A.1 Description
Suppose the value function is \(V\). State variable:
\(x = (\log k, z, y, n)\) (realization of
\(\log K, \log N, Y, Z\)), control variable : \((\tilde e, i)\).
Suppose the value function is separable as follows:
\[V(\log k, n, y, z) = v_k \log k + \zeta(z) + \phi(y) + v_n n\]
Consider the model without jump misspecification with damage function
\[\Lambda (y) = \gamma_1 y + \frac{\gamma_2}{2}y^2 + \frac{\gamma_3}{2} (y - \bar y)^2 \mathbb{I}\{y > \bar y\}\]
For models with jump misspecification, the modification can be make to
\((y,n)\) without affecting \((\log k, z)\), as the HJB is
separable. The complete HJB is as follows:
\[\begin{split}\begin{aligned}
0 = \max_{\tilde e, i }\min_{\omega_\ell:\sum_{\ell=1}^L \omega_\ell = 1} \min_{h}\quad & - \delta V + (1 - \eta) [\log(\alpha - i) + \log k - n] + \eta \log \tilde e \\
& + v_k \cdot \left[\mu_k(z) + i - \frac{\kappa}{2} i^2 - \frac{|\sigma_k(z)|^2}{2} + \sigma_k(z)' h \right] \\
& + \frac{\partial \zeta }{\partial z}(z)\cdot \left[\mu_z(z) + \sigma_z(z)'h \right] + \frac{1}{2} trace\left[\sigma_z(z)' \frac{\partial^2 \zeta(z)}{\partial z\partial z'} \sigma_z(z)'\right] \\
& + \frac{d \phi(y)}{dy} \sum_{\ell=1}^L \omega_\ell\cdot \tilde e\cdot\theta_\ell + \frac12 \frac{d^2 \phi(y)}{dy^2} (\tilde e)^2 |\varsigma|^2\\
& + v_n \left[(\gamma_1 + \gamma_2 y + \gamma_3 (y-\bar y)\mathbb{I}\{y > \bar y\}) (\sum_{\ell=1}^L \omega_\ell \theta_\ell \tilde e + \tilde e \varsigma' h )+ \frac12 \left(\gamma_2 + \gamma_3 \mathbb{I}\{y > \bar y\} \right)\cdot |\varsigma|^2 (\tilde e)^2 \right] \\
& + \frac{\xi_b}{2} h'h + \xi_a \sum_{\ell=1}^L \omega_\ell \left( \log \omega_\ell - \log \pi^a_\ell \right)
\end{aligned}\end{split}\]
A.2 Consumption-capital dynamics
The undamaged version of consumption capital model has a straightforward
solution. The HJB equation for this component is:
\[\begin{split}\begin{aligned}
0 = \max_{ i } \min_{h}\quad & - \delta \left[ v_k \log k + \zeta(z)\right] + (1 - \eta) [\log(\alpha - i) + \log k - n] + \frac{\xi_b}{2} h'h \\
& + v_k \cdot \left[\mu_k(z) + i - \frac{\kappa}{2} i^2 - \frac{|\sigma_k(z)|^2}{2} + \sigma_k(z)' h \right] \\
& + \frac{\partial \zeta }{\partial z}(z)\cdot \left[\mu_z(z) + \sigma_z(z)'h \right] + \frac{1}{2} trace\left[\sigma_z(z)' \frac{\partial^2 \zeta(z)}{\partial z\partial z'} \sigma_z(z)'\right] \\
\end{aligned}\end{split}\]
Coefficients of \(\log k\) satisfy that
\[-\delta v_k + (1 - \eta) = 0 \quad \Longrightarrow \quad v_k = \frac{1 - \eta}{\delta}\]
The first order condition for the investment-capital ratio is
\[- (1 - \eta) \frac{1}{\alpha - i} + v_k (1 - \kappa i) = 0\quad \Longrightarrow \quad - \frac{1}{\alpha -i} + \frac{1 - \kappa i}{\delta} = 0\]
The first order condition for \(h\) is:
\[\xi_b h + \sigma_k v_k + \sigma_z \frac{\partial \zeta}{\partial z} = 0\]
and is therefore:
\[h = - \frac{1}{\xi_b} \left[ \sigma_k v_k + \sigma_k \frac{\partial \zeta}{\partial z}\right]\]
