Notes: Hassan and Mertens (2011)

1. Introduction

In this note, I outline the main results in Hassan and Mertens (2011)¹ for use in a 5min presentation in Prof. Sargent‘s reading group.

This paper asks the question: “Suppose that you air dropped a bit of noise into equity prices, how much worse off would everyone be in the context of a macroeconomic model with production?” As it turns out, adding a bit of noise can have a first order effect on welfare as noising up equity prices causes agents to disinvest in firms in favor of riskless debt, leading to lower production and thus lower consumption as agents can only consume what has been produced. To answer this question, the authors overcome $2$ main challenges. First, they define a new equilibrium concept that allows for heterogeneous beliefs. Second, they solve for the equilibrium asset prices which vary non-linearly with their information set.²

To my knowledge, this paper is the first to address the costs of noisy asset prices in a dynamic general equilibrium setting; however, other papers such as Angeletos and Pavan (2007) have looked at the social costs and benefits of correct prices. Veldkamp (2011, Ch 8) also provides a good reference for solving noisy rational expectations models.

2. Model

Consider a standard linear homogeneous production function that produces output $Y_t$ using inputs of capital $K_t$ and labor $L_t$ . $\eta_t$ denotes the total factor productivity and evolves according to a diffusion process with mean $- \sigma_\eta^2/2$ and variance $\sigma_\eta^2$ :

(1) $\begin{align*} Y_t &= e^{\eta_t} \cdot F(K_t, L_t) \end{align*}$

Capital evolves according to the law of motion below where $\delta$ is the depreciation rate and $I_t$ is the aggregate investment level:

(2) $\begin{align*} K_{t+1} &= K_t \cdot \left( 1 - \delta \right) + I_t \end{align*}$

There are a continuum of households indexed on the real line by $i \in [0,1]$ . At the beginning of every period, each household $i$ gets a signal about tomorrow’s TFP with noise $\nu_t(i)$ that is distributed normally with mean $0$ and variance $\sigma_\nu^2$ :

(3) $\begin{align*} s_t(i) &= \eta_{t+1} + \nu_t(i) \end{align*}$

I refer to $s_t(i)$ as the “knowledge” of agent $i$ about tomorrow’s productivity. Given this knowledge, at time $0$ each agent then has a value function $V_t(i)$ over consumption streams $\left\{ C_t(i) \right\}_{t=0}^\infty$ given his relative investment in stocks rather than bonds $\left\{ \omega_t(i) \right\}_{t=0}^\infty$ :

(4) $\begin{align*} \begin{split} V_t(i) &= \max_{ \left\{ C_t(i) \right\}_{t=0}^\infty, \left\{ \omega_t(i) \right\}_{t=0}^\infty} \mathcal{E}_{i,t} \left[ \sum_{s=t}^\infty \beta^{s-t} \cdot \log C_s(i) \right] \\ &\text{s.t.} \\ W_{t+1} &= \left[ (1 - \omega_t(i)) \cdot (1 + r) + \omega_t(i) \cdot \left( 1 + \tilde{r}_{t+1} \right) \right] \\ &\qquad \times \left( W_t(i) + w_t \cdot L - C_t(i) \right) + \tau_t(i) \end{split} \end{align*}$

In the optimization program above, $\mathcal{E}_{i,t}$ denotes agent $i$ ‘s subjective views on how the future will unfold given his knowledge $s_t(i)$ , $W_t(i)$ denotes agent $i$ ‘s financial wealth at time $t$ , $\tilde{r}_{t+1}$ denotes the return on stocks, $\omega_t$ denotes the wage rate for all agents, $L$ denotes the amount of labor supplied by each household and $\tau_t(i)$ denotes the payments from the contingent claims trading. Equity prices are given by $Q_t$ which is related to dividends $D_t$ via the market return $\tilde{r}_{t+1}$ :

(5) $\begin{align*} 1 + \tilde{r}_{t+1} &= \frac{Q_{t+1} \cdot (1 - \delta) + D_{t+1}}{Q_t} \end{align*}$

Let $\mathbb{E}_{i,t}$ be the “true” rational expectations operator which is linked to each agent’s subjective expectations via a common mean $0$ shock $\tilde{\varepsilon}_t$ which has volatility $\sigma_{\varepsilon}^2$ :

(6) $\begin{align*} \begin{split} \mathbb{E}_{i,t} \left[ \cdot \right] &= \mathbb{E} \left[ \cdot \mid s_t(i), Q_t, K_t, B_{t-1}, \eta_t \right] \\ \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] &= \mathbb{E}_{i,t} \left[ \eta_{t+1} \right] + \tilde{\varepsilon}_t \end{split} \end{align*}$

The representative firm solves the maximization program below and tries to produce as much output as possible given the production technology $e^{\eta_t} \cdot F(K_t,L_t)$ and factor prices $w_t$ and $R_t$ :

(7) $\begin{align*} \Pi_t &= \max_{K_t^d,L_t^d} \left\{ e^{\eta_t} \cdot F\left( K_t^d, L_t^d \right) - w_t \cdot L_t^d - R_t \cdot K_t^d \right\} \end{align*}$

All profits are dispersed immediately to shareholders. Firms take the market price of capital $Q_t$ as given and then invest in capital accumulation according to the rule below where $Q_t \cdot I_t$ represents the value of acquiring $I_t$ units of capital, $-I_t$ represents cost from selling $I_t$ units of consumption good and $- (\chi/2) \cdot (I_t^2/K_t)$ represents an adjustment cost:

(8) $\begin{align*} A_t &= \max_{I_t} \left\{ Q_t \cdot I_t - I_t - \left( \frac{\chi}{2} \right) \cdot \frac{I_t^2}{K_t} \right\} \end{align*}$

Thus, we can characterize the dividends paid per share as:

(9) $\begin{align*} \begin{split} D_t &= R_t + \left( \frac{\chi}{2} \right) \cdot \frac{I_{t+1}^2}{K_{t+1}^2} \\ I_t &= \left( \frac{K_t}{\chi} \right) \cdot \left( Q_t - 1 \right) \end{split} \end{align*}$

With this machinery in place, I can now define an equilibrium in which agents have heterogeneous beliefs about firm productivity:

Definition (Equilibrium): Given a time path of shocks $\left\{ \eta_t, \tilde{\varepsilon}_t, \nu_t(i) \right\}$ an equilibrium in this economy is a time path of quantities $\left\{ C_t(i), B_t(i), W_t(i), \omega_t(i), \tau_t(i) \right\}$ $\left\{ C_t, B_t, W_t, \omega_t, K_t^d, L_t^d, Y_t, K_t, I_t \right\}$ , signals $\left\{ s_t(i) \right\}$ and prices $\left\{ Q_t, r, R_t, w_t \right\}$ with the properies:

$\left\{ C_t(i), \omega_t(i) \right\}$ solve the households’ maximization problem given the vector of prices, initial wealth and the random sequences $\left\{ \tilde{\varepsilon}_t, \tilde{\nu}_t(i) \right\}$ ;

$\left\{ K_t^d, L_t^d \right\}$ solve the representative firm’s maximization problem given the vector of prices;

$\left\{ I_t \right\}$ is the investment sector’s optimal policy given the vector of prices;

$\left\{ w_t \right\}$ clears the labor market, $\left\{ Q_t \right\}$ clears the stock market and $\left\{ R_t \right\}$ clears the market for capital services;

There is a perfectly elastic supply of the consumption good and of bonds in world markets where bonds pay the interest rate $r$ and the price of the consumption good is normalized to $1$ ;

$\left\{ \tau_t(i) \right\}$ are defined such that all households enter each period with the same amount of wealth; and

$\left\{ B_t(i), C_t, B_t, W_t, \omega_t \right\}$ are given by the identities:
(10) $\begin{align*} B_t(i) &= \left[ 1 - \omega_t(i) \right] \cdot \left( W_t(i) - C_t(i) \right) \\ X_t &= \int_0^1 X_t(i) \cdot di, \ \ \forall X \in \left\{ B, C, W \right\} \\ \omega_t &= \frac{Q_t \cdot K_{t+1}}{W_t - C_t} \end{align*}$

This definition collapses down to the standard noisy rational expectation equilibrium in the case of $\sigma_{\tilde{\varepsilon}} = 0$ :

Definition (Rational Expectations Equilibrium): A rational expectations equilibrium is an equilibrium as defined above in which the volatility of inference shocks $\sigma_{\tilde{\varepsilon}} = 0$ so that $\mathcal{E}_{i,t} = \mathbb{E}_{i,t}$ .

However, it also has a sensible interpretation even when agent’s have a common component to their belief heterogeneity:

Definition (Near Rational Expectations Equilibrium): A near rational expectations equilibrium allows for $\sigma_{\tilde{\varepsilon}} > 0$ . Such an equilibrium is $k\%$ stable if the welfare gain to an individual household of obtaining rational expectations is less that $k\%$ of consumption.

3. Financial Economy

In this section, I consider a world in which all agents hold beliefs that are slightly wrong in the same way. To formalize this idea, I define a new error term $\varepsilon_t$ which represents the difference between the average expectation of the total factor productivity at time $t$ in a world where $\sigma_{{\varepsilon}} > 0$ and the average expectation in a world where $\sigma_{\tilde{\varepsilon}} = 0$ :

(11) $\begin{align*} \varepsilon_t &= \gamma \cdot \tilde{\varepsilon}_t \\ &= \int_{\tilde{\varepsilon}_t>0} \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di - \int_{\tilde{\varepsilon}_t=0} \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di \end{align*}$

In particular, by expressing $\gamma = \varepsilon/\tilde{\varepsilon}$ as a ratio we can ask: How much market forces amplify ( $\gamma > 1$ ) the existing common error?

To solve for the equilibrium values in the financial economy, I start with the $2$ Euler equations linking consumption in subsequent periods to the riskless rate of return and the rate of return on equities. Let $\Theta_t = \left\{ K_t, B_{t-1}, \eta_t, \eta_{t+1}, \tilde{\varepsilon}_t \right\}$ denote the state space. Then, I can express these $2$ Euler equations as:

(12) $\begin{align*} \begin{split} C_t \left( \Theta_t, \nu_t(i) \right)^{-1} &= \beta \cdot \mathcal{E}_{i,t} \left[ \left( 1 + \tilde{r}_{t+1} \right) \cdot C_{t+1} \left( \Theta_{t+1}, \nu_{t+1}(i) \right)^{-1} \right] \\ C_t \left( \Theta_t, \nu_t(i) \right)^{-1} &= \beta \cdot \mathcal{E}_{i,t} \left[ \left( 1 + r \right) \cdot C_{t+1} \left( \Theta_{t+1}, \nu_{t+1}(i) \right)^{-1} \right] \end{split} \end{align*}$

Next, given these $2$ Euler equations, I now face a complicated problem of trying to figure out households’ optimal allocation of consumption, stock purchases and bond purchases given that $1)$ households’ expectations depend non-linearly on their beliefs about the current $\eta_t$ and $2)$ asset prices depend non-linearly on the average belief about future productivity. In a related paper, Mertens (2009) shows how to execute a non-linear change of variables that transforms the problem into a optimization program that resembles a more standard noisy rational expectations equilibrium. After this change of variables, I write the stock price as:

(13) $\begin{align*} \begin{split} \hat{q}_t &= \int \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di \\ &= \int \mathbb{E}_{i,t} \left[ \eta_{t+1} \mid \hat{q}_t , s_t(i) \right] \cdot di + \tilde{\varepsilon}_t \end{split} \end{align*}$

i.e., the stock price equals the market expectation of $\eta_{t+1}$ .

I now proceed down the lines of the solution to a standard noisy rational expectations model with value $\eta_{t+1}$ and noise term $\tilde{\varepsilon}_t$ . I conjecture that the price $\hat{q}_t$ is an affine function of the value and the noise term with parameters $\pi_0$ , $\pi_1$ and $\gamma$ :

(14) $\begin{align*} \hat{q}_t &= \pi_0 + \pi_1 \cdot \eta_{t+1} + \gamma \cdot \tilde{\varepsilon}_t \end{align*}$

Thus, we can interpret $\gamma$ as the loading on the noise term in this pricing rule. This is quite natural as the price $\hat{q}_t$ is the average expected value of the total factor productivity $\eta_{t+1}$ and $\int_{\tilde{\varepsilon}_t=0} \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di$ is the price in a world with no common error $\sigma_{\tilde{\varepsilon}} = 0$ . Thus, the loading on the noise term should be $\gamma$ , the amplification parameter defined above.

Next, using results from Mertens (2009), I write the true rational expectation of tomorrow’s total factor productivity $\mathbb{E}_{i,t} \left[ \eta_{t+1} \right]$ as an affine function of each agent’s beliefs and the prevailing price using the coefficients $A_0$ , $A_1$ and $A_2$ which derive from the change of variables procedure which I do not outline here:

(15) $\begin{align*} \mathbb{E}_{i,t} \left[ \eta_{t+1} \right] &= A_0 + A_1 \cdot s_t(i) + A_2 \cdot \hat{q}_t \end{align*}$

By adding in the common noise term and then summing across agents $i \in [0,1]$ , I get another affine expression below which links the average expectation of tomorrow’s total factor productivity to its true value and the amplified noise term:

(16) $\begin{align*} \begin{split} \int \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di &= \left( A_0 + A_2 \cdot \pi_0 \right) \\ &\qquad \qquad + \left( A_1 + A_2 \cdot \pi_1 \right) \cdot \eta_{t+1} \\ &\qquad \qquad \qquad \qquad + \left( A_2 \cdot \gamma + 1 \right) \cdot \tilde{\varepsilon}_t \end{split} \end{align*}$

Using this expression, I can then show that as households look more and more to the price of capital when forming prices, the amplification of common shocks to expectations via the pricing mechanism grows:

Proposition (Inference Shock Amplifier): The more weight the households place on the market price of capital when forming their expectations about $\eta_{t+1}$ , the larger is the error in market expectations relative to $\tilde{\varepsilon}_t$ :

(17) $\begin{align*} \gamma &= \frac{1}{1 - A_2} \end{align*}$

The proof of this proposition follows directly from taking first order conditions:

Proof: Combining the equations for $\hat{q}_t$ and $\int \mathcal{E}_{i,t} \left[ \eta_{t+1} \right] \cdot di$ above yields a system of $3$ equations and $3$ unknowns:

(18) $\begin{align*} \pi_0 &= A_0 + A_2 \cdot \pi_0 \\ \pi_1 &= A_1 + A_2 \cdot \pi_1 \\ \gamma &= 1 + A_2 \cdot \gamma \end{align*}$

Solving for the unknowns yields:

(19) $\begin{align*} \pi_0 &= \frac{A_0}{1-A_2} \\ \pi_1 &= \frac{A_1}{1 - A_2} \\ \gamma &= \frac{1}{1 - A_2} \end{align*}$

Next, I look at how much information prices can hold in regimes that vary based on the the magnitude of the underlying common shock. For instance, consider the following thought experiment. Think about a world in which there was no common shock. If you knew nothing about asset values and then looked at prices, how precisely could you now pin down those values? To within $\$5$ ? $\$1$ ? $\$10$ ? Now consider a world in which $\tilde{\varepsilon} > 0$ so that agents experience a common shock to their preferences. Now, for instance, agents might revise their expectations upward, hold more stocks as a result, and then influence others to do the same for capital gains purposes as the price depends on the average value of agent’s expectations. In such a world with a positive feedback loop, how much worse are prices at conveying the true value of the asset? If in the world with no common shock, you could see a price of $\$30$ and know that the asset was worth something between $\$29$ and $\$31$ , in a world with a common shock have the bounds now expanded to $\$30 \pm \$5$ ?

The proposition below dictates that the bounds will be increasing in the volatility of the common shock:

Proposition (Information Absorbency): The amount of information aggregated in the stock prices decreases with the volatility of the common shock $\sigma_{\varepsilon}$ :

(20) $\begin{align*} 0 &> \frac{\partial \pi_1}{\partial \sigma_{\tilde{\varepsilon}}} \end{align*}$

The proof comes via Mathematica by taking partial derivative of $\pi_1 = A_1/(1 - A_2)$ with respect to $\sigma_{\varepsilon}$ where $\sigma_{\varepsilon}$ is bound up in the $A_1$ and $A_2$ terms so I omit it here.

4. Real Economy

In the previous section I showed how to derive equilibrium values for the financial economy and that the market became substantially less efficient as the volatility of the common shock increased due to the feedback loop between beliefs and asset prices. I now turn to the real economy and show that common errors in expectations distort aggregate consumption in the steady state as well. First, suppose that in equilibrium stock prices are log-normally distributed with variance $\sigma^2$ . Then we know that the optimal consumption plan will be myopically proportional to his wealth as agent’s have log utility:

(21) $\begin{align*} C_t(i) &= \left( 1 - \beta \right) \cdot W_t(i) \end{align*}$

What’s more, his optimal stock holdings will also follow the standard log utility rule:

(22) $\begin{align*} \omega_t(i) &= \frac{\mathcal{E}_{i,t} \left[ 1 + \tilde{r}_{t+1} \right] - \left( 1 + r\right)}{\sigma^2} \end{align*}$

Next, I characterize the steady state levels of stock holdings as well as the marginal product of capital:

Proposition (Steady State): The equilibrium has a unique stochastic steady state if and only if $\beta < (1 + r)^{-1}$ . At the steady state, the aggregate degree of stock holdings is:

(23) $\begin{align*} \omega_{ss} &= \sqrt{\frac{1}{\sigma^2} \cdot \left( \frac{1 - \beta}{\beta} - r \right)} \end{align*}$

and the stochastic steady state capital stock is characterized by:

(24) $\begin{align*} F_K(K_{ss},L) &= \left( 1 + \delta \cdot \chi \right) \cdot \left( r + \omega_{ss} \cdot \sigma^2 + \delta \right) \end{align*}$

The restriction on the time preference parameter $\beta$ makes sure that agents do not want to perpetually accumulate capital. The steady state stock holdings links the gap between the time preference parameter and the riskless rate to agent’s depend to hold capital with a volatility of $\sigma$ . The full proof of this result is given below:

Proof: First, I characterize the steady state marginal product of capital. To do this, I start by linking it to the dividend payout rate:

(25) $\begin{align*} D_{t+1} &= e^{\eta_{t+1}} \cdot \left( \frac{d}{dK} F(K_{t+1}, L) \right) \\ \mathbb{E}_{ss} \left[ D_{ss} \right] &= \left( \frac{d}{dK} F(K_{ss}, L) \right) \end{align*}$

Next, I observe that the excess return on capital can be linked to the marginal product of capital via the relationship:

(26) $\begin{align*} r + \omega_{ss} \cdot \sigma^2 &= - \delta + \left( \frac{1}{1 + \delta \cdot \chi} \right) \cdot \left( \frac{d}{dK} F(K_{ss},L) \right) \end{align*}$

This last statement yield the desired result after rearranging. Next, I turn to the steady state asset holdings which falls out of the budget constraint. First, I rewrite the budget constraint so that it only contains terms from $2$ periods:

(27) $\begin{align*} \left( 1 + r \right) \cdot B_{t-1} + &\left( Q_t \cdot \left( 1 - \delta \right) + D_t \right) \cdot K_t = Q_t \cdot K_{t+1} + B_t + C_t \\ &= Q_t \cdot \left( 1 - \delta \right) \cdot K_t + Q_t \cdot I_t + B_t + C_t \end{align*}$

Next, I write the aggregate wealth, consumption and borrowing terms as functions of the steady state asset holdings:

(28) $\begin{align*} C_{ss} &= (1 - \beta) \cdot W_{ss} \\ \beta \cdot W_{ss} &= K_{ss} \cdot (1 + \delta \cdot \chi) + B_{ss} \\ B_{ss} &= \beta \cdot W_{ss} \cdot \left( 1 - \omega_{ss} \right) \end{align*}$

Finally, I substitute these terms back in and rearrange to yield a result linking the steady state asset holdings to the constants:

(29) $\begin{align*} \left( 1 + \delta \cdot \chi \right) \cdot K_{ss} &= \beta \cdot W_{ss} \cdot \omega_{ss} \\ B_{ss} &= \left( \frac{1 - \omega_{ss}}{\omega_{ss}} \right) \cdot \left( 1 + \delta \cdot \chi \right) \cdot K_{ss} \end{align*}$

Next, I look at the steady state capital accumulation rate of the economy conditional on the volatility of the stock market:

Proposition (Capital Accumulation): An increase in the conditional variance of stock returns decreases the level of capital stock and total output:

(30) $\begin{align*} 0 &> \frac{\partial K_{ss}}{\partial \sigma} \end{align*}$

This result says that the economy will acquire less and less capital as the volatility of stock returns increases. This result stems directly from the full derivative of the steady state capital level:

Proof: First, start with the equation above characterizing the steady state stock holdings $\omega_{ss}$ and then take the full derivative:

(31) $\begin{align*} \frac{d}{d\sigma} K_{ss} &= \frac{1 + \delta \cdot \chi}{\frac{d^2}{(dK)^2} F(K_{ss},L)} \cdot \sqrt{\frac{1 - \beta}{\beta} - r} \end{align*}$

Rearranging terms and signing the RHS yields the necessary relationship.

Thus, in this section I showed that the economy is worse off when stock market volatility increases as a result of a common error in expectations.

See Tarek Hassan and Thomas Mertens ↩
By contrast, in the standard noisy rational expectations setup (e.g., see here) all outcomes are linearly related. ↩