finish adding wealth tax

docs/book/content/theory/households.md

@@ -121,7 +121,7 @@ In this section, we describe what is arguably the most important economic agent
 
   ```{math}
   :label: EqTaxCalcLiabETR
-    T_{j,s,t} = \tau^{etr}_{s,t}(x_{j,s,t}, y_{j,s,t})\left(x_{j,s,t} + y_{j,s,t}\right) + \frac{h^{w}p_{w}b_{j,s,t}}{h^{w}b_{j,s,t} + m^{w}} \quad\forall j,s,t
+    T_{j,s,t} = \tau^{etr}_{s,t}(x_{j,s,t}, y_{j,s,t})\left(x_{j,s,t} + y_{j,s,t}\right) + \left(\frac{h^{w}p_{w}b_{j,s,t}}{h^{w}b_{j,s,t} + m^{w}}\right)b{j,s,t} \quad\forall j,s,t
   ```
 
   where the effective income tax rate can be a function of both labor income and capital income $\tau^{etr}_{s,t}(x_{j,s,t},y_{j,s,t})$. The calibration chapter on the microsimulation model and tax function estimation in the country-specific repository documentation details exactly how the model estimates the income tax functions from microsimulation model data.
@@ -205,7 +205,7 @@ In this section, we describe what is arguably the most important economic agent
   ```{math}
   :label: EqHHBC2
     \text{s.t.}\quad &p_t c_{j,s,t} + \sum_{i=1}^I (1 + \tau^{c}_{i,t})p_{i,t}c_{min,i} + b_{j,s+1,t+1} = \\
-    &\quad (1 + r_{p,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \zeta_{j,s}\frac{BQ_t}{\lambda_j\omega_{s,t}} + \eta_{j,s,t}\frac{TR_{t}}{\lambda_j\omega_{s,t}} + ubi_{j,s,t} - T_{s,t} \\
+    &\quad (1 + r_{p,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \zeta_{j,s}\frac{BQ_t}{\lambda_j\omega_{s,t}} + \eta_{j,s,t}\frac{TR_{t}}{\lambda_j\omega_{s,t}} + ubi_{j,s,t} - T_{j,s,t} \\
     &\qquad\text{and}\quad c_{j,s,t}\geq 0,\: n_{j,s,t} \in[0,\tilde{l}],\:\text{and}\: b_{j,1,t}=0 \quad\forall j, t, \:\text{and}\: E+1\leq s\leq E+S \nonumber
   ```
 
@@ -221,7 +221,7 @@ In this section, we describe what is arguably the most important economic agent
 
   ```{math}
   :label: EqHHeul_b
-    &\frac{(c_{j,s,t})^{-\sigma}}{p_t} = \chi^b_j\rho_s(b_{j,s+1,t+1})^{-\sigma} + \beta_j\bigl(1 - \rho_s\bigr)\left(\frac{1 + r_{p,t+1}\bigl[1 - \tau^{mtry}_{s+1,t+1}\bigr]}{p_{t+1}}\right)(c_{j,s+1,t+1})^{-\sigma} \\
+    &\frac{(c_{j,s,t})^{-\sigma}}{p_t} = \chi^b_j\rho_s(b_{j,s+1,t+1})^{-\sigma} + \beta_j\bigl(1 - \rho_s\bigr)\left(\frac{1 + r_{p,t+1}\bigl[1 - \tau^{mtry}_{s+1,t+1}\bigr] - MTR^w_{j,s+1,t+1}}{p_{t+1}}\right)(c_{j,s+1,t+1})^{-\sigma} \\
     &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall j,t, \quad\text{and}\quad E+1\leq s\leq E+S-1 \\
   ```
 
@@ -234,12 +234,15 @@ In this section, we describe what is arguably the most important economic agent
 
   ```{math}
   :label: EqMTRx_derive
-    \frac{\partial T_{s,t}}{\partial n_{j,s,t}}  = \frac{\partial T_{s,t}}{\partial w_t e_{j,s}n_{j,s,t}}\frac{\partial w_{t}e_{j,s}n_{j,s,t}}{\partial n_{j,s,t}} = \frac{\partial T_{s,t}}{\partial w_{t}e_{j,s}n_{j,s,t}}w_t e_{j,s} = \tau^{mtrx}_{s,t}w_t e_{j,s}
+    \frac{\partial T_{j,s,t}}{\partial n_{j,s,t}}  = \frac{\partial T_{s,t}}{\partial w_t e_{j,s}n_{j,s,t}}\frac{\partial w_{t}e_{j,s}n_{j,s,t}}{\partial n_{j,s,t}} = \frac{\partial T_{s,t}}{\partial w_{t}e_{j,s}n_{j,s,t}}w_t e_{j,s} = \tau^{mtrx}_{s,t}w_t e_{j,s}
   ```
 
   ```{math}
   :label: EqMTRy_derive
-    \frac{\partial T_{s,t}}{\partial b_{j,s,t}} = \frac{\partial T_{s,t}}{\partial r_{p,t}b_{j,s,t}}\frac{\partial r_{p,t} b_{j,s,t}}{\partial b_{j,s,t}} = \frac{\partial T_{s,t}}{\partial r_{p,t} b_{j,s,t}}r_{p,t} = \tau^{mtry}_{s,t}r_{p,t}
+    \frac{\partial T_{j,s,t}}{\partial b_{j,s,t}} &= \frac{\partial T_{s,t}}{\partial r_{p,t}b_{j,s,t}}\frac{\partial r_{p,t} b_{j,s,t}}{\partial b_{j,s,t}} + \left(\frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\left[2 -
+        \frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\right] \right) \\ &= \frac{\partial T_{s,t}}{\partial r_{p,t} b_{j,s,t}}r_{p,t} + \left( \frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\left[2 -
+        \frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\right]\right) \\ &= \tau^{mtry}_{s,t}r_{p,t} + \underbrace{\left( \frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\left[2 -
+        \frac{h^{w}p_{w}b_{j,s,t}}{(b_{j,s,t}h^{w}+m^{w})}\right]\right)}_{\equiv MTR^w_{j,s,t}}
   ```
 
 

docs/book/content/theory/stationarization.md

@@ -105,7 +105,7 @@ The usual definition of equilibrium would be allocations and prices such that ho
   ```{math}
   :label: EqStnrz_eul_b
     \frac{(\hat{c}_{j,s,t})^{-\sigma}}{p_t} &= e^{-\sigma g_y}\Biggl[\chi^b_j\rho_s(\hat{b}_{j,s+1,t+1})^{-\sigma} + \\
-    &\qquad\qquad\quad \beta_j\bigl(1 - \rho_s\bigr)\left(\frac{1 + r_{p,t+1}\bigl[1 - \tau^{mtry}_{s+1,t+1}\bigr]}{p_{t+1}}\right)(\hat{c}_{j,s+1,t+1})^{-\sigma}\Biggr] \\
+    &\qquad\qquad\quad \beta_j\bigl(1 - \rho_s\bigr)\left(\frac{1 + r_{p,t+1}\bigl[1 - \tau^{mtry}_{s+1,t+1}\bigr] - \hat{MTR}^w_{j,s+1,t+1}}{p_{t+1}}\right)(\hat{c}_{j,s+1,t+1})^{-\sigma}\Biggr] \\
     &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\forall j,t, \quad\text{and}\quad E+1\leq s\leq E+S-1 \\
   ```
 
@@ -114,6 +114,9 @@ The usual definition of equilibrium would be allocations and prices such that ho
     \frac{(\hat{c}_{j,E+S,t})^{-\sigma}}{p_t} = e^{-\sigma g_y}\chi^b_j(\hat{b}_{j,E+S+1,t+1})^{-\sigma} \quad\forall j,t \quad\text{and}\quad s = E+S
   ```
 
+Where $\hat{MTR}^w_{j,s,t} = \left( \frac{h^{w}p_{w}\hat{b}_{j,s,t}}{(\hat{b}_{j,s,t}h^{w}+m^{w})}\left[2 -
+        \frac{h^{w}p_{w}\hat{b}_{j,s,t}}{(\hat{b}_{j,s,t}h^{w}+m^{w})}\right]\right)$
+
 (SecStnrzFirms)=
 ## Stationarized Firms Equations