Geometric Interpretation of Noisy Rational Expectations Equilibrium

1. Introduction

In this post, I solve a simple noisy rational expectations equilibrium model from Grossman and Stiglitz (1980) and then give a geometric interpretation of their result. First, in Section 2 I set up and solve a noisy rational expectations model. Then, in Section 3 I show how to display the $2$ linear projections embedded in the model on a 3D figure. I found the figure to be a useful way of keeping track of the assumptions in more complicated settings.

2. Solution

Consider a world with a single period and only $1$ asset with price $p$ and aggregate demand $x$ :

(1) $\begin{align*} p &= a + b \cdot x \\ &= a + b \cdot \left( z + \varepsilon \right) \end{align*}$

The coefficient $a$ denotes the average price while the coefficient $b$ represents the price’s responsiveness to aggregate demand changes. i.e., $b$ represents the amount by which a restaurant would change its prices if all of the sudden twice as many people started showing up each evening. Suppose that there are some traders with knowledge of the true value of the asset $v$ , but also other traders who trade randomly and demand an amount $\varepsilon \sim \mathtt{N}(0,\sigma_{\varepsilon}^2)$ . Suppose that the value of the asset is drawn from a distribution $\mathtt{N}(\mu_v,\sigma_v^2)$ . The coefficients $a$ and $b$ in the equation above are equilibrium objects which I will solve for below as is the value $z$ which is the amount demanded by the informed agents which is also linear in the asset value with coefficients $c$ and $d$ :

(2) $\begin{align*} z = c + d \cdot v \end{align*}$

There is a market maker who sets the equilibrium price in order to break even. Let $\Pi$ denote the informed agent’s utility from trading:

(3) $\begin{align*} \Pi &= \max_{z} \left\{ \mathbb{E} \left[ (v - p) \cdot z \mid v \right] \right\} \\ &= \max_{z} \left\{ \mathbb{E} \left[ (v - a - b \cdot z - b \cdot \varepsilon) \cdot z \mid v \right] \right\} \end{align*}$

Differentiating yields an expression for the optimal holdings of the informed traders given the true asset value:

(4) $\begin{align*} z &= - \frac{a}{2 \cdot b} + \left( \frac{1}{2 \cdot b} \right) \cdot v \end{align*}$

Next, to solve for the coefficient values $(a, b, c, d)$ as functions of the model primitives $(\mu_v, \sigma_v, \sigma_{\varepsilon})$ , I enforce the break even condition for the market maker which demands that the price of the asset be equal to the expected value of the asset conditional on observing the aggregate asset demand:

(5) $\begin{align*} p &= \mathbb{E} \left[ v \mid x \right] \\ &= \mathbb{E} \left[ v \right] + \frac{\mathbb{C} \left[ x, v\right]}{\mathbb{V}\left[ x \right]} \cdot \left( x - \mathbb{E}[x] \right) \\ &= \mu_v + \frac{\mathbb{C} \left[ - \frac{a}{2 \cdot b} + \left( \frac{1}{2 \cdot b} \right) \cdot v + \varepsilon, v\right]}{\mathbb{V}\left[ - \frac{a}{2 \cdot b} + \left( \frac{1}{2 \cdot b} \right) \cdot v + \varepsilon \right]} \cdot \left( x - \left[ - \frac{a}{2 \cdot b} + \frac{\mu_v}{2 \cdot b} \right] \right) \\ &= \mu_v + \left( \frac{\left( \frac{1}{2 \cdot b} \right) \cdot \sigma_v^2}{ \left( \frac{1}{2 \cdot b} \right)^2 \cdot \sigma_v^2 + \sigma_{\varepsilon}^2} \right) \cdot \left( \frac{a}{2 \cdot b} - \frac{\mu_v}{2 \cdot b} \right) \\ &\qquad \qquad + \left( \frac{\left( \frac{1}{2 \cdot b} \right) \cdot \sigma_v^2}{ \left( \frac{1}{2 \cdot b} \right)^2 \cdot \sigma_v^2 + \sigma_{\varepsilon}^2} \right) \cdot x \end{align*}$

Enforcing this condition leads to an expression for $p$ that is linear in the aggregate asset demand $x$ . Thus, I observe the $b$ must be equal to the coefficient on $x$ from the equation above and solve for $b$ :

(6) $\begin{align*} b &= \frac{\left( \frac{1}{2 \cdot b} \right) \cdot \sigma_v^2}{ \left( \frac{1}{2 \cdot b} \right)^2 \cdot \sigma_v^2 + \sigma_{\varepsilon}^2} \\ &= \frac{\sigma_v}{2 \cdot \sigma_{\varepsilon}} \end{align*}$

We see that the market maker tends to change prices more in response to an aggregate demand shock when the asset value is more volatile and when the noise trader demand is less volatile–i.e., when asset value changes aer more likely and when there is less demand noise for the informed traders to hide behind.

3. Geometric Interpretation

The plot below captures the essence of the intuition embedded in the noisy rational expectations model. The core idea is that the market maker cannot precisely distinguish a random fluctuation in demand from a shift in demand due to a shift in the asset value. Thus, when the market maker sets the price, he looks at the aggregate demand and shades the price a bit higher if he observes a high demand or a bit lower if he observes a surprisingly low demand schedule, but does not do so on a one for one schedule: $0 < b < 1$ . By using normally distributed random variables as well as linear pricing and informed demand rules, we can then get nice expressions for the coefficients of interest.

To read this plot, step into the shoes of the market maker and have a look at the blue side of the figure which shows the relationship between the price (i.e., the market maker’s expectation of the value on the y-axis) and the aggregate demand (x-axis). The line $p = a + b \cdot x$ shows the price you will set if you observe an aggregate demand of $x$ . Note that is $x = 0$ , you will set the price of $a$ –the y-intercept.

Where does this pricing rule come from? When you observe the aggregate demand of $x$ , your best guess for the informed demand schedule is $z$ as $\mathbb{E} [ \varepsilon ] = 0$ . This best guess is displayed in the plot by the mapping through the $z = u + v \cdot x$ line on the green floor of the figure over to the z-axis. On the red wall of the figure, we see that this choice of $z$ has to map over to a realized value $v$ that is a linear function of $z$ and is equal to your choice of $p$ . This is the double projection and fixed point problem that pins down the equilibrium values. For instance, note that at $z=0$ , it must be the case that both the $v$ and $p$ functionals cross the y-axis at the same place as $\mathbb{E}[\varepsilon]=0$ .

1. Introduction

2. Solution

3. Geometric Interpretation

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