$|\theta^1_h - \theta^1_l|$ is large:

* Compute agent 1’s valuation of the equity claim with a

fixed-point iteration:

$q_1 = \beta \int \frac{u^\prime(c^1_1(\epsilon))}{u^\prime(c^1_0)} d^e(k,b;\epsilon) g(\epsilon) \ d\epsilon$

where

$c^1_1(\epsilon) = w^1_1(\epsilon) + \theta^1 d^e(k,b;\epsilon)$

and

$c^1_0 = w^1_0 + \theta^1_0V - q_1\theta^1$

* Compute agent 2’s valuation of the bond claim with a

fixed-point iteration:

$p = \beta \int \frac{u^\prime(c^2_1(\epsilon))}{u^\prime(c^2_0)} d^b(k,b;\epsilon) g(\epsilon) \ d\epsilon$

where

$c^2_1(\epsilon) = w^2_1(\epsilon) + \theta^2 d^e(k,b;\epsilon) + b$

and

$c^2_0 = w^2_0 + \theta^2_0 V - q_1 \theta^2 - pb$

* Compute agent 2’s valuation of the equity claim with a

fixed-point iteration:

$q_2 = \beta \int \frac{u^\prime(c^2_1(\epsilon))}{u^\prime(c^2_0)} d^e(k,b;\epsilon) g(\epsilon) \ d\epsilon$

where

$c^2_1(\epsilon) = w^2_1(\epsilon) + \theta^2 d^e(k,b;\epsilon) + b$

and

$c^2_0 = w^2_0 + \theta^2_0 V - q_2 \theta^2 - pb$

* If $q_1 > q_2$, Set $\theta_l = \theta^1$;

otherwise, set $\theta_h = \theta^1$.

* Repeat steps 6Aa through 6Ad until

$|\theta^1_h - \theta^1_l|$ is small.

1. Set bond price as $p$ and equity price as $q = \max(q_1,q_2)$.

1. Compute optimal choices of consumption:

$$

\begin{aligned}

c^1_0 &= w^1_0 + \theta^1_0V - q\theta^1 \\

c^2_1(\epsilon) &= w^2_1(\epsilon) + \theta^2 d^e(k,b;\epsilon) + b

\end{aligned}

$$

1. (Here we confess to abusing notation again, but now in a different

way. In step 7, we interpret frozen $c^i$s as Big

$C^i$. We do this to solve the firm’s problem.) Fixing the

values of $c^i_0$ and $c^i_1(\epsilon)$, compute optimal

choices of capital $k$ and debt level $b$ using the

firm’s first order necessary conditions.

1. Compute deviations from the firm’s FONC for capital $k$ as:

$kfoc = \beta \alpha A k^{\alpha - 1} \left( \int \frac{u^\prime(c^2_1(\epsilon))}{u^\prime(c^2_0)} e^\epsilon g(\epsilon) \ d\epsilon \right) - 1$

- If $kfoc > 0$, Set $k_l = k$; otherwise, set

$k_h = k$.

- Repeat steps 4 through 7A until $|k_h-k_l|$ is small.

1. Compute deviations from the firm’s FONC for debt level $b$ as:

$bfoc = \beta \left[ \int_{\epsilon^*}^\infty \left( \frac{u^\prime(c^1_1(\epsilon))}{u^\prime(c^1_0)} \right) g(\epsilon) \ d\epsilon - \int_{\epsilon^*}^\infty \left( \frac{u^\prime(c^2_1(\epsilon))}{u^\prime(c^2_0)} \right) g(\epsilon) \ d\epsilon \right]$

- If $bfoc > 0$, Set $b_h = b$; otherwise, set

$b_l = b$.

- Repeat steps 3 through 7B until $|b_h-b_l|$ is small.

1. Given prices $q$ and $p$ from step 6, and the firm

choices of $k$ and $b$ from step 7, compute the synthetic

firm value:

$V_x = -k + q + pb$

- If $V_x > V$, then set $V_l = V$; otherwise, set

$V_h = V$.

- bound: Bound for truncated normal distribution. Default value is 3.

```{code-cell} ipython

import pandas as pd

import numpy as np

from scipy.stats import norm

from scipy.stats import truncnorm

from scipy.integrate import quad

from scipy.optimize import bisect

from numba import njit

from interpolation import interp

```

Taken together with our earlier plot of equity holdings, these graphs confirm our earlier conjecture that while both type

of agents hold equities, only agents of type 2 holds bonds.