longTermDebtModel.tex

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\title{Long-Term Debt Model}
\author{Gabriel Mihalache}
\date{September 2022}
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\begin{document}

% \maketitle

\begin{align}
V\left(y, B\right) = \rho_D \log \left\{ \exp\left[ \dfrac{V^d(y)}{\rho_D} \right] +
\exp \left[ \dfrac{V^r(y, B)}{\rho_D} \right] \right\}
\end{align}

\begin{align}
\Pr\left(d = 1 \middle| y, B\right) = \dfrac{
  \exp\left[ \dfrac{V^d(y)}{\rho_D} \right]
}{\exp\left[ \dfrac{V^d(y)}{\rho_D} \right] +
\exp \left[ \dfrac{V^r(y, B)}{\rho_D} \right]}
= \dfrac{1}{1 + \exp \left[ \dfrac{V^r(y, B) - V^d(y)}{\rho_D} \right]}
\end{align}

\begin{align}
V^d\left(y\right) = u\left[h\left(y\right)\right] + \beta \E_{y'|y} \left\{ \gamma V\left( y', 0 \right) +
(1-\gamma) V^d\left(y'\right) \right\}
\end{align}

\begin{align}
W\left(y, B, B'\right) = u\left[ y - \kappa B + q\left(y, B'\right) \left( B' - (1-\delta) B \right) \right] + \beta \E_{y'|y} V\left(y', B'\right)
\end{align}

\begin{align}
V^r\left(y, B\right) = \rho_B \log \sum_{B'} \exp \left[ \dfrac{W\left(y, B, B'\right))}{\rho_B} \right]
\end{align}

\begin{align}
\Pr\left(B' = x \middle| y, B \right) = \dfrac{
  \exp \left[ \dfrac{W\left(y, B, x\right))}{\rho_B} \right]
}{\sum_{i} \exp \left[ \dfrac{W\left(y, B, i\right))}{\rho_B} \right]}
\end{align}

\begin{align}
q\left(y, B'\right) = \dfrac{1}{1+r} \E_{y'|y} \Pr\left(d=0 \middle| y', B'\right)
\left[ \kappa + (1-\delta) \sum_{B''} \Pr\left( B'' \middle| y', B'\right) q\left(y', B''\right) \right]
\end{align}

\begin{align}
u\left(c\right) = \dfrac{c^{1-\sigma} - 1}{1-\sigma}
\end{align}

\begin{align}
h\left(y\right) = y - \max\{ 0, \lambda_0 y + \lambda_1 y^2 \}
\end{align}

\begin{align}
\log y' = - (1-\rho) \dfrac{\sigma_y^2}{2 \left( 1 - \rho^2\right)} + \rho \log y + \sigma_y \varepsilon, \quad \varepsilon \sim \mathcal{N}(0, 1)
\end{align}

\end{document}