From f4ccaef1acd2b6deae0ddb1e016716f50886a5d3 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Wed, 22 May 2024 16:16:34 +1000 Subject: [PATCH 1/5] [unpleasant] Update editorial suggestions --- lectures/unpleasant.md | 55 +++++++++++++++++++----------------------- 1 file changed, 25 insertions(+), 30 deletions(-) diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index 3e57c1bf..7792201f 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -11,14 +11,14 @@ kernelspec: name: python3 --- -# Unpleasant Monetarist Arithmetic +# Some Unpleasant Monetarist Arithmetic ## Overview This lecture builds on concepts and issues introduced in our lecture on **Money Supplies and Price Levels**. -That lecture describes stationary equilibria that reveal a **Laffer curve** in the inflation tax rate and the associated stationary rate of return +That lecture describes stationary equilibria that reveal a [**Laffer curve**](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return on currency. In this lecture we study a situation in which a stationary equilibrium prevails after date $T > 0$, but not before then. @@ -75,6 +75,8 @@ $$ b_t = \gamma_1 - \gamma_2 R_t^{-1} . $$ (eq:up_bdemand) +where $\gamma_1 > \gamma_2 > 0$. + ## Monetary-Fiscal Policy To the basic model of our lecture on **Money Supplies and Price Levels**, we add inflation-indexed one-period government bonds as an additional way for the government to finance government expenditures. @@ -91,8 +93,8 @@ $$ Just before the beginning of time $0$, the public owns $\check m_0$ units of currency (measured in dollars) and $\widetilde R \check B_{-1}$ units of one-period indexed bonds (measured in time $0$ goods); these two quantities are initial conditions set outside the model. -Notice that $\check m_0$ is a **nominal** quantity, being measured in dollar, while -$\widetilde R \check B_{-1}$ is a **real** quantity, being measured in time $0$ goods. +Notice that $\check m_0$ is a *nominal* quantity, being measured in dollar, while +$\widetilde R \check B_{-1}$ is a *real* quantity, being measured in time $0$ goods. ### Open market operations @@ -109,8 +111,8 @@ $$ B_{-1} - \check B_{-1} = \frac{1}{p_0 \widetilde R} \left( \check m_0 - m_0 \right) $$ (eq:openmarketconstraint) -This equation says that the government (e.g., the central bank) can **decrease** $m_0$ relative to -$\check m_0$ by **increasing** $B_{-1}$ relative to $\check B_{-1}$. +This equation says that the government (e.g., the central bank) can *decrease* $m_0$ relative to +$\check m_0$ by *increasing* $B_{-1}$ relative to $\check B_{-1}$. This is a version of a standard constraint on a central bank's **open market operations** in which it expands the stock of money by buying government bonds from the public. @@ -152,7 +154,7 @@ running monetary and fiscal policies. Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\{g_t\}_{t=0}^\infty$ that must be financed by issuing to the public either money or interest bearing bonds. -By **monetary policy** or **debt-management polcy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). +By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). By an **open market operation**, we mean a government monetary policy action in which the government (or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation. @@ -171,7 +173,7 @@ $$ (eq:up_steadyquadratic) Quadratic equation {eq}`eq:up_steadyquadratic` has two roots, $R_l < R_u < 1$. -For reasons described at the end of **this lecture**, we select the larger root $R_u$. +For reasons described at the end of *this lecture*, we select the larger root $R_u$. Next, we compute @@ -211,7 +213,7 @@ We want to compute $$ \begin{aligned} -p_0 & = \gamma_1^{-1} \left[ \sum_{j=0}^\infty \lambda^j m_{1+j} \right] \cr +p_0 & = \gamma_1^{-1} \left[ \sum_{j=0}^\infty \lambda^j m_{j} \right] \cr & = \gamma_1^{-1} \left[ \sum_{j=0}^{T-1} \lambda^j m_{0} + \sum_{j=T}^\infty \lambda^j m_{1+j} \right] \end{aligned} $$ @@ -252,9 +254,9 @@ Python coder. To compute an equilibrium, we deploy the following algorithm. -Given **parameters** include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $ +Given **parameters** include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $. -We define a mapping from $p_0$ to $p_0$ as follows. +We define a mapping from $p_0$ to $\widehat p_0$ as follows. * Set $m_0$ and then compute $B_{-1}$ to satisfy the constraint on time $0$ **open market operations** @@ -469,25 +471,16 @@ def simulate(m0, model, length=15, p0_guess=1): def plot_path(m0_arr, model, length=15): fig, axs = plt.subplots(2, 2, figsize=(8, 5)) - + titles = ['$p_t$', '$m_t$', '$b_t$', '$R_t$'] + for m0 in m0_arr: - paths = simulate(m0, msm, length=length) - - axs[0, 0].plot(paths[0]) - axs[0, 0].set_title('$p_t$') - - axs[0, 1].plot(paths[1]) - axs[0, 1].set_title('$m_t$') - - axs[1, 0].plot(paths[2]) - axs[1, 0].set_title('$b_t$') - - axs[1, 1].plot(paths[3]) - axs[1, 1].set_title('$R_t$') - - axs[0, 1].hlines(model.m0_check, 0, length, - color='r', linestyle='--') - axs[0, 1].text(length*0.8, model.m0_check*0.9, '$\check{m}_0$') + paths = simulate(m0, model, length=length) + for i, ax in enumerate(axs.flat): + ax.plot(paths[i]) + ax.set_title(titles[i]) + + axs[0, 1].hlines(model.m0_check, 0, length, color='r', linestyle='--') + axs[0, 1].text(length * 0.8, model.m0_check * 0.9, '$\check{m}_0$') plt.show() ``` @@ -508,4 +501,6 @@ Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`. * The lower is the post-open-market-operation money supply at time $0$, lower is the price level at time $0$. -* An open market operation that reduces the post-open-market-operation money supply at time $0$ also **lowers** the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross-of-interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$. +* An open market operation that reduces the post-open-market-operation money supply at time $0$ also *lowers* the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross-of-interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$. + +* $R$ is important in the context of maintaining monetary stability and addressing the consequences of increased inflation due to government deficits. Thus, a larger $R$ might be chosen to mitigate the negative impacts on the real rate of return caused by inflation. From da69308e97920ed32c583b48820a8e60faa4e068 Mon Sep 17 00:00:00 2001 From: Humphrey Yang Date: Thu, 27 Jun 2024 21:57:24 +0800 Subject: [PATCH 2/5] update links between lectures --- lectures/unpleasant.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index 7792201f..ba36669d 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -162,7 +162,7 @@ By an **open market operation**, we mean a government monetary policy action in ## Algorithm (basic idea) -We work backwards from $t=T$ and first compute $p_T, R_u$ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium of our lecture on the dynamic Laffer curve for the inflation tax. +We work backwards from $t=T$ and first compute $p_T, R_u$ associated with the low-inflation, low-inflation-tax-rate stationary equilibrium in {doc}`money_inflation_nonlinear`. To start our description of our algorithm, it is useful to recall that a stationary rate of return on currency $\bar R$ solves the quadratic equation @@ -173,7 +173,7 @@ $$ (eq:up_steadyquadratic) Quadratic equation {eq}`eq:up_steadyquadratic` has two roots, $R_l < R_u < 1$. -For reasons described at the end of *this lecture*, we select the larger root $R_u$. +For reasons described at the end of {doc}`money_inflation`, we select the larger root $R_u$. Next, we compute @@ -303,7 +303,7 @@ where $\theta \in [0,1)$ is a relaxation parameter. ## Example Calculations We'll set parameters of the model so that the steady state after time $T$ is initially the same -as in our lecture on "Money and Inflation". +as in {doc}`money_inflation_nonlinear` In particular, we set $\gamma_1=100, \gamma_2 =50, g=3.0$. We set $m_0 = 100$ in that lecture, but now the counterpart will be $M_T$, which is endogenous. @@ -494,7 +494,7 @@ mystnb: plot_path([80, 100], msm) ``` -Figure {numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of +{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`. * An open market operation that reduces the supply of money at time $t=0$ reduces the price level at time $t=0$ From 414d880de6496b38bc796eb7a291b52ce7fb22b6 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Sun, 30 Jun 2024 17:14:46 +1000 Subject: [PATCH 3/5] Update unpleasant.md --- lectures/unpleasant.md | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index ba36669d..eb8f5e31 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -18,7 +18,7 @@ kernelspec: This lecture builds on concepts and issues introduced in our lecture on **Money Supplies and Price Levels**. -That lecture describes stationary equilibria that reveal a [**Laffer curve**](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return +That lecture describes stationary equilibria that reveal a [*Laffer curve*](https://en.wikipedia.org/wiki/Laffer_curve) in the inflation tax rate and the associated stationary rate of return on currency. In this lecture we study a situation in which a stationary equilibrium prevails after date $T > 0$, but not before then. @@ -154,7 +154,7 @@ running monetary and fiscal policies. Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\{g_t\}_{t=0}^\infty$ that must be financed by issuing to the public either money or interest bearing bonds. -By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). +By **monetary policy** or *debt-management policy*, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). By an **open market operation**, we mean a government monetary policy action in which the government (or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation. @@ -254,7 +254,7 @@ Python coder. To compute an equilibrium, we deploy the following algorithm. -Given **parameters** include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $. +Given *parameters* include $g, \check m_0, \check B_{-1}, \widetilde R >1, T $. We define a mapping from $p_0$ to $\widehat p_0$ as follows. @@ -501,6 +501,6 @@ Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`. * The lower is the post-open-market-operation money supply at time $0$, lower is the price level at time $0$. -* An open market operation that reduces the post-open-market-operation money supply at time $0$ also *lowers* the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross-of-interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$. +* An open market operation that reduces the post open market operation money supply at time $0$ also *lowers* the rate of return on money $R_u$ at times $t \geq T$ because it brings a higher gross of interest government deficit that must be financed by printing money (i.e., levying an inflation tax) at time $t \geq T$. * $R$ is important in the context of maintaining monetary stability and addressing the consequences of increased inflation due to government deficits. Thus, a larger $R$ might be chosen to mitigate the negative impacts on the real rate of return caused by inflation. From e47d444495dba7ac29715ec42275a961fce78da1 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 1 Jul 2024 10:08:11 +1000 Subject: [PATCH 4/5] Update unpleasant.md --- lectures/unpleasant.md | 3 +-- 1 file changed, 1 insertion(+), 2 deletions(-) diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index eb8f5e31..551ca8e2 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -494,8 +494,7 @@ mystnb: plot_path([80, 100], msm) ``` -{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of -Sargent and Wallace's **unpleasant monetarist arithmetic** {cite}`sargent1981`. +{numref}`fig:unpl1` summarizes outcomes of two experiments that convey messages of {cite}`sargent1981`. * An open market operation that reduces the supply of money at time $t=0$ reduces the price level at time $t=0$ From 87b01e4c3f9e4c81a28197d5d12bdd7d380dc307 Mon Sep 17 00:00:00 2001 From: JingkunZhao Date: Mon, 1 Jul 2024 10:18:37 +1000 Subject: [PATCH 5/5] Update unpleasant.md --- lectures/unpleasant.md | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/lectures/unpleasant.md b/lectures/unpleasant.md index 551ca8e2..f5232b82 100644 --- a/lectures/unpleasant.md +++ b/lectures/unpleasant.md @@ -93,7 +93,7 @@ $$ Just before the beginning of time $0$, the public owns $\check m_0$ units of currency (measured in dollars) and $\widetilde R \check B_{-1}$ units of one-period indexed bonds (measured in time $0$ goods); these two quantities are initial conditions set outside the model. -Notice that $\check m_0$ is a *nominal* quantity, being measured in dollar, while +Notice that $\check m_0$ is a *nominal* quantity, being measured in dollars, while $\widetilde R \check B_{-1}$ is a *real* quantity, being measured in time $0$ goods. @@ -154,7 +154,7 @@ running monetary and fiscal policies. Here, by **fiscal policy** we mean the collection of actions that determine a sequence of net-of-interest government deficits $\{g_t\}_{t=0}^\infty$ that must be financed by issuing to the public either money or interest bearing bonds. -By **monetary policy** or *debt-management policy*, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). +By **monetary policy** or **debt-management policy**, we mean the collection of actions that determine how the government divides its portolio of debts to the public between interest-bearing parts (government bonds) and non-interest-bearing parts (money). By an **open market operation**, we mean a government monetary policy action in which the government (or its delegate, say, a central bank) either buys government bonds from the public for newly issued money, or sells bonds to the public and withdraws the money it receives from public circulation.