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het_firm.py
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import numpy as np
from numba import njit
import utils
from het_block import het
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI, epsilon, P_index, Z):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
#V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_exit(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI, epsilon, P_index, Z):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
#V = W + div_ind
V = W + Wd
Vd = div_ind
# V = W + np.maximum(Wd, sell_next / (1 + rf))
V = W + np.maximum(Wd, 0)
Vj = div_ind + V
p[Vj < 0] = p_grid[0]
k[Vj < 0] = k_grid[0]
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
Vd = div_ind
# V[Vj < sell_ind] = 0
V[Vj < 0] = 0
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
# here distribution of firms is calculated too
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_Bertrand_ss(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI, epsilon, P_index, Z, D):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
D_swaped = np.swapaxes(D, 0, 1)
der_pP = p_grid[:, np.newaxis, np.newaxis, np.newaxis] ** (- epsilon) / P_index ** (- epsilon) * D_swaped[np.newaxis, :, :, :]
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) * (1 - p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index * der_pP) - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1) * (1 - p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index * der_pP)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
#V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
# here distribution of firms is passed as a parameter
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_Bertrand(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI,
epsilon, P_index, Z, Df):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (
k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
D_swaped = np.swapaxes(Df, 0, 1)
der_pP = p_grid[:, np.newaxis, np.newaxis, np.newaxis] ** (- epsilon) / P_index ** (- epsilon) * D_swaped[
np.newaxis, :, :,
:]
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) * (1 - p_grid[:, np.newaxis, np.newaxis,
np.newaxis] / P_index * der_pP) - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1) * (
1 - p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index * der_pP)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
# V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_open(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI, epsilon, P_index, Z, p_busket):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / p_busket * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / p_busket * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
#V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_open_Bertrand_ss(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI,
epsilon, P_index, Z, p_busket, gamma_hb, eta_hh, D):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
D_swaped = np.swapaxes(D, 0, 1)
der_pP = p_grid[:, np.newaxis, np.newaxis, np.newaxis] ** (- epsilon) / P_index ** (- epsilon) * D_swaped[np.newaxis, :, :, :]
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / p_busket * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) - epsilon * (p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
1-epsilon) / p_busket * Y * der_pP + (1 - gamma_hb) * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (1-epsilon) * P_index / p_busket ** 2 * (
P_index / p_busket) ** (-eta_hh) * der_pP - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1) * (
1 - p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index * der_pP)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / p_busket * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
#V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['p', 'k'], backward=['Vp', 'Vk', 'V', 'Vd']) # order as in grid!
def firm_open_Bertrand(Vp_p, Vk_p, V_p, Vd_p, Pif_p, p_grid, k_grid, ef_grid, rf, w, Y, alpha, kappap, delta, epsI,
epsilon, P_index, Z, p_busket, gamma_hb, eta_hh, Df):
# get grid dimensions
nZ, nP, nK = Vp_p.shape
# step 2: calculate Wp(z, p', k'), Wk(z, p', k'), W(z, p', k') for convenience
Wp, Wk, W = post_decision_vfun(Vp_p, Vk_p, V_p, Pif_p, rf)
Wd = (Vd_p.T @ (1 / (1 + rf) * Pif_p.T)).T
# step 3: obtain k(z, p', k) and Wp(z, p', k) and W(z, p', k) through Wk(z, p', k') = f(k', k)
k_endo, Wp_endo, W_endo, Wd_endo = foc_capital_d(Wk - 1,
1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nP, nK, Wp, W, Wd)
# mc(z, p', k)
mc = w / (1 - alpha) / Z / ef_grid[np.newaxis, np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis,
np.newaxis, :] ** alpha * (
(p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon) * Y / Z / ef_grid[np.newaxis,
np.newaxis, :,
np.newaxis] /
k_grid[np.newaxis, np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
D_swaped = np.swapaxes(Df, 0, 1)
der_pP = p_grid[:, np.newaxis, np.newaxis, np.newaxis] ** (- epsilon) / P_index ** (- epsilon) * D_swaped[np.newaxis, :, :, :]
# step 4: p(z, p, k), k(z, p, k) and W(z, p, k) through rhs(p', p, z, k) = Wp(z, p', k)
rhs = epsilon / kappap * np.log(
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / p_grid[np.newaxis, :, np.newaxis, np.newaxis]
) * Y / p_grid[:, np.newaxis, np.newaxis, np.newaxis] - (1 - epsilon) * Y / p_busket * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
-epsilon) - epsilon * (p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
1 - epsilon) / p_busket * Y * der_pP + (1 - gamma_hb) * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (
1 - epsilon) * P_index / p_busket ** 2 * (
P_index / p_busket) ** (-eta_hh) * der_pP - epsilon * mc * Y / P_index * (
p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index) ** (-epsilon - 1) * (
1 - p_grid[:, np.newaxis, np.newaxis, np.newaxis] / P_index * der_pP)
p, k, W, Wd = foc_price_d(Wp_endo, rhs, p_grid, nZ, nP, nK, k_endo, W_endo, Wd_endo)
y_ind = (p / P_index) ** (-epsilon) * Y
n_ind = (y_ind / (Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)) ** (
1 / (1 - alpha))
mc_ind = w * n_ind ** alpha / (
(1 - alpha) * Z * ef_grid[:, np.newaxis, np.newaxis] * k_grid[np.newaxis, np.newaxis, :] ** alpha)
psi_ind = epsilon * 1 / (2 * kappap) * (np.log(p / p_grid[np.newaxis, :, np.newaxis])) ** 2 * Y
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, np.newaxis,
:]
i_ind = k - (1 - delta) * k_grid[np.newaxis, np.newaxis, :] + k_adj_ind
div_ind = p / p_busket * y_ind - w * n_ind - i_ind - psi_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis, np.newaxis] * alpha * k_grid[np.newaxis, np.newaxis, :] ** (
alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, np.newaxis, :] - 1) * k / k_grid[np.newaxis, np.newaxis, :])
Vp = epsilon * Y / kappap * np.log(p / p_grid[np.newaxis, :, np.newaxis]) / p_grid[np.newaxis, :, np.newaxis]
# V = W + div_ind
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vp, Vk, V, Vd, p, k, i_ind, div_ind, y_ind, psi_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['k'], backward=['Vk', 'V', 'Vd']) # order as in grid!
def firm_no_price_adj_costs(Vk_p, V_p, Vd_p, Pif_p, k_grid, ef_grid, rf, w, Y, alpha, delta, epsI, epsilon, P_index, Z, r):
# get grid dimensions
nZ, nK = Vk_p.shape
# step 2: calculate Wk(z, k'), W(z, k') for convenience
Wk, W, Wd = post_decision_vfun(Vk_p, V_p, Vd_p, Pif_p, rf)
# step 3: obtain k(z, k) and W(z, k) through Wk(z, k') = f(k', k)
k, W, Wd = foc_capital_no_p_d(Wk - 1, 1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nK, W, Wd)
# p(z, k)
p = (epsilon / (epsilon - 1) * w / (1 - alpha) / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha *
(Y / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
) ** (1 / (1 + epsilon * alpha / (1 - alpha))) * P_index
# mc(z, k)
mc_ind = p / P_index * (epsilon - 1) / epsilon
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
V = W + Wd
Vd = div_ind
V_agg = V
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
return Vk, V, Vd, p, k, i_ind, div_ind, y_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['k'], backward=['Vk', 'V', 'Vd']) # order as in grid!
def firm_no_price_adj_costs_exit(Vk_p, V_p, Vd_p, Pif_p, k_grid, ef_grid, rf, w, Y, alpha, delta, epsI, epsilon, P_index, Z):
# get grid dimensions
nZ, nK = Vk_p.shape
# step 2: calculate Wk(z, k'), W(z, k') for convenience
Wk, W, Wd = post_decision_vfun(Vk_p, V_p, Vd_p, Pif_p, rf)
# step 3: obtain k(z, k) and W(z, k) through Wk(z, k') = f(k', k)
k, W, Wd = foc_capital_no_p_d(Wk - 1, 1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nK, W, Wd)
# p(z, k)
p = (epsilon / (epsilon - 1) * w / (1 - alpha) / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha *
(Y / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
) ** (1 / (1 + epsilon * alpha / (1 - alpha))) * P_index
# mc(z, k)
mc_ind = p / P_index * (epsilon - 1) / epsilon
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
k_adj_ind1 = 1 / delta / epsI / 2 * (k_grid[0] / k - 1) ** 2 * k
sell_next = -k_grid[0] + (1 - delta) * k - k_adj_ind1
#V = W + np.maximum(Wd, sell_next / (1 + rf))
V = W + np.maximum(Wd, 0)
Vj = div_ind + V
k_adj_ind2 = 1 / delta / epsI / 2 * (k_grid[0] / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
sell_ind = (-k_grid[0] + (1 - delta) * k_grid[np.newaxis, :] - k_adj_ind2) * np.ones((nZ, nK))
p[Vj < 0] = 0
k[Vj < 0] = k_grid[0]
# p[Vj < sell_ind] = 0
# k[Vj < sell_ind] = k_grid[0]
mc_ind = p / P_index * (epsilon - 1) / epsilon
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
Vd = div_ind
#V[Vj < sell_ind] = 0
V[Vj < 0] = 0
V_agg = V
return Vk, V, Vd, p, k, i_ind, div_ind, y_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['k'], backward=['Vk', 'V', 'Vd']) # order as in grid!
def firm_no_price_adj_costs_exit_Bertrand_ss(Vk_p, V_p, Vd_p, Pif_p, k_grid, ef_grid, rf, w, Y, alpha, delta, epsI, epsilon, P_index, Z, D):
# get grid dimensions
nZ, nK = Vk_p.shape
# step 2: calculate Wk(z, k'), W(z, k') for convenience
Wk, W, Wd = post_decision_vfun(Vk_p, V_p, Vd_p, Pif_p, rf)
# step 3: obtain k(z, k) and W(z, k) through Wk(z, k') = f(k', k)
k, W, Wd = foc_capital_no_p_d(Wk - 1, 1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nK, W, Wd)
# p(z, k) from rhs(p', z, k) = lhs(p', z, k)
nP = 100
p_grid = utils.agrid(amax=1.5, n=nP, amin=0.7)
rhs = (epsilon * (1 - (p_grid[:, np.newaxis, np.newaxis] / P_index) ** (1 - epsilon) * D[np.newaxis, :, :]) / (
epsilon * (1 - (p_grid[:, np.newaxis, np.newaxis] / P_index) ** (1 - epsilon)) * D[np.newaxis, :, :] - 1) *
w / (1 - alpha) / Z / ef_grid[np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis, :] ** alpha * (
Y / Z / ef_grid[np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
) ** (1 / (1 + epsilon * alpha / (1 - alpha)))
p = foc_price_RBC_Bertrand(rhs, p_grid[:, np.newaxis, np.newaxis] / P_index, p_grid, nZ, nP, nK)
epsilon_modified = epsilon * (1 - (p / P_index) ** (1 - epsilon) * D)
# mc(z, k)
mc_ind = p / P_index * (epsilon_modified - 1) / epsilon_modified
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
k_adj_ind1 = 1 / delta / epsI / 2 * (k_grid[0] / k - 1) ** 2 * k
sell_next = -k_grid[0] + (1 - delta) * k - k_adj_ind1
# V = W + np.maximum(Wd, sell_next / (1 + rf))
V = W + np.maximum(Wd, 0)
Vj = div_ind + V
k_adj_ind2 = 1 / delta / epsI / 2 * (k_grid[0] / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
sell_ind = (-k_grid[0] + (1 - delta) * k_grid[np.newaxis, :] - k_adj_ind2) * np.ones((nZ, nK))
p[Vj < 0] = 0
k[Vj < 0] = k_grid[0]
# p[Vj < sell_ind] = 0
# k[Vj < sell_ind] = k_grid[0]
mc_ind = p / P_index * (epsilon - 1) / epsilon
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
Vd = div_ind
# V[Vj < sell_ind] = 0
V[Vj < 0] = 0
V_agg = V
return Vk, V, Vd, p, k, i_ind, div_ind, y_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['k'], backward=['Vk', 'V', 'Vd']) # order as in grid!
def firm_no_price_adj_costs_exit_Bertrand(Vk_p, V_p, Vd_p, Pif_p, k_grid, ef_grid, rf, w, Y, alpha, delta, epsI, epsilon, P_index, Z, Df):
# get grid dimensions
nZ, nK = Vk_p.shape
# step 2: calculate Wk(z, k'), W(z, k') for convenience
Wk, W, Wd = post_decision_vfun(Vk_p, V_p, Vd_p, Pif_p, rf)
# step 3: obtain k(z, k) and W(z, k) through Wk(z, k') = f(k', k)
k, W, Wd = foc_capital_no_p_d(Wk - 1, 1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nK, W, Wd)
# p(z, k) from rhs(p', z, k) = lhs(p', z, k)
nP = 100
p_grid = utils.agrid(amax=1.5, n=nP, amin=0.7)
rhs = (epsilon * (1 - (p_grid[:, np.newaxis, np.newaxis] / P_index) ** (1 - epsilon)) / (
epsilon * (1 - (p_grid[:, np.newaxis, np.newaxis] / P_index) ** (1 - epsilon)) - 1) * w / (1 - alpha) / Z /
ef_grid[np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis, :] ** alpha * (
Y / Z / ef_grid[np.newaxis, :, np.newaxis] / k_grid[np.newaxis, np.newaxis, :] ** alpha) ** (alpha / (1 - alpha))
) ** (1 / (1 + epsilon * alpha / (1 - alpha)))
p = foc_price_RBC_Bertrand(rhs, p_grid[:, np.newaxis, np.newaxis] / P_index, p_grid, nZ, nP, nK)
epsilon_modified = epsilon * (1 - (p / P_index) ** (1 - epsilon))
# mc(z, k)
mc_ind = p / P_index * (epsilon_modified - 1) / epsilon_modified
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / P_index * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
V = W + Wd
Vd = div_ind
V_agg = V
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
return Vk, V, Vd, p, k, i_ind, div_ind, y_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
@het(exogenous='Pif', policy=['k'], backward=['Vk', 'V', 'Vd']) # order as in grid!
def firm_no_price_adj_costs_open(Vk_p, V_p, Vd_p, Pif_p, k_grid, ef_grid, rf, w, Y, alpha, delta, epsI, epsilon, P_index, Z, p_busket):
# get grid dimensions
nZ, nK = Vk_p.shape
# step 2: calculate Wk(z, k'), W(z, k') for convenience
Wk, W, Wd = post_decision_vfun(Vk_p, V_p, Vd_p, Pif_p, rf)
# step 3: obtain k(z, k) and W(z, k) through Wk(z, k') = f(k', k)
k, W, Wd = foc_capital_no_p_d(Wk - 1, 1 / delta / epsI * (k_grid[:, np.newaxis] / k_grid[np.newaxis, :] - 1),
k_grid, nZ, nK, W, Wd)
# p(z, k)
p = (epsilon / (epsilon - 1) * w / (1 - alpha) / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha *
(Y / Z / ef_grid[:, np.newaxis] / k_grid[np.newaxis, :] ** alpha) ** (alpha / (1 - alpha)) *
P_index ** (epsilon * alpha / (1-alpha)) * p_busket) ** (1 / (1 + epsilon * alpha / (1 - alpha)))
# mc(z, k)
mc_ind = p / p_busket * (epsilon - 1) / epsilon
# calculation of other variables of interest
n_ind = ((1 - alpha) * mc_ind * Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha / w) ** (1 / alpha)
y_ind = Z * ef_grid[:, np.newaxis] * k_grid[np.newaxis, :] ** alpha * n_ind ** (1 - alpha)
k_adj_ind = 1 / delta / epsI / 2 * (k / k_grid[np.newaxis, :] - 1) ** 2 * k_grid[np.newaxis, :]
i_ind = k - (1 - delta) * k_grid[np.newaxis, :] + k_adj_ind
div_ind = p / p_busket * y_ind - w * n_ind - i_ind
markup_ind = p / P_index / mc_ind
q_ind = (k / k_grid[np.newaxis, :] - 1) / (delta * epsI) + 1
# envelope conditions and update of the value function guess
Vk = (Z * mc_ind * ef_grid[:, np.newaxis] * alpha * k_grid[np.newaxis, :] ** (alpha - 1) * n_ind ** (1 - alpha) +
1 - delta - 1 / delta / 2 / epsI * (k / k_grid[np.newaxis, :] - 1) ** 2 +
1 / delta / epsI * (k / k_grid[np.newaxis, :] - 1) * k / k_grid[np.newaxis, :])
V = W + Wd
Vd = div_ind
pp = p ** (1 - epsilon)
yy_ind = y_ind ** ((epsilon - 1) / epsilon)
V_agg = V
return Vk, V, Vd, p, k, i_ind, div_ind, y_ind, n_ind, mc_ind, markup_ind, pp, yy_ind, k_adj_ind, V_agg, q_ind
def post_decision_vfun(Vp_p, Vk_p, V_p, Pi, rf):
Wp = (Vp_p.T @ (1 / (1 + rf) * Pi.T)).T
Wk = (Vk_p.T @ (1 / (1 + rf) * Pi.T)).T
W = (V_p.T @ (1 / (1 + rf) * Pi.T)).T
return Wp, Wk, W
def post_decision_vfun_no_p(Vk_p, V_p, Pi, rf):
Wk = (Vk_p.T @ (1 / (1 + rf) * Pi.T)).T
W = (V_p.T @ (1 / (1 + rf) * Pi.T)).T
return Wk, W
@njit
def foc_capital(lhs, rhs, k_grid, nZ, nP, nK, Wp, W):
k = np.empty((nZ, nP, nK))
Wp_endo = np.empty((nZ, nP, nK))
W_endo = np.empty((nZ, nP, nK))
for iz in range(nZ):
for ip in range(nP):
ikp = 0 # use mononicity in k
for ik in range(nK):
while True:
if lhs[iz, ip, ikp] < rhs[ikp, ik]:
break
elif ikp < nK - 1:
ikp += 1
else:
break
if ikp == 0:
k[iz, ip, ik] = k_grid[0]
Wp_endo[iz, ip, ik] = Wp[iz, ip, 0]
W_endo[iz, ip, ik] = W[iz, ip, 0]
elif ikp == nK:
k[iz, ip, ik] = k_grid[ikp]
Wp_endo[iz, ip, ik] = Wp[iz, ip, ikp]
W_endo[iz, ip, ik] = W[iz, ip, ikp]
else:
y0 = lhs[iz, ip, ikp - 1] - rhs[ikp - 1, ik]
y1 = lhs[iz, ip, ikp] - rhs[ikp, ik]
k[iz, ip, ik] = k_grid[ikp - 1] - y0 * (k_grid[ikp] - k_grid[ikp - 1]) / (y1 - y0)
Wp_endo[iz, ip, ik] = Wp[iz, ip, ikp - 1] + (
k[iz, ip, ik] - k_grid[ikp - 1]) * (
Wp[iz, ip, ikp] - Wp[iz, ip, ikp - 1]) / (
k_grid[ikp] - k_grid[ikp - 1])
W_endo[iz, ip, ik] = W[iz, ip, ikp - 1] + (
k[iz, ip, ik] - k_grid[ikp - 1]) * (
W[iz, ip, ikp] - W[iz, ip, ikp - 1]) / (
k_grid[ikp] - k_grid[ikp - 1])
return k, Wp_endo, W_endo
@njit
def foc_capital_d(lhs, rhs, k_grid, nZ, nP, nK, Wp, W, Wd):
k = np.empty((nZ, nP, nK))
Wp_endo = np.empty((nZ, nP, nK))
W_endo = np.empty((nZ, nP, nK))
Wd_endo = np.empty((nZ, nP, nK))
for iz in range(nZ):
for ip in range(nP):