diff --git a/lectures/rational_expectations.md b/lectures/rational_expectations.md index c153ca79b..593028ed3 100644 --- a/lectures/rational_expectations.md +++ b/lectures/rational_expectations.md @@ -77,7 +77,7 @@ We'll also use the LQ class from `QuantEcon.py`. from quantecon import LQ ``` -### The Big Y, little y Trick +### The big Y, little y trick This widely used method applies in contexts in which a **representative firm** or agent is a "price taker" operating within a competitive equilibrium. @@ -107,7 +107,7 @@ Please watch for how this strategy is applied as the lecture unfolds. We begin by applying the Big $Y$, little $y$ trick in a very simple static context. -#### A Simple Static Example of the Big Y, little y Trick +#### A simple static example of the big Y, little y trick Consider a static model in which a unit measure of firms produce a homogeneous good that is sold in a competitive market. @@ -175,7 +175,7 @@ to be solved for the competitive equilibrium market-wide output $Y$. After solving for $Y$, we can compute the competitive equilibrium price $p$ from the inverse demand curve {eq}`ree_comp3d_static`. -### Related Planning Problem +### Related planning problem Define **consumer surplus** as the area under the inverse demand curve: @@ -207,7 +207,7 @@ References for this lecture include * {cite}`Sargent1987`, chapter XIV * {cite}`Ljungqvist2012`, chapter 7 -## Rational Expectations Equilibrium +## Rational expectations equilibrium ```{index} single: Rational Expectations Equilibrium; Definition ``` @@ -228,7 +228,7 @@ law of motion generated by production choices induced by this belief. We formulate a rational expectations equilibrium in terms of a fixed point of an operator that maps beliefs into optimal beliefs. (ree_ce)= -### Competitive Equilibrium with Adjustment Costs +### Competitive equilibrium with adjustment costs ```{index} single: Rational Expectations Equilibrium; Competitive Equilbrium (w. Adjustment Costs) ``` @@ -251,7 +251,7 @@ where * $Y_t = \int_0^1 y_t(\omega) d \omega = y_t$ is the market-wide level of output (ree_fp)= -#### The Firm's Problem +#### The firm's problem Each firm is a price taker. @@ -287,7 +287,7 @@ This includes ones that the firm cares about but does not control like $p_t$. We turn to this problem now. -#### Prices and Aggregate Output +#### Prices and aggregate output In view of {eq}`ree_comp3d`, the firm's incentive to forecast the market price translates into an incentive to forecast aggregate output $Y_t$. @@ -297,7 +297,7 @@ The output $y_t(\omega)$ of a single firm $\omega$ has a negligible effect on ag That justifies firms in regarding their forecasts of aggregate output as being unaffected by their own output decisions. -#### Representative Firm's Beliefs +#### Representative firm's beliefs We suppose the firm believes that market-wide output $Y_t$ follows the law of motion @@ -311,7 +311,9 @@ where $Y_0$ is a known initial condition. The **belief function** $H$ is an equilibrium object, and hence remains to be determined. -#### Optimal Behavior Given Beliefs +Because of this, at this stage $Y_{t+1}$ only means the perceived output in the next period, $Y^e_{t+1}$. + +#### Optimal behavior given beliefs For now, let's fix a particular belief $H$ in {eq}`ree_hlom` and investigate the firm's response to it. @@ -344,7 +346,7 @@ h(y, Y) := \textrm{argmax}_{y'} Evidently $v$ and $h$ both depend on $H$. -#### Characterization with First-Order Necessary Conditions +#### Characterization with first-order necessary conditions In what follows it will be helpful to have a second characterization of $h$, based on first-order conditions. @@ -364,12 +366,14 @@ $$ v_y(y,Y) = a_0 - a_1 Y + \gamma (y' - y) $$ +and equivalently, $v_y(y', H(Y)) = a_0 - a_1 H(Y) +\gamma (y'' - y')$ + Substituting this equation into {eq}`comp5` gives the **Euler equation** ```{math} :label: ree_comp7 --\gamma (y_{t+1} - y_t) + \beta [a_0 - a_1 Y_{t+1} + \gamma (y_{t+2} - y_{t+1} )] =0 +-\gamma (y_{t+1} - y_t) + \beta [a_0 - a_1 H(Y_t) + \gamma (y_{t+2} - y_{t+1} )] =0 ``` The firm optimally sets an output path that satisfies {eq}`ree_comp7`, taking {eq}`ree_hlom` as given, and subject to @@ -384,7 +388,7 @@ A representative firm's decision rule solves the difference equation {eq}`ree_c Note that solving the Bellman equation {eq}`comp4` for $v$ and then $h$ in {eq}`ree_opbe` yields a decision rule that automatically imposes both the Euler equation {eq}`ree_comp7` and the transversality condition. -#### The Actual Law of Motion for Output +#### The actual law of motion for output As we've seen, a given belief translates into a particular decision rule $h$. @@ -399,17 +403,19 @@ Y_{t+1} = h(Y_t, Y_t) Thus, when firms believe that the law of motion for market-wide output is {eq}`ree_hlom`, their optimizing behavior makes the actual law of motion be {eq}`ree_comp9a`. (ree_def)= -### Definition of Rational Expectations Equilibrium +### Definition of rational expectations equilibrium +```{prf:definition} A **rational expectations equilibrium** or **recursive competitive equilibrium** of the model with adjustment costs is a decision rule $h$ and an aggregate law of motion $H$ such that 1. Given belief $H$, the map $h$ is the firm's optimal policy function. 1. The law of motion $H$ satisfies $H(Y)= h(Y,Y)$ for all $Y$. +``` Thus, a rational expectations equilibrium equates the perceived and actual laws of motion {eq}`ree_hlom` and {eq}`ree_comp9a`. -#### Fixed Point Characterization +#### Fixed point characterization As we've seen, the firm's optimum problem induces a mapping $\Phi$ from a perceived law of motion $H$ for market-wide output to an actual law of motion $\Phi(H)$. @@ -417,14 +423,14 @@ The mapping $\Phi$ is the composition of two mappings, the first of which maps a The $H$ component of a rational expectations equilibrium is a fixed point of $\Phi$. -## Computing an Equilibrium +## Computing an equilibrium ```{index} single: Rational Expectations Equilibrium; Computation ``` Now let's compute a rational expectations equilibrium. -### Failure of Contractivity +### Failure of contractivity Readers accustomed to dynamic programming arguments might try to address this problem by choosing some guess $H_0$ for the aggregate law of motion and then iterating with $\Phi$. @@ -434,6 +440,10 @@ Indeed, there is no guarantee that direct iterations on $\Phi$ converge [^fn_im] There are examples in which these iterations diverge. +To see this intuively from Blackwell's sufficient condition, let us assume there are two beliefs $H_a(Y) > H_b(Y)$ for any $Y$. + +Then by Euler equation {eq}`ree_comp7`, the optimal $y_{t+1} = h(Y_t, Y_t)$ decreases as $H$ increases, which indicates the monotoncity required in the Blackwell's condition is not satisfied. + Fortunately, another method works here. The method exploits a connection between equilibrium and Pareto optimality expressed in @@ -444,7 +454,7 @@ Lucas and Prescott {cite}`LucasPrescott1971` used this method to construct a rat Some details follow. (ree_pp)= -### A Planning Problem Approach +### A planning problem approach ```{index} single: Rational Expectations Equilibrium; Planning Problem Approach ``` @@ -477,7 +487,7 @@ $$ subject to an initial condition for $Y_0$. -### Solution of Planning Problem +### Solution of planning problem Evaluating the integral in {eq}`comp10` yields the quadratic form $a_0 Y_t - a_1 Y_t^2 / 2$. @@ -514,7 +524,7 @@ equation \beta a_0 + \gamma Y_t - [\beta a_1 + \gamma (1+ \beta)]Y_{t+1} + \gamma \beta Y_{t+2} =0 ``` -### Key Insight +### Key insight Return to equation {eq}`ree_comp7` and set $y_t = Y_t$ for all $t$. @@ -533,7 +543,7 @@ It follows that for this example we can compute equilibrium quantities by formin The optimal policy function for the planning problem is the aggregate law of motion $H$ that the representative firm faces within a rational expectations equilibrium. -#### Structure of the Law of Motion +#### Structure of the law of motion As you are asked to show in the exercises, the fact that the planner's problem is an LQ control problem implies an optimal policy --- and hence aggregate law @@ -590,8 +600,7 @@ If there were a unit measure of identical competitive firms all behaving accord :class: dropdown ``` -To map a problem into a [discounted optimal linear control -problem](https://python.quantecon.org/lqcontrol.html), we need to define +To map a problem into a {doc}`discounted optimal linear control problem`, we need to define - state vector $x_t$ and control vector $u_t$ - matrices $A, B, Q, R$ that define preferences and the law of @@ -700,6 +709,14 @@ Y_{t+1} = n 96.949 + (1 - n 0.046) Y_t $$ +For the case of a unit measure of firms, +$$ +\begin{aligned} +\int_0^1 y_{t+1}(\omega)\, d\omega &= h_0 + h_1 \int_0^1 y_{t}(\omega)\, dω + h_2 Y_t \\ +Y_{t+1} &= h_0 + h_1 Y_t + h_2 Y_t \\ +Y_{t+1} &= 96.949 + (1 - 0.046) Y_t +\end{aligned} +$$ ```{solution-end} ```