS model, we find it useful to adopt a recursive formulation
 using techniques like those in our lectures on {doc}`dynamic Stackelberg models <dyn_stack>` and {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`.
 
-## Recursive Version of AMSS Model
+## Recursive version of AMSS model
 
 We now describe a recursive formulation of the AMSS economy.
 
@@ -424,7 +424,7 @@ We now explore how these constraints alter  Bellman equations for a time
 $0$ Ramsey planner and for time $t \geq 1$, history $s^t$
 continuation Ramsey planners.
 
-### Recasting State Variables
+### Recasting state variables
 
 In the AMSS setting, the government faces a sequence of budget constraints
 
@@ -487,7 +487,7 @@ history $s^t$ as
 
 for $t \geq 1$.
 
-### Measurability Constraints
+### Measurability constraints
 
 Write equation {eq}`eqn:AMSSapp2` as
 
@@ -510,7 +510,7 @@ That implies that it has to be *measurable* with respect to $s^{t-1}$.
 Equations {eq}`eqn:AMSSapp2b` are the *measurability constraints* that the AMSS model adds to the single time $0$ implementation
 constraint imposed in the Lucas and Stokey model.
 
-### Two Bellman Equations
+### Two Bellman equations
 
 Let $\Pi(s|s_-)$ be a Markov transition matrix whose entries tell probabilities of moving from state $s_-$ to state $s$ in one period.
 
@@ -567,7 +567,7 @@ where maximization is subject to
 u_{c,0} b_0 = u_{c,0} (n_0-g_0) - u_{l,0} n_0 + x_0
 ```
 
-### Martingale Supercedes State-Variable Degeneracy
+### Martingale supercedes state-variable degeneracy
 
 Let $\mu(s|s_-) \Pi(s|s_-)$ be a Lagrange multiplier on the constraint {eq}`eqn:AMSSapp6`
 for state $s$.
@@ -620,7 +620,7 @@ that for each $s_-$, the sum over $s$ equals unity.
 ```{exercise-end}
 ```
 
-### Absence of State Variable Degeneracy
+### Absence of state variable degeneracy
 
 Along a Ramsey plan, the state variable $x_t = x_t(s^t, b_0)$
 becomes a function of the history $s^t$ and initial
@@ -642,7 +642,7 @@ In the AMSS model, both $x$ and $s$ are needed to describe the state.
 This property of the AMSS model  transmits a twisted martingale
 component to consumption, employment, and the tax rate.
 
-### Digression on Non-negative Transfers
+### Digression on non-negative transfers
 
 Throughout this lecture, we have imposed that transfers $T_t = 0$.
 
@@ -686,7 +686,7 @@ The recursive formulation is implemented as follows
 
 We now turn to some examples.
 
-### Anticipated One-Period War
+### Anticipated one-period war
 
 In our lecture on {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`
 we studied how the government manages uncertainty in a simple setting.
@@ -874,7 +874,7 @@ Without state-contingent debt, the optimal tax rate is history dependent.
 * A war at time $t=3$ causes a permanent **increase** in the tax rate.
 * Peace at time $t=3$ causes a permanent **reduction** in the tax rate.
 
-#### Perpetual War Alert
+#### Perpetual war alert
 
 History dependence occurs more dramatically in a case in which the government
 perpetually faces the prospect  of war.

lectures/amss2.md

@@ -88,7 +88,7 @@ import matplotlib.pyplot as plt
 from scipy.optimize import fsolve, fmin
 ```
 
-## Forces at Work
+## Forces at work
 
 The forces  driving asymptotic  outcomes here are examples of dynamics present in a more general class of incomplete markets models analyzed in {cite}`BEGS1` (BEGS).
 
@@ -112,7 +112,7 @@ rates  rather than fluctuations in  par values of debt to insure against shocks
   shutting down the stochastic component of debt dynamics.
 - At that point, the tail of the par value of government debt becomes a trivial martingale: it is constant over time.
 
-## Logical Flow of Lecture
+## Logical flow of lecture
 
 We present ideas  in the following order
 
@@ -126,7 +126,7 @@ We present ideas  in the following order
     - we verify that the LS Ramsey planner chooses to purchase **identical** claims to time $t+1$ consumption for all Markov states tomorrow for each Markov state today.
 * We compute the BEGS approximations to check how accurately they describe the dynamics of the long-simulation.
 
-### Equations from Lucas-Stokey (1983) Model
+### Equations from Lucas-Stokey (1983) model
 
 Although we are studying an AMSS {cite}`aiyagari2002optimal` economy,  a Lucas-Stokey {cite}`LucasStokey1983` economy plays
 an important  role in the reverse-engineering calculation to be described below.
@@ -189,7 +189,7 @@ $$
 It is useful to transform  some of the above equations to forms that are more natural for analyzing the
 case of a CRRA utility specification that we shall use in our example economies.
 
-### Specification with CRRA Utility
+### Specification with CRRA utility
 
 As in lectures {doc}`optimal taxation without state-contingent debt <amss>` and {doc}`optimal taxation with state-contingent debt <opt_tax_recur>`,
 we assume that the representative agent has utility function
@@ -244,7 +244,7 @@ The CRRA utility function is represented in the following class.
 :load: _static/lecture_specific/amss2/crra_utility.py
 ```
 
-## Example Economy
+## Example economy
 
 We set the following parameter values.
 
@@ -292,7 +292,7 @@ tags: [collapse-20]
 ---
 ```
 
-## Reverse Engineering Strategy
+## Reverse engineering strategy
 
 We can reverse engineer a value $b_0$ of initial debt due   that renders the AMSS measurability constraints not binding from time $t =0$ onward.
 
@@ -337,7 +337,7 @@ state $s_t=s$ from $b(s) = {\frac{x(s)}{u_c(s)}}$ or the matrix equation
 **Step 7:** At the value of $\Phi$ and the value of $\bar b$ that emerged from step 6, solve equations
 {eq}`amss2_TS_barg11` and {eq}`eqn_AMSS2_10` jointly for $c_0, b_0$.
 
-## Code for Reverse Engineering
+## Code for reverse engineering
 
 Here is code to do the calculations for us.
 
@@ -420,7 +420,7 @@ c0, b0
 
 Thus, we have reverse engineered an initial $b0 = -1.038698407551764$ that ought to render the AMSS measurability constraints slack.
 
-## Short Simulation for Reverse-engineered: Initial Debt
+## Short simulation for reverse-engineered: initial debt
 
 The following graph shows simulations of outcomes for both a Lucas-Stokey economy and for an AMSS economy starting from initial government
 debt equal to $b_0 = -1.038698407551764$.
@@ -477,7 +477,7 @@ Notice how for $t \geq 1$, the tax rate is a constant - so is the par value of g
 
 However, output and labor supply are both nontrivial time-invariant functions of the Markov state.
 
-## Long Simulation
+## Long simulation
 
 The following graph shows the par value of government debt and the flat-rate tax on labor income  for a long simulation for our sample economy.
 
@@ -523,7 +523,7 @@ plt.tight_layout()
 plt.show()
 ```
 
-### Remarks about Long Simulation
+### Remarks about long simulation
 
 As remarked above, after $b_{t+1}(s^t)$ has converged to a constant, the measurability constraints in the AMSS model cease to bind
 
@@ -536,7 +536,7 @@ This leads us to seek an initial value of government debt $b_0$ that renders the
 
 We  now describe how to find such an initial level of government debt.
 
-## BEGS Approximations of Limiting Debt and Convergence Rate
+## BEGS approximations of limiting debt and convergence rate
 
 It is useful to link the outcome of our reverse engineering exercise to limiting approximations constructed by BEGS {cite}`BEGS1`.
 
@@ -573,7 +573,7 @@ BEGS interpret random variations in the right side of {eq}`eq_fiscal_risk` as a
   ${\mathcal R}_\tau(s, s_{-}) {\mathcal B}_{-}$,  and
 - fluctuations in the effective government deficit ${\mathcal X}_t$
 
-### Asymptotic Mean
+### Asymptotic mean
 
 BEGS give conditions under which the ergodic mean of ${\mathcal B}_t$ is
 
@@ -607,7 +607,7 @@ Expressing formula {eq}`prelim_formula` in terms of  our notation tells us that
 \hat b = \frac{\mathcal B^*}{\beta E_t u_{c,t+1}}
 ```
 
-### Rate of Convergence
+### Rate of convergence
 
 BEGS also derive the following  approximation to the rate of convergence to ${\mathcal B}^{*}$ from an arbitrary initial condition.
 
@@ -619,7 +619,7 @@ BEGS also derive the following  approximation to the rate of convergence to ${\m
 
 (See the equation above equation (47) in {cite}`BEGS1`)
 
-### Formulas and Code Details
+### Formulas and code details
 
 For our example, we describe some code that we use to compute the steady state mean and the rate of convergence to it.