Merge remote-tracking branch 'upstream/master' into calib_tables

Author: jdebacker

CHANGELOG.md

@@ -5,6 +5,25 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.16.1] - 2026-06-04 12:00:00
+
+### Added
+
+- Adds a `use_sparse_FOC_jac` `Specifications` parameter (default `True`) that accelerates the time path iteration (TPI) household solve. With it on, `scipy.optimize.root` is given a sparse (banded) finite-difference Jacobian for the stacked household Euler and labor first order conditions: the sparsity pattern is auto-detected once per problem size and the solver then needs far fewer function evaluations per Jacobian build (about 20x fewer on the default S=80 cohort solve), with an automatic fallback to dense finite differences if the Jacobian is not sparse enough to benefit or if a solve fails. The result matches the legacy dense-finite-difference solution to within the model's resource-constraint accuracy floor on every calibration tested (OG-Core standard example, OG-ETH, OG-ZAF, OG-PHL, OG-IDN), giving roughly a 1.9-2.4x TPI speedup at no accuracy cost. Set `use_sparse_FOC_jac=False` to recover bit-identical agreement with v0.16.0 and earlier.
+
+## [0.16.0] - 2026-06-02 12:00:00
+
+### Added
+
+- Adds a new parameter, `alpha_FA`, that allows the user to specify the level of direct foreign aid as a percentage of GDP.  See PR [#1126](https://github.com/PSLmodels/OG-Core/pull/1126).
+
+## [0.15.13] - 2026-05-15 06:00:00
+
+### Added
+
+- Increases the maximum value of the `mindist_TPI` parameter to 0.01 in `default_parameters.json`.
+- Updates the `uv.lock` with package updates.
+
 ## [0.15.12] - 2026-05-14 12:00:00
 
 ### Added
@@ -568,6 +587,9 @@ and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0
 - Any earlier versions of OG-USA can be found in the [`OG-Core`](https://github.com/PSLmodels/OG-Core) repository [release history](https://github.com/PSLmodels/OG-Core/releases) from [v.0.6.4](https://github.com/PSLmodels/OG-Core/releases/tag/v0.6.4) (Jul. 20, 2021) or earlier.
 
 
+[0.16.0]: https://github.com/PSLmodels/OG-Core/compare/v0.15.13...v0.16.0
+[0.15.13]: https://github.com/PSLmodels/OG-Core/compare/v0.15.12...v0.15.13
+[0.15.12]: https://github.com/PSLmodels/OG-Core/compare/v0.15.11...v0.15.12
 [0.15.11]: https://github.com/PSLmodels/OG-Core/compare/v0.15.10...v0.15.11
 [0.15.10]: https://github.com/PSLmodels/OG-Core/compare/v0.15.9...v0.15.10
 [0.15.9]: https://github.com/PSLmodels/OG-Core/compare/v0.15.8...v0.15.9

docs/book/content/intro/parameters.md

@@ -279,6 +279,15 @@ _Valid Range:_ min = 0.0 and max = 5.0
 _Out-of-Range Action:_ error  
 
 
+### Fiscal Policy Parameters
+
+####  `alpha_FA`  
+_Description:_ Foreign aid payments to domestic government as a share of GDP.  
+_Value Type:_ float  
+_Valid Range:_ min = 0.0 and max = 1.0  
+_Out-of-Range Action:_ error  
+
+
 ### Government Pension Parameters
 
 ####  `retirement_age`  
@@ -556,7 +565,7 @@ _Out-of-Range Action:_ error
 _Description:_ Consumption tax rate.  Set value for base year, click '+' to add value for next year.  All future years not specified are set to last value entered.  
 _Notes:_ This policy parameter represents the effective consumption tax rate from sales taxes, VATs, and excise taxes by consumption good. Tax rates cab vary over time. It is thus a TxI array.  
 _Value Type:_ float  
-_Valid Range:_ min = 0.0 and max = 5.0  
+_Valid Range:_ min = -5.0 and max = 5.0  
 _Out-of-Range Action:_ error  
 
 
@@ -879,7 +888,7 @@ _Out-of-Range Action:_ error
 ####  `mindist_TPI`  
 _Description:_ Tolerance for convergence of time path solution.  
 _Value Type:_ float  
-_Valid Range:_ min = 1e-13 and max = 0.001  
+_Valid Range:_ min = 1e-13 and max = 0.01  
 _Out-of-Range Action:_ error  
 
 

docs/book/content/theory/derivations.md

@@ -114,3 +114,16 @@ In the Cobb-Douglas unit elasticity case ($\varepsilon=1$) of the CES production
 ```
 
 Again, even if this simple case, we cannot solve for $r$ as a function of $w$ for the reasons above.
+
+
+(SecAppDerivHHjac)=
+## Sparsity of the household equation Jacobian
+
+  Holding fixed the prices and policies a type-$j$ cohort faces, its $2S$ stationarized necessary conditions {eq}`EqStnrz_eul_n`, {eq}`EqStnrz_eul_b`, and {eq}`EqStnrz_eul_bS` in the $2S$ unknowns $\{n_{j,s},\hat b_{j,s+1}\}_{s=E+1}^{E+S}$ have a banded Jacobian. From the budget constraint {eq}`EqStnrzHHBC`, stationarized consumption at age $s$ depends on only three unknowns,
+
+  ```{math}
+  :label: EqAppDerivHHjac_cons
+    \hat c_{j,s} = \frac{1}{p}\Bigl[(1+r_p)\hat b_{j,s} + \hat w\,e_{j,s}\,n_{j,s} - \widehat{tax}_{j,s} - e^{g_y}\hat b_{j,s+1}\Bigr] + X_{j,s},
+  ```
+
+  where $\widehat{tax}_{j,s}$ depends only on $(\hat b_{j,s}, n_{j,s})$ through labor and capital income (already in the active set, so it adds no further coupling), and $X_{j,s}$ collects terms fixed in the inner solve (bequests $\hat{bq}_{j,s}$, remittances $\hat{rm}_{j,s}$, government transfers $\hat{tr}_{j,s}$, UBI $\hat{ubi}_{j,s}$, the pension benefit $\theta_j$, and the $\hat c_{min,i}$ terms). The labor Euler equation {eq}`EqStnrz_eul_n` at age $s$ therefore depends on $\{\hat b_{j,s},\hat b_{j,s+1},n_{j,s}\}$ alone, and the savings Euler equation {eq}`EqStnrz_eul_b`---which links $\hat c_{j,s}$ to $\hat b_{j,s+1}$ and $\hat c_{j,s+1}$---depends on $\{\hat b_{j,s},\hat b_{j,s+1},\hat b_{j,s+2},n_{j,s},n_{j,s+1}\}$. The marginal tax rates $\tau^{mtrx}_s$ and $\tau^{mtry}_{s+1}$ are functions of own-age income (already in these sets), so they add no further coupling, and the terminal condition {eq}`EqStnrz_eul_bS` is sparser still. Each of the $2S$ equations therefore depends on at most five of the $2S$ unknowns, regardless of $S$, so the Jacobian has at most $10S$ nonzero entries rather than the $(2S)^2 = 4S^2$ of a fully coupled system. This is the per-cohort counterpart to the dense $2JS$ system noted at the start of Chapter {ref}`Chap_Eqm`: cohorts couple only through prices, which are held fixed in the inner solve. A finite-difference Jacobian can then be built from a number of evaluations set by the bandwidth---about seven at $S = 80$---rather than $2S$, by probing together unknowns that affect no common equation (Figure {numref}`FigHHjacSparsity`).

docs/book/content/theory/equilibrium.md

@@ -25,6 +25,15 @@ In all of the specifications of `OG-Core`, we use a two-stage fixed point algori
 
 Our approach is to choose the minimum number of macroeconomic variables in an outer loop in order to be able to solve the household's $2JS$ Euler equations in terms of only the $\bar{n}_{j,s}$ and $\bar{b}_{j,s+1}$ variables directly, holding all other variables constant. The household system of Euler equations has a provable root solution and is orders of magnitude more tractable (less nonlinear) to solve holding these outer loop variables constant.
 
+Moreover, with the outer-loop variables held fixed, each cohort's system of $2S$ Euler equations is not only less nonlinear but structurally sparse: every equation involves at most five of the $2S$ unknowns---a household's own age and its immediate neighbors. The root finder normally probes each unknown separately when building each step ($2S = 160$ evaluations of the system when $S = 80$), but with most equations depending on only a handful of unknowns, those affecting no common equation can be probed together, cutting the count to about seven at $S = 80$---a number set by how many neighbors couple, not by $S$. The parameter `use_sparse_FOC_jac` (default `True`) controls this; set it to `False` to use the legacy dense-finite-difference Jacobian on every call. The structure is derived in Appendix {ref}`SecAppDerivHHjac`.
+
+```{figure} ./images/HH_jac_sparsity.png
+---
+name: FigHHjacSparsity
+---
+Sparsity pattern of the household equation Jacobian, at $S = 12$. Left: the standard finite-difference solve treats every entry of the $2S\times 2S$ matrix as live ($(2S)^2 = 576$ entries). Right: the actual structure---each Euler equation depends only on a household's own age and its immediate neighbors, leaving most entries zero (92 of 576 here; 636 of 25{,}600 at the default $S = 80$).
+```
+
 The steady-state solution method for each of the cases above is associated with a solution method that has a subset of the following outer-loop variables $\Bigl\{\bar{r}_p, \bar{r}, \bar{w}, \{\bar{p}_m\}_{m=1}^{M-1}, \bar{Y}, \overline{TR}, \overline{BQ}, factor\Bigr\}$.
 
 

docs/book/content/theory/government.md

@@ -615,11 +615,11 @@ Note that the budget closure rule (described in Section ref{`SecUnbalGBCcloseRul
 (SecUnbalGBCbudgConstr)=
 ## Government Budget Constraint
 
-  Let the level of government debt in period $t$ be given by $D_t$. The government budget constraint requires that government revenue $Rev_t$ plus the budget deficit ($D_{t+1} - D_t$) equal expenditures on interest on the debt, government spending on public goods $G_t$, total infrastructure investments $I_{g,t}$, total pension outlays, total transfer payments to households $TR_t$, and $UBI_t$ every period $t$,
+  Let the level of government debt in period $t$ be given by $D_t$. The government budget constraint requires that government revenue $Rev_t$, external foreign assistance, $FA_t$, plus the budget deficit ($D_{t+1} - D_t$) equal expenditures on interest on the debt, government spending on public goods $G_t$, total infrastructure investments $I_{g,t}$, total pension outlays, total transfer payments to households $TR_t$, and $UBI_t$ every period $t$,
 
   ```{math}
   :label: EqUnbalGBCbudgConstr
-    D_{t+1} + Rev_t = (1 + r_{gov,t})D_t + G_t + I_{g,t} + Pensions_t + TR_t + UBI_t  \quad\forall t
+    D_{t+1} + Rev_t + FA_t = (1 + r_{gov,t})D_t + G_t + I_{g,t} + Pensions_t + TR_t + UBI_t  \quad\forall t
   ```
 
   where $r_{gov,t}$ is the interest rate paid by the government defined in equation {eq}`EqUnbalGBC_rate_wedge` below, $G_{t}$ is government spending on public goods, $I_{g,t}$ is total government spending on infrastructure investment, $TR_{t}$ are non-pension government transfers, and $UBI_t$ is the total UBI transfer outlays across households in time $t$. All variables in {eq}`EqUnbalGBCbudgConstr` are real variables denominated in units of current-period output in industry $M$ the numeraire ($p_{M,t}=1$ for all $t$).
@@ -713,8 +713,8 @@ Note that the budget closure rule (described in Section ref{`SecUnbalGBCcloseRul
     &\text{where}\quad g_{g,t} =
       \begin{cases}
         1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\:\text{if}\quad t < T_{G1} \\
-        \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - I_{g,t} - TR_{t} - UBI_{t} + Rev_{t}}{\alpha_g Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
-        \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - I_{g,t} - TR_{t} - UBI_{t} + Rev_{t}}{\alpha_g Y_t} \qquad\qquad\quad\:\:\,\text{if}\quad t \geq T_{G2}
+        \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - I_{g,t} - TR_{t} - UBI_{t} + Rev_{t} + FA_{t}}{\alpha_g Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
+        \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - I_{g,t} - TR_{t} - UBI_{t} + Rev_{t} + FA_{t}}{\alpha_g Y_t} \qquad\qquad\quad\:\:\,\text{if}\quad t \geq T_{G2}
       \end{cases} \\
     &\text{and}\quad g_{tr,t} = 1 \quad\forall t
   \end{split}
@@ -735,8 +735,8 @@ Note that the budget closure rule (described in Section ref{`SecUnbalGBCcloseRul
     &\text{where}\quad g_{tr,t} =
       \begin{cases}
         1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\:\,\text{if}\quad t < T_{G1} \\
-        \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - G_{t} - I_{g,t} -  UBI_{t} + Rev_{t}}{\alpha_{tr} Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
-        \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - G_{t} - I_{g,t} - UBI_{t} + Rev_{t}}{\alpha_{tr}Y_t} \qquad\qquad\quad\:\:\:\text{if}\quad t \geq T_{G2}
+        \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - G_{t} - I_{g,t} -  UBI_{t} + Rev_{t} + FA_{t}}{\alpha_{tr} Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
+        \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - G_{t} - I_{g,t} - UBI_{t} + Rev_{t} + FA_{t}}{\alpha_{tr}Y_t} \qquad\qquad\quad\:\:\:\text{if}\quad t \geq T_{G2}
       \end{cases} \\
     &\text{and}\quad g_{g,t} = 1 \quad\forall t
   \end{split}
@@ -766,8 +766,8 @@ Note that the budget closure rule (described in Section ref{`SecUnbalGBCcloseRul
     &\text{where}\quad g_{trg,t} =
     \begin{cases}
       1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\:\,\text{if}\quad t < T_{G1} \\
-      \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - I_{g,t} - UBI_{t} + Rev_{t}}{\left(\alpha_g + \alpha_{tr}\right)Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
-      \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - I_{g,t} - UBI_{t} + Rev_{t}}{\left(\alpha_g + \alpha_{tr}\right)Y_t} \qquad\qquad\quad\:\:\:\text{if}\quad t \geq T_{G2}
+      \frac{\left[\rho_{d}\alpha_{D}Y_{t} + (1-\rho_{d})D_{t}\right] - (1+r_{gov,t})D_{t} - I_{g,t} - UBI_{t} + Rev_{t} + FA_{t}}{\left(\alpha_g + \alpha_{tr}\right)Y_t} \quad\text{if}\quad T_{G1}\leq t<T_{G2} \\
+      \frac{\alpha_{D}Y_{t} - (1+r_{gov,t})D_{t} - I_{g,t} - UBI_{t} + Rev_{t} + FA_{t}}{\left(\alpha_g + \alpha_{tr}\right)Y_t} \qquad\qquad\quad\:\:\:\text{if}\quad t \geq T_{G2}
     \end{cases}
   \end{split}
   ```

docs/book/content/theory/images/HH_jac_sparsity.png

@@ -244,7 +244,7 @@ The usual definition of equilibrium would be allocations and prices such that ho
 
   ```{math}
   :label: EqStnrzGovBC
-    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{I}_{g,t} + \hat{Pensions}_t + \hat{TR}_t + \hat{UBI}_t \quad\forall t
+    e^{g_y}\left(1 + \tilde{g}_{n,t+1}\right)\hat{D}_{t+1} + \hat{Rev}_t + \hat{FA}_t = (1 + r_{gov,t})\hat{D}_t + \hat{G}_t + \hat{I}_{g,t} + \hat{Pensions}_t + \hat{TR}_t + \hat{UBI}_t \quad\forall t
   ```
 
   The stationarized versions of the rule for total government infrastructure investment spending $I_{g,t}$ in {eq}`EqUnbalGBC_Igt` and the rule for government investment spending in each industry in {eq}`EqUnbalGBC_Igt` are found by dividing both sides of the respective equations by $e^{g_y t}\tilde{N}_t$.
@@ -298,8 +298,8 @@ The usual definition of equilibrium would be allocations and prices such that ho
     &\text{where}\quad g_{g,t} =
     \begin{cases}
       1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \:\text{if}\: 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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_g \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
+      \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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_t}{\alpha_g \hat{Y}_t} \:\text{if}\: 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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_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
   \end{split}
@@ -313,8 +313,8 @@ The usual definition of equilibrium would be allocations and prices such that ho
     &\text{where}\quad g_{tr,t} =
     \begin{cases}
       1 \qquad\qquad\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{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_{tr} \hat{Y}_t} \:\text{if}\: 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{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\alpha_{tr} \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
+      \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{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_t}{\alpha_{tr} \hat{Y}_t} \:\text{if}\: 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{G}_{t} - \hat{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_t}{\alpha_{tr} \hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
     \end{cases} \\
     &\text{and}\quad g_{g,t} = 1 \quad\forall t
   \end{split}
@@ -328,8 +328,8 @@ The usual definition of equilibrium would be allocations and prices such that ho
     &\text{where}\quad g_{trg,t} =
     \begin{cases}
       1 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\quad \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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \:\text{if}\: 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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t}}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
+      \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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_t}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \:\text{if}\: 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{I}_{g,t} - \hat{UBI}_t + \hat{Rev}_{t} + \hat{FA}_t}{\left(\alpha_g + \alpha_{tr}\right)\hat{Y}_t} \qquad\qquad\quad\,\text{if}\quad t \geq T_{G2}
     \end{cases}
   \end{split}
   ```

docs/book/content/theory/stationarization.md

@@ -1115,7 +1115,8 @@ def SS_solver(
     net_capital_outflows_vec[-1] = net_capital_outflows
     RM_vec_ss = np.zeros(p.M)
     RM_vec_ss[-1] = RM_ss
-
+    foreign_aid_vec_ss = np.zeros(p.M)
+    foreign_aid_vec_ss[-1] = p.alpha_FA[-1] * Yss
     RC = aggr.resource_constraint(
         Y_vec_ss,
         C_m_vec_ss,
@@ -1124,6 +1125,7 @@ def SS_solver(
         I_g_vec_ss,
         net_capital_outflows_vec,
         RM_vec_ss,
+        foreign_aid_vec_ss,
     )
     logger.info(f"Foreign debt holdings = {D_f_ss}")
     logger.info(f"Foreign capital holdings = {K_f_ss}")