"""
------------------------------------------------------------------------
Fiscal policy functions for unbalanced budgeting. In particular, some
functions require time-path calculation.
------------------------------------------------------------------------
"""
# Packages
import numpy as np
"""
------------------------------------------------------------------------
Functions
------------------------------------------------------------------------
"""
[docs]
def D_G_path(r_gov, dg_fixed_values, p):
r"""
Calculate the time paths of debt and government spending
.. math::
\begin{split}
&e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\hat{D}_{t+1} + \hat{Rev}_t = (1 + r_{gov,t})\hat{D}_t + \hat{G}_t + \hat{TR}_t + \hat{UBI}_t \quad\forall t \\
&\hat{G}_t = g_{g,t}\:\alpha_{g}\: \hat{Y}_t \\
&\text{where}\quad g_{g,t} =
\begin{cases}
1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\text{if}\quad t < T_{G1} \\
\frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\left[\rho_{d}\alpha_{D}\hat{Y}_{t} + (1-\rho_{d})\hat{D}_{t}\right] - (1+r_{gov,t})\hat{D}_{t} - \hat{TR}_{t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
\frac{e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\alpha_{D}\hat{Y}_{t} - (1+r_{gov,t})\hat{D}_{t} - \hat{TR}_{t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
\end{cases} \\
&\text{and}\quad g_{tr,t} = 1 \quad\forall t \\
&e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}^{f}_{t+1} = \hat{D}^{f}_{t} + \zeta_{D}\Bigl(e^{g_y}\bigl[1 + \tilde{g}_{n,t+1}\bigr]\hat{D}_{t+1} - \hat{D}_{t}\Bigr) \quad\forall t
\end{split}
Args:
r_gov (Numpy array): interest rate on government debt over the
time path
dg_fixed_values (tuple): (Y, total_tax_revenue,
agg_pension_outlays, UBI_outlays,TR, D0, G0) values of variables
that are taken as given in the government budget constraint
Gbaseline (Numpy array): government spending over the time path
in the baseline equilibrium, used only if
baseline_spending=True
p (OG-Core Specifications object): model parameters
Returns:
(tuple): fiscal variable path output:
* D (Numpy array): government debt over the time path
* G (Numpy array): government spending over the time path
* D_d (Numpy array): domestic held government debt over the
time path
* D_f (Numpy array): foreign held government debt over the
time path
* new_borrowing_f: new borrowing from foreigners
"""
(
Y,
total_tax_revenue,
agg_pension_outlays,
UBI_outlays,
TR,
I_g,
Gbaseline,
D0_baseline,
) = dg_fixed_values
growth = (1 + p.g_n) * np.exp(p.g_y)
D = np.zeros(p.T + 1)
if p.baseline:
D[0] = p.initial_debt_ratio * Y[0]
else:
D[0] = D0_baseline
if p.baseline_spending:
G = p.alpha_bs_G[: p.T] * Gbaseline[: p.T]
else:
G = p.alpha_G[: p.T] * Y[: p.T]
if p.budget_balance:
D = np.zeros(p.T + 1)
G = p.alpha_G[: p.T] * Y[: p.T]
D_f = np.zeros(p.T)
D_d = np.zeros(p.T)
new_borrowing = np.zeros(p.T)
debt_service = np.zeros(p.T)
new_borrowing_f = np.zeros(p.T)
else:
t = 1
while t < p.T - 1:
D[t] = (1 / growth[t]) * (
(1 + r_gov[t - 1]) * D[t - 1]
+ G[t - 1]
+ I_g[t - 1]
+ TR[t - 1]
+ UBI_outlays[t - 1]
+ agg_pension_outlays[t - 1]
- total_tax_revenue[t - 1]
)
if (t >= p.tG1) and (t < p.tG2):
G[t] = (
growth[t + 1]
* (p.rho_G * p.debt_ratio_ss * Y[t] + (1 - p.rho_G) * D[t])
- (1 + r_gov[t]) * D[t]
+ total_tax_revenue[t]
- agg_pension_outlays[t]
- I_g[t - 1]
- TR[t]
- UBI_outlays[t]
)
elif t >= p.tG2:
G[t] = (
growth[t + 1] * (p.debt_ratio_ss * Y[t])
- (1 + r_gov[t]) * D[t]
+ total_tax_revenue[t]
- agg_pension_outlays[t]
- I_g[t - 1]
- TR[t]
- UBI_outlays[t]
)
t += 1
# in final period, growth rate has stabilized, so we can replace
# growth[t+1] with growth[t]
t = p.T - 1
D[t] = (1 / growth[t]) * (
(1 + r_gov[t - 1]) * D[t - 1]
+ G[t - 1]
+ I_g[t - 1]
+ TR[t - 1]
+ UBI_outlays[t - 1]
+ agg_pension_outlays[t - 1]
- total_tax_revenue[t - 1]
)
G[t] = (
growth[t] * (p.debt_ratio_ss * Y[t])
- (1 + r_gov[t]) * D[t]
+ total_tax_revenue[t]
- agg_pension_outlays[t]
- I_g[t - 1]
- TR[t]
- UBI_outlays[t]
)
D[t + 1] = (1 / growth[t + 1]) * (
(1 + r_gov[t]) * D[t]
+ G[t]
+ I_g[t - 1]
+ TR[t]
+ UBI_outlays[t]
+ agg_pension_outlays[t]
- total_tax_revenue[t]
)
D_ratio_max = np.amax(D[: p.T] / Y[: p.T])
print("Maximum debt ratio: ", D_ratio_max)
# Find foreign and domestic debt holding
# Fix initial amount of foreign debt holding
D_f = np.zeros(p.T + 1)
D_f[0] = p.initial_foreign_debt_ratio * D[0]
for t in range(0, p.T):
D_f[t + 1] = (D_f[t] / growth[t + 1]) + (
p.zeta_D[t] * (D[t + 1] - (D[t] / growth[t + 1]))
)
D_d = D[: p.T] - D_f[: p.T]
new_borrowing = (
D[1 : p.T + 1] * np.exp(p.g_y) * (1 + p.g_n[1 : p.T + 1])
- D[: p.T]
)
debt_service = r_gov[: p.T] * D[: p.T]
new_borrowing_f = (
D_f[1 : p.T + 1] * np.exp(p.g_y) * (1 + p.g_n[1 : p.T + 1])
- D_f[: p.T]
)
return (
D,
G,
D_d,
D_f[: p.T],
new_borrowing,
debt_service,
new_borrowing_f,
)
[docs]
def get_D_ss(r_gov, Y, p):
r"""
Calculate the steady-state values of government spending and debt
.. math::
\begin{split}
\bar{D} &= \alpha_D \bar{Y}\\
\bar{D_d} &= \bar{D} - \bar{D}^{f}\\
\bar{D_f} &= \zeta_{D}\bar{D} \\
\overline{\text{new borrowing}} &= (e^{g_{y}}(1 + \bar{g}_n) - 1)\bar{D}\\
\overline{\text{debt service}} &= \bar{r}_{gov}\bar{D} \\
\overline{\text{new foreign borrowing}} &=
(e^{g_{y}}(1 + \bar{g}_n) - 1)\bar{D_f}\\
\end{split}
Args:
r_gov (scalar): steady-state interest rate on government debt
Y (scalar): steady-state GDP
p (OG-Core Specifications object): model parameters
Returns:
(tuple): steady-state fiscal variables:
* D (scalar): steady-state government debt
* D_d (scalar): steady-state domestic held government debt
* D_f (scalar): steady-state foreign held government debt
* new_borrowing: steady-state new borrowing
* debt_service: steady-state debt service costs
* new_borrowing_f: steady-state borrowing from foreigners
"""
if p.budget_balance:
D = 0.0
else:
D = p.debt_ratio_ss * Y
debt_service = r_gov * D
new_borrowing = D * ((1 + p.g_n_ss) * np.exp(p.g_y) - 1)
D_f = p.zeta_D[-1] * D
D_d = D - D_f
new_borrowing_f = D_f * (np.exp(p.g_y) * (1 + p.g_n_ss) - 1)
return D, D_d, D_f, new_borrowing, debt_service, new_borrowing_f
[docs]
def get_G_ss(
Y,
total_tax_revenue,
agg_pension_outlays,
TR,
UBI_outlays,
I_g,
new_borrowing,
debt_service,
p,
):
r"""
Calculate the steady-state values of government spending.
.. math::
\bar{G} = \bar{Rev} + \bar{D}\bigl[(1 + \bar{g}_n)e^{g_y} -
(1 + \bar{r}_{gov})\bigr] - \bar{I}_g - \bar{TR} - \overline{UBI}
Args:
Y (scalar): aggregate output
total_tax_revenue (scalar): steady-state tax revenue
agg_pension_outlays (scalar): steady-state pension outlays
TR (scalar): steady-state transfer spending
UBI_outlays (scalar): steady-state total UBI outlays
I_g (scalar): steady-state public infrastructure investment
new_borrowing (scalar): steady-state amount of new borrowing
debt_service (scalar): steady-state debt service costs
p (OG-Core Specifications object): model parameters
Returns:
G (tuple): steady-state government spending
"""
if p.budget_balance:
G = p.alpha_G[-1] * Y
else:
G = (
total_tax_revenue
+ new_borrowing
- (agg_pension_outlays + TR + debt_service + UBI_outlays + I_g)
)
return G
[docs]
def get_debt_service_f(r_p, D_f):
r"""
Function to compute foreign debt service payments.
.. math::
\text{Foreign debt service}_{t} = r_{p,t} * D^{f}_{t}
Args:
Returns:
debt_service_f (array_like): foreign debt service payment amount
"""
debt_service_f = r_p * D_f
return debt_service_f
[docs]
def get_TR(
Y,
TR,
G,
total_tax_revenue,
agg_pension_outlays,
UBI_outlays,
I_g,
p,
method,
):
r"""
Function to compute aggregate transfers. Note that this excludes
transfer spending through the public pension system.
.. math::
TR^{'}_{t}=
\begin{cases}
Revenue,& \text{if balanced budget} \\
TR^{baseline}, & \text{if baseline spending}\\
\alpha_{T,t}Y_{t}, & \text{otherwise}
\end{cases}
Args:
Y (array_like): aggregate output
TR (array_like): aggregate government transfers
G (array_like): total government spending
total_tax_revenue (array_like): total tax revenue net of
government pension benefits
agg_pension_outlays (array_like): total government pension
outlays
UBI_outlays (array_like): total universal basic income (UBI) outlays
I_g (array_like): public infrastructure investement
p (OG-Core Specifications object): model parameters
method (str): whether doing SS or TP calculation
Returns:
new_TR (array_like): new value of aggregate government transfers
"""
if p.budget_balance:
new_TR = (
total_tax_revenue - agg_pension_outlays - G - UBI_outlays - I_g
)
elif p.baseline_spending:
new_TR = p.alpha_bs_T[-1] * TR
else:
if method == "SS":
new_TR = p.alpha_T[-1] * Y
else: # time path case
new_TR = p.alpha_T[: p.T] * Y[: p.T]
return new_TR
[docs]
def get_r_gov(r, p, method):
r"""
Determine the interest rate on government debt
.. math::
r_{gov,t} = \max\{(1-\tau_{d,t}r_{t} - \mu_d, 0.0\}
Args:
r (array_like): interest rate on private capital debt over the
time path or in the steady state
p (OG-Core Specifications object): model parameters
Returns:
r_gov (array_like): interest rate on government debt over the
time path or in the steady-state
"""
if method == "SS":
r_gov = np.maximum(p.r_gov_scale[-1] * r - p.r_gov_shift[-1], 0.00)
else:
r_gov = np.maximum(p.r_gov_scale * r - p.r_gov_shift, 0.00)
return r_gov
[docs]
def get_I_g(Y, Ig_baseline, p, method="SS"):
r"""
Find investment in public capital
.. math::
I_{g,t} = \alpha_{I,t}Y_{t}
Args:
Y (array_like): aggregate output
Ig_baseline (array_like): public infrastructure investment in
the baseliine simulation
p (OG-Core Specifications object): model parameters
method (str): either 'SS' for steady-state or 'TPI' for transition path
Returns
I_g (array_like): investment in public capital
"""
if p.baseline_spending:
if method == "SS":
I_g = p.alpha_bs_I[-1] * Ig_baseline
else:
I_g = p.alpha_bs_I[: p.T] * Ig_baseline
else:
if method == "SS":
I_g = p.alpha_I[-1] * Y
else:
I_g = p.alpha_I[: p.T] * Y
return I_g
[docs]
def get_K_g(K_g0, I_g, p, method):
r"""
Law of motion for the government capital stock
.. math::
K_{g,t+1} = \frac{(1 - \delta_g)K_{g,t} + I_{g,t}}
{(1 + \tilde{g}_{n,t+1})e^{g_y}}
Args:
K_g0 (scalar): initial stock of public capital
I_g (array_like): government infrastructure investment
p (ParamTools object): model parameters
method (str): either 'SS' for steady-state or 'TPI' for transition path
Returns
K_g (array_like): stock of public capital
"""
if method == "TPI":
K_g = np.zeros(p.T)
K_g[0] = K_g0
for t in range(p.T - 1): # TODO: numba jit this
growth = (1 + p.g_n[t + 1]) * np.exp(p.g_y)
K_g[t + 1] = ((1 - p.delta_g) * K_g[t] + I_g[t]) / growth
else: # SS
growth = (1 + p.g_n_ss) * np.exp(p.g_y)
K_g = I_g / (growth - (1 - p.delta_g))
return K_g
