diff --git a/lectures/calvo.md b/lectures/calvo.md index ae6d55ce..886b2fa5 100644 --- a/lectures/calvo.md +++ b/lectures/calvo.md @@ -64,7 +64,7 @@ We'll use ideas from papers by Cagan {cite}`Cagan`, Calvo {cite}`Calvo1978`, an well as from chapter 19 of {cite}`Ljungqvist2012`. In addition, we'll use ideas from linear-quadratic dynamic programming -described in [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg problems `. +described in [Linear Quadratic Control](https://python-intro.quantecon.org/lqcontrol.html) as applied to Ramsey problems in {doc}`Stackelberg plans `. We specify model fundamentals in ways that allow us to use linear-quadratic discounted dynamic programming to compute an optimal government @@ -104,8 +104,7 @@ Let: - $\theta_t = p_{t+1} - p_t$ be the net rate of inflation between $t$ and $t+1$ - $\mu_t = m_{t+1} - m_t$ be the net rate of growth of nominal balances -The demand for real balances is governed by a perfect foresight -version of a Cagan {cite}`Cagan` demand function for real balances: +The demand for real balances is governed by a discrete time version of Sargent and Wallace's {cite}`sargent1973stability` perfect foresight version of a Cagan {cite}`Cagan` demand function for real balances: ```{math} :label: eq_old1 @@ -119,9 +118,9 @@ Equation {eq}`eq_old1` asserts that the demand for real balances is inversely related to the public's expected rate of inflation, which equals the actual rate of inflation because there is no uncertainty here. -(When there is no uncertainty, an assumption of **rational expectations** that becomes equivalent to **perfect foresight**). +(When there is no uncertainty, an assumption of **rational expectations** becomes equivalent to **perfect foresight**). -(See {cite}`Sargent77hyper` for a rational expectations version of the model when there is uncertainty.) +({cite}`Sargent77hyper` presents a rational expectations version of the model when there is uncertainty.) Subtracting the demand function {eq}`eq_old1` at time $t$ from the demand function at $t+1$ gives: @@ -204,7 +203,7 @@ as it ordinarily would be in the state-space model described in our lecture on -We use form {eq}`eq_old4` because we want to apply an approach described in our lecture on {doc}`Stackelberg problems `. +We use form {eq}`eq_old4` because we want to apply an approach described in our lecture on {doc}`Stackelberg plans `. Notice that $\frac{1+\alpha}{\alpha} > 1$ is an eigenvalue of transition matrix $A$ that threatens to destabilize the state-space system. @@ -224,14 +223,9 @@ $$ U(-\alpha \theta_t) = u_0 + u_1 (-\alpha \theta_t) -\frac{u_2}{2}(-\alpha \theta_t)^2 . $$ (eq_old5a) -The ``bliss level`` of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains -it is $-\frac{u_1}{u_2 \alpha}$. - -(TO TOM: the first sentece in the next section is very similar to the sentence above.) - ## Friedman's Optimal Rate of Deflation -According to {eq}`eq_old5a`, the "bliss level" of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains it is +According to {eq}`eq_old5a`, the ``bliss level`` of real balances is $\frac{u_1}{u_2}$ and the inflation rate that attains it is $$ @@ -324,9 +318,10 @@ for all $t \geq 0$. Values of $V(\bar \mu)$ computed according to formula {eq}`eq:barvdef` for three different values of $\bar \mu$ will play important roles below. * $V(\mu^{MP})$ is the value of attained by the government in a **Markov perfect equilibrium** +* $V(\mu^R_\infty)$ is the value that a continuation Ramsey planner attains at $t \rightarrow +\infty$ + * We shall discover that $V(\mu^R_\infty)$ is the worst continuation value attained along a Ramsey plan * $V(\mu^{CR})$ is the value of attained by the government in a **constrained to constant $\mu$ equilibrium** -* $V(\mu^R_\infty)$ is the limiting value attained by a continuation Ramsey planner under a Ramsey plan. - * We shall see that $V(\mu^R_\infty)$ is a worst continuation value attained along a Ramsey plan + ## Structure @@ -414,7 +409,7 @@ The models are distinguished by their having either $\{\mu_t\}_{t=0}^\infty$ once and for all at time $0$ subject to the constraint that $\mu_t = \mu$ for all $t \geq 0$; or -- A sequence indexed by $t =0, 1, 2, \ldots$ of separate policymakers +- A sequence of distinct policymakers indexed by $t =0, 1, 2, \ldots$ - a time $t$ policymaker chooses $\mu_t$ only and forecasts that future government decisions are unaffected by its choice. @@ -438,7 +433,7 @@ The relationship between outcomes in the first (Ramsey) timing protocol and th We'll begin with the timing protocol associated with a Ramsey plan and deploy an application of what we nickname **dynamic programming squared**. -The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a second Bellman equation. +The nickname refers to the feature that a value satisfying one Bellman equation appears as an argument in a value function associated with a second Bellman equation. Thus, our models have involved two Bellman equations: @@ -455,7 +450,7 @@ Here we consider a Ramsey planner that chooses $\{\mu_t, \theta_t\}_{t=0}^\infty$ to maximize {eq}`eq_old7` subject to the law of motion {eq}`eq_old4`. -We can split this problem into two stages, as in {doc}`Stackelberg problems ` and {cite}`Ljungqvist2012` Chapter 19. +We can split this problem into two stages, as in the lecture {doc}`Stackelberg plans ` and {cite}`Ljungqvist2012` Chapter 19. In the first stage, we take the initial inflation rate $\theta_0$ as given and solve what looks like an ordinary LQ discounted dynamic programming problem. @@ -491,7 +486,7 @@ $$ x' = Ax + B\mu $$ -As in {doc}`Stackelberg problems `, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$. +As in the lecture {doc}`Stackelberg plans `, we can map this problem into a linear-quadratic control problem and deduce an optimal value function $J(x)$. Guessing that $J(x) = - x'Px$ and substituting into the Bellman equation gives rise to the algebraic matrix Riccati equation: @@ -698,7 +693,7 @@ about dynamic or time inconsistency. ## Time inconsistency -As discussed in {doc}`Stackelberg problems ` and {doc}`Optimal taxation with state-contingent debt `, a continuation Ramsey plan is not a Ramsey plan. +As discussed in {doc}`Stackelberg plans ` and {doc}`Optimal taxation with state-contingent debt `, a continuation Ramsey plan is not a Ramsey plan. This is a concise way of characterizing the time inconsistency of a Ramsey plan. @@ -1351,11 +1346,10 @@ $$ \begin{aligned} \theta^{CR} & = - \frac{\alpha u_1}{\alpha^2 u_2 + c } \\ \theta^{MPE} & = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c} \\ -\theta^{MPE} & = - \frac{\alpha u_1}{\alpha^2 u_2 + (1+\alpha)c} +\theta^{*} & = -\frac{u_1}{u_2 \alpha} \end{aligned} $$ -(TO TOM: $\theta^{MPE}$ is repeated in the above equations. Should one of them be $\theta^*$?) But let's see what happens when we change $c$. @@ -1376,7 +1370,7 @@ generate_table(clqs, dig=4) The above table and figures show how changes in $c$ alter $\theta_\infty^R$ and $\theta_0^R$ as well as $\theta^{CR}$ and $\theta^{MPE}$, but not - $\theta^*$, again in accord with formulas + $\theta^*,$ again in accord with formulas {eq}`eq:Friedmantheta`, {eq}`eq:muRamseyconstrained`, and {eq}`eq:Markovperfectmu`. Notice that as $c $ gets larger and larger, $\theta_\infty^R, \theta_0^R$ @@ -1529,20 +1523,16 @@ A constrained-to-constant-$\mu$ Ramsey plan is time consistent by constructio ### Implausibility of Ramsey Plan -In settings in which governments actually choose sequentially, many economists -regard a time inconsistent plan as implausible because of the incentives to -deviate that are presented along the plan. - -(TO TOM: In our meeting, you suggested that we can improve the sentence above.) +Many economists regard a time inconsistent plan as implausible because they question the plausibility of timing protocol in +which a plan for setting a sequence of policy variables is chosen once-and-for-all at time $0$. -A way to state this reaction is to say that a Ramsey plan is not credible because there are persistent incentives for policymakers to deviate from it. For that reason, the Markov perfect equilibrium concept attracts many economists. -* A Markov perfect equilibrium plan is constructed to insure that government policymakers who choose sequentially do not want to deviate from it. +* A Markov perfect equilibrium plan is constructed to insure that a sequence of government policymakers who choose sequentially do not want to deviate from it. -The *no incentive to deviate from the plan* property is what makes the Markov perfect equilibrium concept attractive. +The property of a Markov perfect equilibrium that there is *no incentive to deviate from the plan* makes it attractive. ## Comparison of Equilibrium Values