diff --git a/lectures/cagan_adaptive.md b/lectures/cagan_adaptive.md index 4b3d9724..88d13626 100644 --- a/lectures/cagan_adaptive.md +++ b/lectures/cagan_adaptive.md @@ -16,13 +16,13 @@ kernelspec: ## Introduction -This lecture is a sequel or prequel to another lecture {doc}`monetarist theory of price levels `. +This lecture is a sequel or prequel to the lecture {doc}`cagan_ree`. -We'll use linear algebra to do some experiments with an alternative "monetarist" or "fiscal" theory of price levels". +We'll use linear algebra to do some experiments with an alternative "monetarist" or "fiscal" theory of price levels. -Like the model in this lecture {doc}`monetarist theory of price levels `, the model asserts that when a government persistently spends more than it collects in taxes and prints money to finance the shortfall, it puts upward pressure on the price level and generates persistent inflation. +Like the model in {doc}`cagan_ree`, the model asserts that when a government persistently spends more than it collects in taxes and prints money to finance the shortfall, it puts upward pressure on the price level and generates persistent inflation. -Instead of the "perfect foresight" or "rational expectations" version of the model in this lecture {doc}`monetarist theory of price levels `, our model in the present lecture is an "adaptive expectations" version of a model that Philip Cagan {cite}`Cagan` used to study the monetary dynamics of hyperinflations. +Instead of the "perfect foresight" or "rational expectations" version of the model in {doc}`cagan_ree`, our model in the present lecture is an "adaptive expectations" version of a model that Philip Cagan {cite}`Cagan` used to study the monetary dynamics of hyperinflations. It combines these components: @@ -36,7 +36,7 @@ It combines these components: Our model stays quite close to Cagan's original specification. -As in the {doc}`present values ` and {doc}`consumption smoothing` lectures, the only linear algebra operations that we'll be using are matrix multiplication and matrix inversion. +As in the lectures {doc}`pv` and {doc}`cons_smooth`, the only linear algebra operations that we'll be using are matrix multiplication and matrix inversion. To facilitate using linear matrix algebra as our principal mathematical tool, we'll use a finite horizon version of the model. @@ -54,7 +54,7 @@ Let * $\pi_0^*$ public's initial expected rate of inflation between time $0$ and time $1$. -The demand for real balances $\exp\left(\frac{m_t^d}{p_t}\right)$ is governed by the following version of the Cagan demand function +The demand for real balances $\exp\left(m_t^d-p_t\right)$ is governed by the following version of the Cagan demand function $$ m_t^d - p_t = -\alpha \pi_t^* \: , \: \alpha > 0 ; \quad t = 0, 1, \ldots, T . @@ -88,7 +88,7 @@ $$ (eq:adaptexpn) As exogenous inputs into the model, we take initial conditions $m_0, \pi_0^*$ and a money growth sequence $\mu = \{\mu_t\}_{t=0}^T$. -As endogenous outputs of our model we want to find sequences $\pi = \{\pi_t\}_{t=0}^T, p = \{p_t\}_{t=0}^T$ as functions of the endogenous inputs. +As endogenous outputs of our model we want to find sequences $\pi = \{\pi_t\}_{t=0}^T, p = \{p_t\}_{t=0}^T$ as functions of the exogenous inputs. We'll do some mental experiments by studying how the model outputs vary as we vary the model inputs. @@ -278,7 +278,7 @@ $$ (eq:notre) This outcome is typical in models in which adaptive expectations hypothesis like equation {eq}`eq:adaptexpn` appear as a component. -In this lecture {doc}`monetarist theory of the price level `, we studied a version of the model that replaces hypothesis {eq}`eq:adaptexpn` with +In {doc}`cagan_ree` we studied a version of the model that replaces hypothesis {eq}`eq:adaptexpn` with a "perfect foresight" or "rational expectations" hypothesis. @@ -296,7 +296,7 @@ import matplotlib.pyplot as plt Cagan_Adaptive = namedtuple("Cagan_Adaptive", ["α", "m0", "Eπ0", "T", "λ"]) -def create_cagan_model(α, m0, Eπ0, T, λ): +def create_cagan_adaptive_model(α, m0, Eπ0, T, λ): return Cagan_Adaptive(α, m0, Eπ0, T, λ) ``` +++ {"user_expressions": []} @@ -314,7 +314,7 @@ m0 = 1 μ0 = 0.5 μ_star = 0 -md = create_cagan_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ) +md = create_cagan_adaptive_model(α=α, m0=m0, Eπ0=μ0, T=T, λ=λ) ``` +++ {"user_expressions": []} @@ -431,7 +431,7 @@ $$ \end{cases} $$ -Notice that we studied exactly this experiment in a rational expectations version of the model in this lecture {doc}`monetarist theory of the price level `. +Notice that we studied exactly this experiment in a rational expectations version of the model in {doc}`cagan_ree`. So by comparing outcomes across the two lectures, we can learn about consequences of assuming adaptive expectations, as we do here, instead of rational expectations as we assumed in that other lecture. @@ -442,7 +442,7 @@ So by comparing outcomes across the two lectures, we can learn about consequence π_seq_1, Eπ_seq_1, m_seq_1, p_seq_1 = solve_and_plot(md, μ_seq_1) ``` -We invite the reader to compare outcomes with those under rational expectations studied in another lecture {doc}`monetarist theory of price levels `. +We invite the reader to compare outcomes with those under rational expectations studied in {doc}`cagan_ree`. Please note how the actual inflation rate $\pi_t$ "overshoots" its ultimate steady-state value at the time of the sudden reduction in the rate of growth of the money supply at time $T_1$.