diff --git a/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png b/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png index a3833f10..3ae6891e 100644 Binary files a/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png and b/lectures/_static/lecture_specific/long_run_growth/tooze_ch1_graph.png differ diff --git a/lectures/money_inflation_nonlinear.md b/lectures/money_inflation_nonlinear.md index e120bdf1..f716916f 100644 --- a/lectures/money_inflation_nonlinear.md +++ b/lectures/money_inflation_nonlinear.md @@ -45,7 +45,7 @@ That lecture will show that * it reverses the perverse dynamics by making the *lower* stationary inflation rate the one to which the system typically converges * a more plausible comparative dynamic outcome emerges in which now inflation can be *reduced* by running *lower* government deficits -## The model +## The Model Let @@ -70,56 +70,9 @@ where $g$ is the part of government expenditures financed by printing money. **Remark:** Please notice that while equation {eq}`eq:mdemand` is linear in logs of the money supply and price level, equation {eq}`eq:msupply` is linear in levels. This will require adapting the equilibrium computation methods that we deployed in {doc}`money_inflation`. -## Computing an equilibrium sequence -We'll deploy a method similar to *Method 2* used in {doc}`money_inflation`. - -We'll take the time $t$ state vector to be $m_t, p_t$. - -* we'll treat $m_t$ as a ''natural state variable'' and $p_t$ as a ''jump'' variable. - -Let - -$$ -\lambda \equiv \frac{\alpha}{1+ \alpha} -$$ - -Let's rewrite equation {eq}`eq:mdemand`, respectively, as - -$$ -p_t = (1-\lambda) m_{t+1} + \lambda p_{t+1} -$$ (eq:mdemand2) - -We'll summarize our algorithm with the following pseudo-code. - -**Pseudo-code** - -* start for $m_0, p_0$ at time $t =0$ - -* solve {eq}`eq:msupply` for $m_{t+1}$ - -* solve {eq}`eq:mdemand2` for $p_{t+1} = \lambda^{-1} p_t + (1 - \lambda^{-1}) m_{t+1}$ - -* compute the inflation rate $\pi_t = p_{t+1} - p_t$ and growth of money supply $\mu_t = m_{t+1} - m_t $ -* iterate on $t$ to convergence of $\pi_t \rightarrow \overline \pi$ and $\mu_t \rightarrow \overline \mu$ - -It will turn out that - -* if they exist, limiting values $\overline \pi$ and $\overline \mu$ will be equal - -* if limiting values exist, there are two possible limiting values, one high, one low - -* for almost all initial log price levels $p_0$, the limiting $\overline \pi = \overline \mu$ is -the higher value - -* for each of the two possible limiting values $\overline \pi$ ,there is a unique initial log price level $p_0$ that implies that $\pi_t = \mu_t = \overline \mu$ for all $t \geq 0$ - - * this unique initial log price level solves $\log(\exp(m_0) + g \exp(p_0)) - p_0 = - \alpha \overline \pi $ - - * the preceding equation for $p_0$ comes from $m_1 - p_0 = - \alpha \overline \pi$ - -## Limiting values of inflation rate +## Limiting Values of Inflation Rate We can compute the two prospective limiting values for $\overline \pi$ by studying the steady-state Laffer curve. @@ -203,7 +156,7 @@ print(f'The two steady state of π are: {π_l, π_u}') We find two steady state $\overline \pi$ values. -## Steady state Laffer curve +## Steady State Laffer curve The following figure plots the steady state Laffer curve together with the two stationary inflation rates. @@ -247,9 +200,16 @@ def plot_laffer(model, πs): plot_laffer(model, (π_l, π_u)) ``` -## Associated initial price levels +## Initial Price Levels + +Now that we have our hands on the two possible steady states, we can compute two functions $\underline p(m_0)$ and +$\overline p(m_0)$, which as initial conditions for $p_t$ at time $t$, imply that $\pi_t = \overline \pi $ for all $t \geq 0$. + +The function $\underline p(m_0)$ will be associated with $\pi_l$ the lower steady-state inflation rate. + +The function $\overline p(m_0)$ will be associated with $\pi_u$ the lower steady-state inflation rate. + -Now that we have our hands on the two possible steady states, we can compute two initial log price levels $p_0$, which as initial conditions, imply that $\pi_t = \overline \pi $ for all $t \geq 0$. ```{code-cell} ipython3 def solve_p0(p0, m0, α, g, π): @@ -312,7 +272,68 @@ eq_g = lambda x: np.exp(-model.α * x) - np.exp(-(1 + model.α) * x) print('eq_g == g:', np.isclose(eq_g(m_seq[-1] - m_seq[-2]), model.g)) ``` -## Slippery side of Laffer curve dynamics +## Computing an Equilibrium Sequence + +We'll deploy a method similar to *Method 2* used in {doc}`money_inflation`. + +We'll take the time $t$ state vector to be the pair $(m_t, p_t)$. + +We'll treat $m_t$ as a ``natural state variable`` and $p_t$ as a ``jump`` variable. + +Let + +$$ +\lambda \equiv \frac{\alpha}{1+ \alpha} +$$ + +Let's rewrite equation {eq}`eq:mdemand` as + +$$ +p_t = (1-\lambda) m_{t+1} + \lambda p_{t+1} +$$ (eq:mdemand2) + +We'll summarize our algorithm with the following pseudo-code. + +**Pseudo-code** + +The heart of the pseudo-code iterates on the following mapping from state vector $(m_t, p_t)$ at time $t$ +to state vector $(m_{t+1}, p_{t+1})$ at time $t+1$. + + +* starting from a given pair $(m_t, p_t)$ at time $t \geq 0$ + + * solve {eq}`eq:msupply` for $m_{t+1}$ + + * solve {eq}`eq:mdemand2` for $p_{t+1} = \lambda^{-1} p_t + (1 - \lambda^{-1}) m_{t+1}$ + + * compute the inflation rate $\pi_t = p_{t+1} - p_t$ and growth of money supply $\mu_t = m_{t+1} - m_t $ + +Next, compute the two functions $\underline p(m_0)$ and $\overline p(m_0)$ described above + +Now initiate the algorithm as follows. + + * set $m_0 >0$ + * set a value of $p_0 \in [\underline p(m_0), \overline p(m_0)]$ and form the pair $(m_0, p_0)$ at time $t =0$ + +Starting from $(m_0, p_0)$ iterate on $t$ to convergence of $\pi_t \rightarrow \overline \pi$ and $\mu_t \rightarrow \overline \mu$ + +It will turn out that + +* if they exist, limiting values $\overline \pi$ and $\overline \mu$ will be equal + +* if limiting values exist, there are two possible limiting values, one high, one low + +* for almost all initial log price levels $p_0$, the limiting $\overline \pi = \overline \mu$ is +the higher value + +* for each of the two possible limiting values $\overline \pi$ ,there is a unique initial log price level $p_0$ that implies that $\pi_t = \mu_t = \overline \mu$ for all $t \geq 0$ + + * this unique initial log price level solves $\log(\exp(m_0) + g \exp(p_0)) - p_0 = - \alpha \overline \pi $ + + * the preceding equation for $p_0$ comes from $m_1 - p_0 = - \alpha \overline \pi$ + + +## Slippery Side of Laffer Curve Dynamics We are now equipped to compute time series starting from different $p_0$ settings, like those in {doc}`money_inflation`.