Merge pull request #818 from jdebacker/io_matrix

Merging

docs/book/content/theory/derivations.md

@@ -5,15 +5,15 @@
 This appendix contains derivations from the theory in the body of this book.
 
 
-(SecAppDerivIndSpecCons)=
+(SecAppDerivHHcons)=
 ## Household first order condition for industry-specific consumption demand
 
-  The derivation for the household first order condition for industry-specific consumption demand {eq}`EqHHFOCcm` is the following:
+  The derivation for the household first order condition for industry-specific consumption demand {eq}`EqHHFOCci` is the following:
   ```{math}
-  :label: EqAppDerivHHIndSpecConsFOC
-    \tilde{p}_{m,t} = \tilde{p}_{j,s,t}\alpha_m(c_{j,m,s,t} - c_{min,m})^{\alpha_m-1}\prod_{u\neq m}^M\left(c_{j,u,s,t} - c_{min,u}\right)^{\alpha_u} \\
-    \tilde{p}_{m,t}(c_{j,m,s,t} - c_{min,m}) = \tilde{p}_{j,s,t}\alpha_m(c_{j,m,s,t} - c_{min,m})^{\alpha_m}\prod_{u\neq m}^M\left(c_{j,u,s,t} - c_{min,u}\right)^{\alpha_u} \\
-    \tilde{p}_{m,t}(c_{j,m,s,t} - c_{min,m}) = \tilde{p}_{j,s,t}\alpha_m\prod_{m=1}^M\left(c_{j,m,s,t} - c_{min,m}\right)^{\alpha_m} = \alpha_m \tilde{p}_{j,s,t}c_{j,s,t}
+  :label: EqAppDerivHHconsFOC
+    \tilde{p}_{i,t} = \tilde{p}_{j,s,t}\alpha_m(c_{i,j,s,t} - c_{min,i})^{\alpha_i-1}\prod_{u\neq i}^I\left(c_{u,j,s,t} - c_{min,u}\right)^{\alpha_u} \\
+    \tilde{p}_{i,t}(c_{i,j,s,t} - c_{min,i}) = \tilde{p}_{j,s,t}\alpha_i(c_{i,j,s,t} - c_{min,i})^{\alpha_i}\prod_{u\neq i}^I\left(c_{u,j,s,t} - c_{min,u}\right)^{\alpha_u} \\
+    \tilde{p}_{i,t}(c_{i,j,s,t} - c_{min,i}) = \tilde{p}_{j,s,t}\alpha_i\prod_{i=1}^I\left(c_{i,j,s,t} - c_{min,i}\right)^{\alpha_i} = \alpha_i \tilde{p}_{j,s,t}c_{j,s,t}
   ```
 
 

docs/book/content/theory/equilibrium.md

@@ -60,10 +60,11 @@ The computational algorithm for solving for the steady-state follows the steps b
 
 2. Choose an initial guess for the values of the steady-state interest rate (the after-tax marginal product of capital) $\bar{r}^i$, wage rate $\bar{w}^i$,  portfolio rate of return $\bar{r}_p^i$, output prices $\boldsymbol{\bar{p}}^i$ (note that $\bar{p}_M =1$ since it's the numeraire good), total bequests $\overline{BQ}^{\,i}$, total household transfers $\overline{TR}^{\,i}$, and income multiplier $factor^i$, where superscript $i$ is the index of the iteration number of the guess.
 
-    1. Given $\boldsymbol{\bar{p}}^i$ find the price of the composite good, $\bar{p}$ using equation {eq}`EqCompPnorm2`
-    2. Using {eq}`Eq_tr` with $\overline{TR}^{\,i}$, find transfers to each household, $\overline{tr}_{j,s}^i$
-    3. Using the bequest transfer process, {eq}`Eq_bq` and aggregate bequests, $\overline{BQ}^{\,i}$, find $bq_{j,s}^i$
-    4. Given values $\bar{p}$, $\bar{r}_{p}^i$, $\bar{w}^i$ $\overline{bq}_{j,s}^i$, $\overline{tr}_{j,s}^i$, and $factor^i$, solve for the steady-state household labor supply $\bar{n}_{j,s}$ and savings $\bar{b}_{j,s+1}$ decisions for all $j$ and $E+1\leq s\leq E+S$.
+    1. Given $\boldsymbol{\bar{p}}^i$ find the price of consumption goods using {eq}`EqHH_pi2`
+    2. From price of consumption goods, determine the price of the composite consmpution good, $\bar{p}$ using equation {eq}`EqCompPnorm2`
+    3. Using {eq}`Eq_tr` with $\overline{TR}^{\,i}$, find transfers to each household, $\overline{tr}_{j,s}^i$
+    4. Using the bequest transfer process, {eq}`Eq_bq` and aggregate bequests, $\overline{BQ}^{\,i}$, find $bq_{j,s}^i$
+    5. Given values $\bar{p}$, $\bar{r}_{p}^i$, $\bar{w}^i$ $\overline{bq}_{j,s}^i$, $\overline{tr}_{j,s}^i$, and $factor^i$, solve for the steady-state household labor supply $\bar{n}_{j,s}$ and savings $\bar{b}_{j,s+1}$ decisions for all $j$ and $E+1\leq s\leq E+S$.
 
         1. Each of the $j\in 1,2,...J$ sets of $2S$ steady-state Euler equations can be solved separately. `OG-Core` parallelizes this process using the maximum number of processors possible (up to $J$ processors). Solve each system of Euler equations using a multivariate root-finder to solve the $2S$ necessary conditions of the household given by the following steady-state versions of stationarized household Euler equations {eq}`EqStnrz_eul_n`, {eq}`EqStnrz_eul_b`, and {eq}`EqStnrz_eul_bS` simultaneously for each $j$.
 
@@ -89,9 +90,9 @@ The computational algorithm for solving for the steady-state follows the steps b
         :label: EqSS_HHeul_bS
           (\bar{c}_{j,E+S})^{-\sigma} = e^{-\sigma g_y}\chi^b_j(\bar{b}_{j,E+S+1})^{-\sigma} \quad\forall j
         ```
-    5. Determine from the quantity of the composite consumption good consumed by each household, $\bar{c}_{j,s}$, use equation {eq}`EqHH_cmDem` to determine consumption of each output good, $\bar{c}_{m,j,s}$
-    6. Using $\bar{c}_{m,j,s}$ in {eq}`EqCmt`, solve for aggregate consumption of each output good, $\bar{C}_{m}$
-    7. Given values for $\bar{n}_{j,s}$ and $\bar{b}_{j,s+1}$ for all $j$ and $s$, solve for steady-state labor supply, $\bar{L}$, savings, $\bar{B}$
+    6. Determine from the quantity of the composite consumption good consumed by each household, $\bar{c}_{j,s}$, use equation {eq}`EqHH_cmDem` to determine consumption of each output good, $\bar{c}_{m,j,s}$
+    7. Using $\bar{c}_{m,j,s}$ in {eq}`EqCmt`, solve for aggregate consumption of each output good, $\bar{C}_{m}$
+    8. Given values for $\bar{n}_{j,s}$ and $\bar{b}_{j,s+1}$ for all $j$ and $s$, solve for steady-state labor supply, $\bar{L}$, savings, $\bar{B}$
         1. Use $\bar{n}_{j,s}$ and the steady-state version of the stationarized labor market clearing equation {eq}`EqStnrzMarkClrLab` to get a value for $\bar{L}^{i}$.
 
            ```{math}
@@ -104,14 +105,14 @@ The computational algorithm for solving for the steady-state follows the steps b
            :label: EqSS_Bt
              \bar{B} \equiv \frac{1}{1 + \bar{g}_{n}}\sum_{s=E+2}^{E+S+1}\sum_{j=1}^{J}\Bigl(\bar{\omega}_{s-1}\lambda_j\bar{b}_{j,s} + i_s\bar{\omega}_{s}\lambda_j\bar{b}_{j,s}\Bigr)
            ```
-    8. Solve for the exogenous government interest rate $\bar{r}_{gov}^{i}$ using equation {eq}`EqUnbalGBC_rate_wedge`.
-    9. Use {eq}`EqStnrzTfer` to find $\bar{Y}^i$ from the guess of $\overline{TR}^i$
-    10. Use {eq}`EqStnrz_DY` to find $\bar{D}^i$ from $\bar{Y}^i$
-    11. Using $\bar{D}^i$, we can find foreign investor holdings of debt, $\bar{D}^{f,i}$ from {eq}`EqMarkClr_zetaD2` and then solve for domestic debt holdings through the debt market clearing condition: $\bar{D}^{d,i} = \bar{D}^i - \bar{D}^{f,i}$
-    12. Using $\bar{Y}^i$, find government infrastructure investment, $\bar{I}_{g}$ from {eq}`EqStnrz_Igt`
-    13. Using the law of motion of the stock of infrastructure, {eq}`EqStnrz_Kgmt`, and $\bar{I}_{g}$, solve for $\bar{K}_{g}^{i}$
-    14. Find output and factor demands for M-1 industries:
-        1. By {eq}`EqMarkClrGoods_Mm1`, $\bar{Y}_{m}=\bar{C}_{m}$
+    9. Solve for the exogenous government interest rate $\bar{r}_{gov}^{i}$ using equation {eq}`EqUnbalGBC_rate_wedge`.
+    10. Use {eq}`EqStnrzTfer` to find $\bar{Y}^i$ from the guess of $\overline{TR}^i$
+    11. Use {eq}`EqStnrz_DY` to find $\bar{D}^i$ from $\bar{Y}^i$
+    12. Using $\bar{D}^i$, we can find foreign investor holdings of debt, $\bar{D}^{f,i}$ from {eq}`EqMarkClr_zetaD2` and then solve for domestic debt holdings through the debt market clearing condition: $\bar{D}^{d,i} = \bar{D}^i - \bar{D}^{f,i}$
+    13. Using $\bar{Y}^i$, find government infrastructure investment, $\bar{I}_{g}$ from {eq}`EqStnrz_Igt`
+    14. Using the law of motion of the stock of infrastructure, {eq}`EqStnrz_Kgmt`, and $\bar{I}_{g}$, solve for $\bar{K}_{g}^{i}$
+    15. Find output and factor demands for M-1 industries:
+        1. By {eq}`EqMarkClrGoods_Mm1`, $\hat{Y}_{m,t}=\hat{C}_{m,t}$, where $\hat{C}_{m,t}$ is determined by {eq}`EqStnrzEqCmt`
         2. The capital-output ratio can be determined from the FOC for the firms' choice of capital: $\frac{\bar{K}_m}{\bar{Y}_m} = \gamma_m\left[\frac{\bar{r} +\bar{\delta}_M - \bar{\tau}^{corp}_m\bar{\delta}^{\tau}_m - \bar{\tau}^{inv}_m\bar{\delta}_M}{\left(1 - \bar{\tau}^{corp}_m\right)\bar{p}_m(\bar{Z}_m)^\frac{\varepsilon_m-1}{\varepsilon_m}}\right]^{-\varepsilon_m}$
         3. Capital demand can thus be found: $\bar{K}_{m} = \frac{\bar{K}_m}{\bar{Y}_m} * \bar{Y}_m$
         4. Labor demand can be found by inverting the production function:
@@ -323,8 +324,10 @@ The stationary non-steady state (transition path) solution algorithm has followi
 4. Using {eq}`Eq_tr` with $\boldsymbol{\hat{TR}}^{\,i}$, find transfers to each household, $\boldsymbol{\hat{tr}}_{j,s}^i$
 5. Using the bequest transfer process, {eq}`Eq_bq` and aggregate bequests, $\boldsymbol{\hat{BQ}}^{\,i}$, find $\boldsymbol{\hat{bq}}_{j,s}^i$
 6. Given time path guesses $\{\boldsymbol{r}_p^i, \boldsymbol{\hat{w}}^i, \boldsymbol{p}^i, \boldsymbol{\hat{bq}}^i, \boldsymbol{\hat{tr}}^i\}$, we can solve for each household's lifetime decisions $\{n_{j,s,t},\hat{b}_{j,s+1,t+1}\}_{s=E+1}^{E+S}$ for all $j$, $E+1\leq s \leq E+S$, and $1\leq t\leq T_2+S-1$.
-   1. The household problem can be solved with a multivariate root finder solving the $2S$ equations and unknowns at once for each $j$ and $1\leq t\leq T+S-1$. The root finder uses $2S$ household Euler equations  {eq}`EqStnrz_eul_n`, {eq}`EqStnrz_eul_b`, and {eq}`EqStnrz_eul_bS` to solve for each household's $2S$ lifetime decisions. The household decision rules for each type and birth cohort are solved separately.
-   2. After solving the first iteration of time path iteration, subsequent initial values for the $J$, $2S$ root finding problems are based on the solution in the prior iteration. This speeds up computation further and makes the initial guess for the highly nonlinear system of equations start closer to the solution value.
+   1. Given $\boldsymbol{p}^i$ find the price of consumption goods using {eq}`EqHH_pi2`
+   2. From price of consumption goods, determine the price of the composite consmpution good, $\bar{p}$ using equation {eq}`EqCompPnorm2`
+   3. The household problem can be solved with a multivariate root finder solving the $2S$ equations and unknowns at once for each $j$ and $1\leq t\leq T+S-1$. The root finder uses $2S$ household Euler equations  {eq}`EqStnrz_eul_n`, {eq}`EqStnrz_eul_b`, and {eq}`EqStnrz_eul_bS` to solve for each household's $2S$ lifetime decisions. The household decision rules for each type and birth cohort are solved separately.
+   4. After solving the first iteration of time path iteration, subsequent initial values for the $J$, $2S$ root finding problems are based on the solution in the prior iteration. This speeds up computation further and makes the initial guess for the highly nonlinear system of equations start closer to the solution value.
 7. Determine from the quantity of the composite consumption good consumed by each household, $\hat{c}_{j,s,t}$, use equation {eq}`EqHH_cmDem` to determine consumption of each output good, $\hat{c}_{m,j,s,t}$
 8. Using $\hat{c}_{m,j,s,t}$ in {eq}`EqCmt`, solve for aggregate consumption of each output good, $\hat{C}_{m,t}$
 9.  Given values for $n_{j,s,t}$ and $\hat{b}_{j,s+1,t+1}$ for all $j$, $s$, and $t$, solve for aggregate labor supply, $\hat{L}_t$, and savings, $B_t$ in each period
@@ -338,7 +341,7 @@ The stationary non-steady state (transition path) solution algorithm has followi
 15. Using $\hat{Y}_t^i$, find government infrastructure investment, $\hat{I}_{g,t}$ from {eq}`EqStnrz_Igt`
 16. Using the law of motion of the stock of infrastructure, {eq}`EqStnrz_Kgmt`, and $\hat{I}_{g,t}$, solve for $\hat{K}_{g,t}^{i}$
 17. Find output and factor demands for M-1 industries:
-    1. By {eq}`EqMarkClrGoods_Mm1`, $\hat{Y}_{m,t}=\hat{C}_{m,t}$
+    1. By {eq}`EqMarkClrGoods_Mm1`, $\hat{Y}_{m,t}=\hat{C}_{m,t}$, where $\hat{C}_{m,t}$ is determined by {eq}`EqStnrzEqCmt`
     2. The capital-output ratio can be determined from the FOC for the firms' choice of capital: $\frac{\hat{K}_{m,t}}{\hat{Y}_{m,t}} = \gamma_m\left[\frac{r_t + \delta_{M,t} - \tau^{corp}_{m,t}\delta^{\tau}_{m,t} - \tau^{inv}_{m,t}\delta_{M,t}}{(1-\tau^{corp}_{m,t})p_{m,t}({Z}_{m,t})^\frac{\varepsilon_m -1}{\varepsilon_m}}\right]^{-\varepsilon_m}$
     3. Capital demand can thus be found: $\hat{K}_{m,t} = \frac{\hat{K}_{m,t}}{\hat{Y}_{m,t}} * \hat{Y}_{m,t}$
     4. Labor demand can be found by inverting the production function:

docs/book/content/theory/government.md

@@ -17,9 +17,10 @@ Government levies taxes on households and firms, funds public pensions, and make
 Income taxes are modeled through the total tax liability function $T_{s,t}$, which can be decomposed into the effective tax rate times total income {eq}`EqTaxCalcLiabETR2`. In this chapter, we detail the household tax component of government activity $T_{s,t}$ in `OG-Core`, along with our method of incorporating detailed microsimulation data into a dynamic general equilibrium model.
 
 ```{math}
-:label: EqHHBC
-  c_{j,s,t} + b_{j,s+1,t+1} &= (1 + r_{hh,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \\
-  &\quad\quad\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}  \\
+:label: EqHHBC2
+  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} = \\
+  &(1 + r_{p,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \\
+  &\quad\quad\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}  \\
   &\quad\forall j,t\quad\text{and}\quad s\geq E+1 \quad\text{where}\quad b_{j,E+1,t}=0\quad\forall j,t
 ```
 
@@ -280,7 +281,7 @@ When computing the role of compliance on the effective tax rate, we take a weigh
 
 #### Consumption taxes
 
-Linear consumption taxes, $\tau^c_{m,t}$ can vary over time and by output good.
+Linear consumption taxes, $\tau^c_{i,t}$ can vary over time and by consumption good.
 
 #### Wealth taxes
 
@@ -389,8 +390,8 @@ Businesses face a linear tax rate $\tau^{b}_{m,t}$, which can vary by industry a
   We see from the household's budget constraint that taxes $T_{j,s,t}$ and transfers $TR_{t}$ enter into the household's decision,
 
   ```{math}
-  :label: EqHHBC2
-    p_t c_{j,s,t} + &\sum_{m=1}^M p_{m,t}c_{min,m} + b_{j,s+1,t+1} = \\
+  :label: EqHHBC3
+    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} = \\
     &(1 + r_{p,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} + \\
     &\quad\quad\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}  \\
     &\quad\forall j,t\quad\text{and}\quad s\geq E+1 \quad\text{where}\quad b_{j,E+1,t}=0\quad\forall j,t