A11 Appendix to Chapter 11

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A11.1 A robust formulation of Borovicka’s heterogeneous beliefs economy

Authors: Lars Peter Hansen and Jiaying (Jessie) Liao

The following is based on Jaroslav Borovicka’s paper [Borovička, 2020].

A11.1.1 Preferences

There are two investors with the same subjective rate of discount \(\delta\) and the same unitary elasticity of intertemporal substitution. Borovicka does not impose the latter restriction and solves the model using a different approach. In what follows we let superscripts index the individuals and include parentheses when we raise variables to powers.

A11.1.2 Beliefs

Investor \(n\) believes that the exogenously specified aggregate state of the economy evolves as:

\[ d \log Y_t = (\mu_y + \sigma_y u^n) dt + \sigma_y dW_t^n. \]

where \(W^n\) is a standard Brownian motion under person \(n\)’s baseline beliefs. Output is split into consumption allocated between the two investors. Let \(\zeta^n = C^n / Y\), where \(C^n\) is the consumption of investor \(n\). Then

(1)\[\zeta^1 + \zeta^2 = 1.\]

A11.1.3 Robust adjustment to the stochastic evolution

Multiplicative martingales capture the twisted beliefs. They evolve as:

\[ d \log M_t^n = -\frac{(H_t^n + u^n)^2}{2} dt + (H_t^n + u^n)dW_t. \]

Construct an endogenous state variable process

\[ X_t = \log M_t^2 - \log M_t^1. \]

A11.1.4 Value function construction for a robust planner

Guess a value function of the form:

\[ (m^1 + m^2)y + m^1 v(x) \]

Construct the planner HJB equation:

\[\begin{split} \begin{aligned} 0 =& \max_{\zeta^1, \zeta^2, \zeta^1 + \zeta^2 = 1} \min_{h^1, h^2} -\delta v + \delta \log(\zeta^1) + \delta \exp(x)\log(\zeta^2) \\ &+ \frac{dv}{dx} \left[ \frac{(h^1 + u^1)^2 - (h^2 + u^2)^2}{2} \right] \\ &+ \left[\mu_y + \sigma_y(h^1 + u^1)\right] + \exp(X)\left[\mu_y + \sigma_y(h^2 + u^2)\right] \\ &+ \frac{d^2 v}{dx^2} \frac{(h^2 + u^2 - h^1 - u^1)^2}{2} + \frac{dv}{dx}(h^2 + u^2 - h^1 - u^1)(h^1 + u^1) \\ &+ \xi^1 \frac{(h^1)^2}{2} + \exp(x)\xi^2\frac{(h^2)^2}{2}. \end{aligned} \end{split}\]

(where we divided through by \(m^1\).)

Combining the first derivative terms:

\[\begin{split} \begin{aligned} 0 =& \max_{\zeta^1, \zeta^2, \zeta^1 + \zeta^2 = 1} \min_{h^1, h^2} -\delta v + \delta \log(\zeta^1) + \delta \exp(X)\log(\zeta^2) \\ &+ \mathcal A v\frac{(h^2 + u^2 - h^1 - u^1)^2}{2} \\ &+ \mu_y + \sigma_y(h^1 + u^1) + \exp(x)\left[\mu_y + \sigma_y(h^2 + u^2)\right] \\ &+ \xi^1\frac{(h^1)^2}{2} + \exp(x)\xi^2\frac{(h^2)^2}{2}. \end{aligned} \end{split}\]

where

\[ \mathcal A v \eqdef \frac{d^2v}{dx^2} - \frac{dv}{dx}. \]

A11.1.5 Solution

Outer Problem: Maximization over \(\zeta^1, \zeta^2\)

The first-order condition with respect to \(\zeta^1\) gives:

\[ \frac{1}{\zeta^1} = \frac{\exp(x)}{\zeta^2}. \]

Combined with the constraint (1), this yields logistic probabilities:

\[ \zeta^{1*} = \frac{1}{1 + \exp(x)}, \qquad \zeta^{2*} = \frac{\exp(x)}{1 + \exp(x)}. \]

Inner Problem: Minimization over \(h^1, h^2\)

The first-order conditions are:

\[ -\mathcal A v(h^2 + u^2 - h^1 - u^1) + \sigma_y + \xi^1 h^1 = 0, \]

and $\( \mathcal A v(h^2 + u^2 - h^1 - u^1) + \exp(x)\sigma_y + \exp(x)\xi^2 h^2 = 0. \)$

Expressed in matrix notation:

\[\begin{split} \begin{bmatrix} \mathcal A v + \xi^1 & -\mathcal A v \\ -\mathcal A v & \mathcal A v + \exp(x)\xi^2 \end{bmatrix} \begin{bmatrix} h^1 \\ h^2 \end{bmatrix} = \begin{bmatrix} -\sigma_y + \mathcal A v(u^2 - u^1) \\ -\exp(x)\sigma_y - \mathcal A v(u^2 - u^1) \end{bmatrix}. \end{split}\]

Rearrange this equation:

\[\begin{split} \begin{bmatrix} \mathcal A v + \xi^1 & -\mathcal A v \\ -\mathcal A v & \mathcal A v + \exp(x)\xi^2 \end{bmatrix} \begin{bmatrix} h^1 + u^1 \\ h^2 + u^2 \end{bmatrix} = \begin{bmatrix} -\sigma_y + \xi^1 u^1 \\ -\exp(x)\sigma_y + \exp(x)\xi^2 u^2 \end{bmatrix}. \end{split}\]

Solve the resulting HJB equation using so-called “natural boundaries”. Construct the allocation rules and minimized twists in the drifts for each of the economic agents. In particular, find: \(g^{1*}(x)\), \(g^{2*}(x)\), \(\zeta^{1*}(x)\), and \(\zeta^{2*}(x)\) where:

\[ g^{*n}(x) \overset{\mathrm{def}}{=} h^{n*}(x) + u^n. \]

A11.1.6 Chernoff entropy

Perform this computation in three steps.

i) Construct a baseline specification of \(X\) to be:

\[ dX_t = \mu_x(X_t)dt + \sigma_x(X_t)dW_t \]

where

\[ \mu_x \eqdef \frac{(g^{*1})^2}{2} - \frac{(g^{*2})^2}{2} \]

\[ \sigma_x \eqdef g^{*2} - g^{*1}. \]

Note that \(\mu_x\) and \(\sigma_x\) inherit state dependence from \(g^{*1}\) and \(g^{*2}\).

ii) For each investor \(n\) and each \(0 < \alpha < 1\), solve the principal eigenvalue equation assuming natural boundaries:

\[ (\mu_x + \alpha \sigma_x g^{*n})\psi' + \frac{(\sigma_x)^2}{2}\psi'' - \frac{\alpha(1-\alpha)(g^{*n})^2}{2}\psi = \lambda \psi \]

for \(\lambda = \lambda_\alpha^n\) and \(\psi = \psi_\alpha^n\).

iii) For each investor \(n\),

\[ \text{Chernoff entropy} = \max_{0 < \alpha < 1} -\lambda_\alpha^n. \]

The principal eigenvalue equation reconsidered

For each \(0 < \alpha < 1\), the solution to the principal eigenvalue for one investor also provides a solution for the other one. To establish this, first write the equation for investor one:

\[ (\mu_x + \alpha \sigma_x g^{*1})\psi' + \frac{(\sigma_x)^2}{2}\psi'' - \frac{\alpha(1-\alpha)(g^{*1})^2}{2}\psi = \lambda\psi. \]

Let

\[ \tilde\psi(x) = \exp(-\alpha x)\psi(x). \]

Then

\[ \tilde\psi'(x) = \exp(-\alpha x)\left[\psi'(x) - \alpha\psi(x)\right]. \]

\[ \tilde\psi''(x) = \exp(-\alpha x)\left[\psi''(x) - 2\alpha\psi'(x) + \alpha^2\psi(x)\right]. \]

We apply investor two’s second-order differential equation to \(\tilde\psi(x)\).

i) Consider the implied \(\psi''\) contribution:

\[ \exp(-\alpha x)\frac{(\sigma_x)^2(x)}{2}\psi''(x). \]

ii) Consider the implied \(\psi'\) contribution. Observe that

\[ \mu_x + \alpha\sigma_x g^{*2} = \mu_x + \alpha(g^{*2})^2 - \alpha g^{*2}g^{*1} \]

\[ -\alpha(\sigma_x)^2 = -\alpha(g^{*2} - g^{*1})^2 = -\alpha(g^{*2})^2 - \alpha(g^{*1})^2 + 2\alpha g^{*2}g^{*1} \]

Thus

\[\begin{split} \begin{aligned} \mu_x + \alpha\sigma_x(g^{*2}) - \alpha(\sigma_x)^2 &= \mu_x - \alpha(g^{*1})^2 + \alpha g^{*2}g^{*1} \\ &= \mu_x + \alpha(g^{*2} - g^{*1})g^{*1} \\ &= \mu_x + \alpha\sigma_x g^{*1}. \end{aligned} \end{split}\]

From this calculation, we see that the first-order term can be expressed as:

\[ \exp(-\alpha x)\left[\mu_x(x) + \alpha\sigma_x(x)g^{*1}(x)\right]\psi'(x). \]

iii) Consider the implied \(\psi\) contribution. Observe that

\[\begin{split} \begin{aligned} &-\alpha\left(\mu_x + \alpha\sigma_x g^{*2}\right) + \frac{\alpha^2(\sigma_x)^2}{2} - \frac{\alpha(1-\alpha)(g^{*2})^2}{2} \\ &= -\alpha\frac{(g^{*1})^2}{2} + \alpha\frac{(g^{*2})^2}{2} - \alpha\frac{(g^{*2})^2}{2} - \alpha^2(g^{*2})^2 + \alpha^2 g^{*2}g^{*1} + \frac{\alpha^2}{2}(g^{*2} - g^{*1})^2 + \frac{\alpha^2}{2}(g^{*2})^2 \\ &= -\alpha\frac{(g^{*1})^2}{2} + \frac{\alpha^2(g^{*1})^2}{2} \\ &= -\frac{\alpha(1-\alpha)}{2}(g^{*1})^2. \end{aligned} \end{split}\]

From this calculation, we see that the level term can be expressed as:

\[ -\exp(-\alpha x)\frac{\alpha(1-\alpha)}{2}\left[g^{*1}(x)\right]^2\psi(x). \]

With these calculations, it follows the \(\lambda^2 = \lambda^1\) and \(\psi^2(x) = \exp(-\alpha x)\psi^1(x)\) and “solve” the principal eigenvalue problem for investor two given that \(\lambda_1\) and \(\psi_1\) solve the principal eigenvalue problem for investor one. For this constructed solution to be of interest, we must verify that the implied \(\alpha\) stochastic evolutions induce stable stochastic dynamics.

To check stochastic stability,

i) Form a drift:

\[ \mu_{\alpha^*}^n = \mu_x + \alpha^*\sigma_x g^{*n} + (\sigma_x)^2(\log \psi^{*n})' \]

and use \((\sigma_x)^2\) as the diffusion coefficient where \(\alpha^*\) solves the maximization problem and \(\psi^{*n}\) is the candidate eigenfunction for person \(n\). Note that since \(\log\psi^{*2}(x) = \log\psi^{*1}(x) - \alpha^*x\),

\[\begin{split} \begin{aligned} \mu_{\alpha^*}^2 &= \mu_x + \alpha^*(g^{*2} - g^{*1})g^{*2} - \alpha^*(g^{*2} - g^{*1})^2 + (\sigma_x)^2(\log\psi^{*1})' \\ &= \mu_x + \alpha^*(g^{*2} - g^{*1})g^{*1} + (\sigma_x)^2(\log\psi^{*1})' \\ &= \mu_x + \alpha^*\sigma_x g^{*1} + (\sigma_x)^2(\log\psi^{*1})' \\ &= \mu_\alpha^{*1}. \end{aligned} \end{split}\]

ii) Check if the implied scalar diffusion has a stationary density for \(n = 1\). Note that the implied scalar diffusion for \(n = 2\) will be the same.

Stationary region in parameter space

A11.1.7 Interactive page for the two investor, heterogenous belief economy

A11.2 Computations for the robust second-order expansion

Write the system of interest, including the state equations (11.31), the consumption equation and static constraint (11.32), the first-order conditions (11.33), and the co-state evolution (11.34) as:

\[\begin{align*} X_{t+1} \left( \mathsf{q} \right) = & \hspace{.2cm} \psi^x \left[D_t \left( \mathsf{q} \right), X_{t} \left( \mathsf{q} \right), {\sf q} W_{t+1}, {\sf q} \right], \cr \log G_{t+1} \left( \mathsf{q} \right) - \log G_t \left( \mathsf{q} \right) = & \hspace{.2cm} \psi^g \left[ D_t \left( \mathsf{q} \right), X_{t} \left( \mathsf{q} \right), {\sf q} W_{t+1}, {\sf q} \right], \cr {\widehat C}_t \left( \mathsf{q} \right) = & \hspace{.2cm} \kappa \left[D_t \left( \mathsf{q} \right), X_{t} \left( \mathsf{q} \right) \right] + {\widehat G}_t \left( \mathsf{q} \right), \cr 0 = & \hspace{.2cm} \phi[D_t({\sf q} ), X_t({\sf q})], \cr Q_t {\mathbb E} \left( N_{t+1} H_{t+1} \mid {\mathfrak A}_t \right) + P_t L_{t} - M_t = & \hspace{.2cm} 0 \end{align*}\]

where

\[\begin{align*} Q_t \eqdef & \hspace{.2cm} \beta \exp\left[(1-\rho) \left( {\widehat R}_t - {\widehat V}_t \right) \right] \cr \cr P_t \eqdef & \hspace{.2cm} (1-\beta) \exp\left[(\rho-1) \left( {\widehat V}_t - {\widehat G}_t \right) \right] \cr \cr H_{t+1} \eqdef & \hspace{.2cm} \begin{bmatrix} \psi_{d'}^x (D_t, X_t, W_{t+1})' & \psi_{d'}^g (D_t, X_t, W_{t+1}) \cr \psi_{x'}^x (D_t, X_t, W_{t+1})' & \psi_{x'}^g (D_t, X_t, W_{t+1})' \cr \psi_{g'}^x (D_t, X_t, W_{t+1})' & 1 \end{bmatrix} \begin{bmatrix} MX_{t+1} \cr MG_{t+1} \end{bmatrix} \ \cr \cr L_t \eqdef & \hspace{.2cm} \begin{bmatrix} \exp \left[ (1-\rho) \left({\widehat C}_t - {\widehat G}_t \right) \right] \begin{bmatrix} \kappa_d(D_t,X_t) \cr \kappa_x(D_t, X_t) \cr 1 \end{bmatrix} \end{bmatrix} \cr M_t \eqdef & \hspace{.2cm} \begin{bmatrix} 0 \cr MX_t \cr MG_t \end{bmatrix} + \begin{bmatrix} \phi_{d'}(D_t, X_t)' \cr \phi_{x'}(D_t,X_t)' \cr 0 \end{bmatrix} MS_t . \end{align*}\]

Here, for computational purposes, we use that \({\widehat C}_t - {\widehat V}_t = \left({\widehat C}_t - {\widehat G}_t\right) - \left({\widehat V}_t - {\widehat G}_t\right)\). We solve for \( {\widehat C}_t - {\widehat G}_t, D_t, MX_t\) as a function of \(X_t\). The objects: \({\widehat C} - {\widehat G}, D\) and \(MX\) are sometimes referred to as jump variables since we do not impose initial conditions for these variables as part of a solution.

Our solution will entail an iteration. We will impose a specification for \(Q^1\), \(Q^2\), and \(N\) and find an approximate solution for the dynamical system. Then given this solution, we will compute a new implied solution for \(Q^1\), \(Q^2\), and \(N\). We then iterate this until we achieve numerical convergence. We use second-order approximations for both steps.

A11.3 Some steady state calculations

Observe from the recursive utility updating that:

\[\begin{align*} {\widehat V}_t^0 - {\widehat G}_t^0 & = \frac 1 {1 - \rho} \log \left[ (1 - \beta) \exp\left[(1 - \rho) \left( {\widehat C}_t^0 - {\widehat G}_t^0\right)\right] + \beta \exp\left[(1 - \rho) \left( {\widehat R}_t^0 - {\widehat G}_t^0\right)\right] \right] \cr {\widehat R}_t^0 - {\widehat G}_t^0 & = {\widehat V}_{t+1}^0 - {\widehat G}_{t+1}^0 + {\widehat G}_{t+1}^0 - {\widehat G}_t^0. \end{align*}\]

In the steady state we view this as two equations in three variables: \({\widehat V}_t^0 - {\widehat G}_t^0 , {\widehat R}_t^0 - {\widehat G}_t^0\) and \({\widehat G}_{t+1}^0 - {\widehat G}_t^0\), each of which we assume is time invariant.
We construct

\[\begin{align*} Q^0_t = & \beta \exp\left[(1-\rho)\left( {\widehat R}_t^0 - {\widehat V}_t^0 \right) \right] \cr = & \beta \exp\left[(1-\rho) \left({\widehat G}_{t+1}^0 - {\widehat G}_t^0 \right) \right], \end{align*}\]

and

\[P_t^0 = (1-\beta) \exp\left[(\rho-1) \left( {\widehat V}_t^0 - {\widehat G}_t^0 \right) \right] \]

in the steady state, and use them to construct the remaining steady-state equations:

\[Q_t^0 H_{t+1}^0 + P_t^0 - M_t^0 = 0.\]

In this equation, \(H_{t+1}^0\), \(L_t^0\), and \(M_t^0\) are constructed from the formulas for \(H_{t+1}\), \(L_t\), and \(M_t\), defined previously, by setting the shock vector to zero and treating the relevant variables as time invariant.

A11.4 \(Q\) and \(P\) derivatives

For the order one, write

\[Q_t^1 \eqdef (1-\rho) Q_t^0 \left( {\widehat R}_t^1 - {\widehat V}_t^1\right)\]

To compute this contribution, we use equation (11.17) to write

\[{\widehat R}_t^1 - {\widehat C}_t^1 = \frac 1 \lambda \left({\widehat V}_t^1 - {\widehat C}_t^1\right).\]

We then construct

\[\begin{align*} {\widehat R}_t^1 - {\widehat V}_t^1 =& \left({\widehat R}_t^1 - {\widehat C}_t^1\right) + \left({\widehat C}_t^1 - {\widehat V}_t^1\right) \cr = & \left(\frac {1 - \lambda } \lambda \right) \left[ \left({\widehat V}_t^1 - {\widehat G}_t^1\right) - \left({\widehat C}_t^1 - {\widehat G}_t^1 \right)\right] . \end{align*}\]

To compute \({\widehat V}_t - {\widehat G}_t \), we rewrite (11.22) as:

\[\begin{align*} & {\widehat V}_t^1 - {\widehat G}_t^1 = (1-\lambda) \left({\widehat C}_t^1 - {\widehat G}_t^1 \right) \cr & + \left( {\frac \lambda {1 - \gamma_o}} \right) \log {\mathbb E} \left( \exp \left[ (1 - \gamma_o) \left[ \left( {\widehat V}_{t+1}^1 - {\widehat G}_{t+1}^1 \right) + \left( {\widehat G}_{t+1}^1 - {\widehat G}_t^1\right) \right] \right] \mid {\mathfrak A}_t \right) \end{align*}\]

and solve this equation forward by first computing the \(\gamma_o=1\) answer and then adjusting this answer for \(\gamma_o > 1\) analogous to the approach described in Remark 11.5.

For the order two approximation,

\[Q_t^2 \eqdef \hspace{.2cm} (1-\rho)^2 Q_t^0 \left( {\widehat R}_t^1 - {\widehat V}_t^1\right)^2 + (1-\rho) Q_t^0 \left( {\widehat R}_t^2 - {\widehat V}_t^2\right). \]

Express:

\[{\widehat R}_t^2 - {\widehat V}_t^2 = \left({\widehat R}_t^2 - {\widehat G}_t^2\right) - \left( {\widehat V}_t^2 - {\widehat G}_t^2\right) .\]

It follows from (11.29) that

\[\begin{align} {\widehat V}_t^2 - {\widehat G}_t^2 = \hspace{.2cm} & \lambda {\mathbb E} \left( N_{t+1}^0 \left[\left({\widehat V}^2_{t+1} - {\widehat G}^2_{t+1} \right) + \left( {\widehat G}^2_{t+1} - {\widehat G}^2_{t}\right) \right] \mid {\mathfrak A}_t \right) \cr \hspace{.2cm} & + (1- \lambda) \left({\widehat C}_t^2 - {\widehat G}_t^2 \right) + (1 - \rho) (1 - \lambda) \lambda \left( {\widehat R}_t^1 - {\widehat G}_t^1 + {\widehat G}_t^1 - {\widehat C}_t^1 \right)^2, \end{align}\]

which we solve this equation forward under the \(N^0\) implied change in probability measure. Form:

\[\begin{align} {\widehat R}_t^2 - {\widehat G}_t^2 = & \left( \frac 1 \lambda \right) \left({\widehat V}_t^2 - {\widehat G}_t^2\right) - \left( \frac {1 - \lambda}{\lambda} \right) \left({\widehat C}_t^2 - {\widehat G}_t^2 \right) \cr & - (1 - \rho) (1 - \lambda) \left( {\widehat R}_t^1 - {\widehat G}_t^1 + {\widehat G}_t^1 - {\widehat C}_t^1 \right)^2. \end{align} \]

The \(P\) approximations use some of these same computations where

\[\begin{align*} P_t^0 \eqdef &(1-\beta) \exp\left[ (\rho-1) \left( {\widehat V}_t^0 - {\widehat G}_t^0 \right) \right]\cr P_t^1 \eqdef & (\rho-1) P_t^0 \left( {\widehat V}_t^1 - {\widehat G}_t^1 \right) \cr P_t^2 \eqdef & (\rho-1)^2 P_t^0 \left( {\widehat V}_t^1 - {\widehat G}_t^1 \right)^2 + (1-\rho) P_t^0\left( {\widehat V}_t^2 - {\widehat G}_t^2 \right). \end{align*}\]

A11.5 \(N\) derivatives

\[N_{t+1}^0 \eqdef \exp\left[(1 - \gamma_o) \left({\widetilde V}_{t+1}^0 - {\widetilde R}_t^0 \right) \right] = \exp\left[ (1-\gamma_o) \left({\widehat V}_{t+1}^1 - {\widehat R}_t^1\right) \right]\]

\[\begin{align} N_{t+1}^1 \eqdef \hspace{.2cm} & \left. \frac d {d{\sf q}} \exp\left[(1 - \gamma_o) \left({\widetilde V}_{t+1} - {\widetilde R}_t \right) \right] \right\vert_{{\sf q} =0} \cr = \hspace{.2cm} & (1 - \gamma_o) N_{t+1}^0\left( {\widetilde V}_{t+1}^1 - {\widetilde R}_t^1 \right) \cr = \hspace{.2cm} & \left(\frac {1-\gamma_o} 2 \right) N_{t+1}^0 \left( {\widehat V}_{t+1}^2 - {\widehat R}_{t}^2\right). \end{align}\]

Form:

\[\begin{align*} {\widehat V}_{t+1}^1 - {\widehat R}_{t}^1 = & \left({\widehat V}_{t+1}^1 - {\widehat G}_{t+1}^1\right) + \left({\widehat G}_{t+1}^1 - {\widehat G}_{t}^1\right) - \left({\widehat R}_{t}^1 - {\widehat G}_{t}^1\right) \cr {\widehat V}_{t+1}^2 - {\widehat R}_{t}^2 = & \left({\widehat V}_{t+1}^2 - {\widehat G}_{t+1}^2\right) + \left({\widehat G}_{t+1}^2 - {\widehat G}_{t}^2\right) - \left({\widehat R}_{t}^2 - {\widehat G}_{t}^2\right). \end{align*} \]

It can be verified that both \(N_{t+1}^1\) and \( N_{t+1}^2\) have conditional expectations equal to zero. Express

(2)\[\begin{align} {\widehat V}_{t+1}^1 - {\widehat R}_{t}^1 = & \left({\frac 1 {1-\gamma_o}} \right) \left[\mu^0 \cdot ( W_{t+1} - \mu^0) + {\frac 1 2} \mu^0 \cdot \mu^0\right] \cr {\widehat V}_{t+1}^2 - {\widehat R}_{t}^2 = & {\frac 1 2} \left(W_{t+1} - \mu^0 \right)'\Upsilon_2^2 \left(W_{t+1} - \mu^0 \right) - {\frac 1 2} \rm{tr}\left( \Upsilon_2^2\right) \cr & - \left(W_{t+1} - \mu^0\right)' \left( \Upsilon_1^2 X_t^1 + \Upsilon_0^2\right). \end{align}\]

In producing these representations, we use that \({\widehat V}_{t+1}^2 - {\widehat R}_{t}^2\) has conditional mean zero under the conditional probability distribution induced by \(N_{t+1}^0.\)

A11.6 Approximation formulas: approach one

Consider the equation:

\[Q_t {\mathbb E} \left( N_{t+1} H_{t+1} \mid {\mathfrak A}_t \right) + P_t L_{t} - M_t = 0 .\]

not including the state evolution equations.

A11.6.1 Order zero

The order zero approximation of the product \( N_{t+1} Q_{t} H_{t+1} + P_t L_{t} - M_t\) is:

\[Q_t^0 N_{t+1}^0 H_{t+1}^0 + P_t^0L_t^0 - M_0 = 0 \]

Thus the order zero approximate equation is:

\[Q_t^0 {\mathbb E} \left[N_{t+1}^0 \left( H_{t+1}^0 \right) \mid {\mathfrak A}_t \right] + P_t^0 L_{t}^0 - M_0= Q_t^0 H_{t+1}^0 + P_t^0 L_{t}^0 - M_0= 0\]

since \( N_{t+1}^0 \) has conditional expectation equal to one.

We add to this subsystem the \({\sf q} = 0\) state dynamic equation inclusive of jump variables, and we compute a stable steady state solution.

A11.6.2 Order one

The order one approximation of the product \( Q_t N_{t+1} H_{t+1}+ P_t L_{t} - M_t \) is:

\[Q_t^1 N_{t+1}^0 H_{t+1}^0 + Q_t^0 N_{t+1}^1 H_{t+1}^0 + Q_t^0 N_{t+1}^0 H_{t+1}^1 + P_t^0L_{t}^1 + P_t^1L_t^0 - M_t^1.\]

Thus the order one approximate equation is:

\[\begin{split}\begin{align} & Q_t^0 {\mathbb E} \left( N_{t+1}^1 H_{t+1}^0 + N_{t+1}^0 H_{t+1}^1 \mid {\mathfrak A}_t \right) + Q_t^1 H_{t+1}^0 + P_t^0 L_t^1 + P_t^1 L_t^0\\ & = Q_t^0 {\mathbb E} \left( N_{t+1}^0 H_{t+1}^1 \mid {\mathfrak A}_t \right) + Q_{t}^1 H_{t+1}^0 + P_t^0 L_t^1 + P_t^1 L_t^0 - M_t^1\\ & = 0 \end{align}\end{split}\]

where we used the implication that \( {\mathbb E} \left( N_{t+1}^1 \mid {\mathfrak A}_t \right) = 0.\)

A11.6.3 Order two

The order two approximation of the product \( Q_t N_{t+1} H_{t+1} + P_t L_{t} - M_t \) is:

\[\begin{split}\begin{align} & Q_t^0 N_{t+1}^0 H_{t+1}^2 + Q_t^0 N_{t+1}^2H_{t+1}^0 + 2 Q_t^0 N_{t+1}^1 H_{t+1}^1 \\ &+ P_t^0 L_t^2 + 2 P_t^1 L_t^1 + P_t^2 L_t^0 - M_t^2 \\ & + 2 N_{t+1}^1 Q_{t}^1 H_{t+1}^0 + 2 N_{t+1}^0 Q_{t}^1 H_{t+1}^1 + N_{t+1}^0 Q_{t}^2 H_{t+1}^0 \end{align}\end{split}\]

The terms \( Q_t^0 N_{t+1}^2H_{t+1}^0\) and \(2 N_{t+1}^1 Q_{t}^1 H_{t+1}^0\) have conditional expectation equal to zero. Thus the approximating equation is:

\[\begin{align} & Q_t^0 {\mathbb E} \left( N_{t+1}^0 H_{t+1}^2 \mid {\mathfrak A}_t \right) + P_t^0 L_t^2 + 2 P_t^1 L_t^1 + P_t^2 L_t^0 \cr & + 2Q_t^0 {\mathbb E}\left( N_{t+1}^1 H_{t+1}^1 \mid {\mathfrak A}_t \right) + 2 Q_t^1 {\mathbb E} \left(N_{t+1}^0 H_{t+1}^1 \mid {\mathfrak A}_t \right) + H_{t+1}^0 Q_t^2 {\mathbb E} \left( N_{t+1}^0 \mid {\mathfrak A}_t \right) \cr & = 0. \end{align} \]

To elaborate on the contributions in the second line, express \( H_{t+1}^1 \) as

(3)\[H_{t+1}^1 = \Theta_0^1 + \Theta_1^1 X_t^1 + \Theta_2^1 \left( W_{t+1} - \mu^0 \right). \]

Then

\[\begin{split}\begin{align} 2Q_t^0 {\mathbb E}\left( N_{t+1}^1 H_{t+1}^1 \mid {\mathfrak A}_t \right) & = (1-\gamma_o) Q_t^0 {\mathbb E} \left[ N_{t+1}^0 \left( {\widehat V}_{t+1}^2 - {\widehat R}_{t}^2\right)H_{t+1}^1 \mid{\mathfrak A}_t \right], \\ & = (\gamma_o - 1) Q_t^0 \Theta_2^1 \left[ \Upsilon_1^2 X_t^1 + \Upsilon_0^2\right] \\ 2 Q_t^1 {\mathbb E} \left(N_{t+1}^0 H_{t+1}^1 \mid {\mathfrak A}_t \right) & = 2 Q_t^0 (1- \rho) \left({\widehat R}_t^1 - {\widehat V}_t^1 \right)\left( \Theta_0^1 + \Theta_1^1 X_t^1 \right), \\ H_{t+1}^0 Q_t^2 {\mathbb E} \left( N_{t+1}^0 \mid {\mathfrak A}_t \right) & = H_{t+1}^0 Q_t^0 \left[(1-\rho)^2 \left( {\widehat R}_t^1 - {\widehat V}_t^1\right)^2 + (1-\rho) \left( {\widehat R}_t^2 - {\widehat V}_t^2\right)\right]. \end{align} \end{split}\]

The formula for the first of these terms follows from (2) and (3), along with the fact that the third central moments of normals are zero.

We add to this second-order subsystem, the second-order approximation of the state dynamics inclusive of the jump variables. We substitute in the solution for the first-order approximation for the jump variables into both the first and second-order approximate state dynamics. In solving the second-order jump variable adjustment we use expectations induced by \( N_{t+1}^0 \) throughout under which \(W_{t+1} \) is conditionally normally distributed with mean \( \mu^0 \) and covariance \( I \).

A11.7 Approximation formulas: approach two

In this approach we use the same order zero approximation. For the order one approximation, we use the formula (11.30) for \({\widetilde N}_{t+1},\) which approximates \(N_{t+1}^*,\) in conjunction with:

(4)\[\begin{align} Q_t^0 {\mathbb E} \left( {\widetilde N}_{t+1} H_{t+1}^1 \mid {\mathfrak A}_t \right) + Q_{t}^1 H_{t+1}^0 + P_t^0 L_t^1 + P_t^1 L_t^0 - M_t^1 = 0. \end{align}\]

From formula (2), it follows that under the \({\widetilde N}_{t+1},\) induced change in probability, \(W_{t+1}\) is normally distributed with conditional mean

\[\mu^0 + (1-\gamma_o) \left(\Upsilon_2^2\right)^{-1} \left(\Upsilon_1^2X_t^1 + \Upsilon_0^2\right) ,\]

and conditional precision:

\[(\gamma_o - 1) \Upsilon_2^2 + {\mathbb I}. \]

For the order two approximation, we use:

\[Q_t^0 {\mathbb E} \left( {\widetilde N}_{t+1} H_{t+1}^2 \mid {\mathfrak A}_t \right) + P_t^0 L_t^2 + 2 P_t^1 L_t^1 + P_t^2 L_t^0 - M_t^2 + 2 Q_t^1 {\mathbb E} \left( {\widetilde N}_{t+1} H_{t+1}^1 \mid {\mathfrak A}_t \right) + H_{t+1}^0 Q_t^2 = 0 . \]

A11.8 Parameter values

To facilitate a comparison to a global solution method, we write down a discrete-time approximation to a continuous time version of such an economy. (See Section 4.4 of [Hansen et al., 2024] for a continuous-time benchmark model that our discrete-time system approximates. Note that we convert the annual parameters in that paper to a quarterly time unit.) The parameter settings we use are:

\(\delta\)	\(\iota_k\)	\(\zeta\)	\(\nu_k\)	\(\nu_1\)	\(\nu_2\)	\(\mu_2\)
0.025	0.01	32	0.01	0.014	0.0485	\(6.3 \times 10^{-6}\)

\[\begin{align*} \begin{bmatrix} \sigma_k \cr \sigma_1 \cr \sigma_2 \end{bmatrix} = \sqrt{3} \begin{bmatrix} .92 & .40 & 0 \cr 0 & 5.7 & 0 \cr 0 & 0 & .00031 \end{bmatrix} \end{align*}\]

\[ \begin{bmatrix} \sigma_k \cr \sigma_1 \end{bmatrix} ,\]

For the extension to the habit persistence model in the habit-persistence section, we use a habit persistence of \(\nu_h = 0.025\).