Updated households.md

docs/book/content/theory/households.md

@@ -111,16 +111,16 @@ In this section, we describe what is arguably the most important economic agent
   where $c_{j,s,t}$ is consumption, $b_{j,s+1,t+1}$ is savings for the next period, $r_{p,t}$ is the normalized interest rate (return) on household savings invested in the financial intermediary, $b_{j,s,t}$ is current period wealth (savings from last period), $w_t$ is the normalized wage, and $n_{j,s,t}$ is labor supply. Equations {eq}`eq_rK` and {eq}`eq_portfolio_return` of Chapter {ref}`Chap_FinInt` show how the rate of return from the financial intermediary $r_{p,t}$ might differ from the marginal product of capital $r_t$ and from the interest rate the government pays $r_{gov,t}$. Note that we must add in the cost of minimum consumption $c_{min,i}$ for all $i$ because that amount is subtracted out of composite consumption in {eq}`EqHHCompCons`.
 
   The last six terms on the right-hand-side of the budget constraint {eq}`EqHHBC` have to do with dfferent types of transfers to households: bequests $bq_{j,s,t}$, remittances $rm_{j,s,t}$, government transfers $tr_{j,s,t}$, universal basic income $ubi_{j,s,t}$, public pension benefits $pension_{j,s,t}$, and net taxes $tax_{j,s,t}$. The first four types of transfers to households are detailed in the section below entitled {ref}`SecHHtransfers`.
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   The third term $bq_{j,s,t}$ on the right-hand-side of the budget constraint {eq}`EqHHBC` represents the portion of total bequests $BQ_t$ that go to the age-$s$, income-group-$j$ household. Section {ref}`SecHHbequests` details how total bequests are distributed among the different households. The next term on the right-hand-side of the budget constraint {eq}`EqHHBC` represents remittances received by the household $rm_{j,s,t}$. We describe these remittances in Section {ref}`SecHHremit` below.
 
   The next term is government transfers $tr_{j,s,t}$ to households of lifetime income group $j$ in age $s$ at time $t$. Section {ref}`SecHHgovtransfers` details how total government transfers to households are distributed among the differenty household types. The term $ubi_{j,s,t}$ the time series of a matrix of universal basic income (UBI) transfers by lifetime income group $j$ and age group $s$ in each period $t$. There is a specification where the time series of this matrix is stationary (growth adjusted) and a specification in which it's stationary value is going to zero in the limit (non-growth-adjusted). The calibration chapter on UBI in the country-specific repository documentation describes the exact way in which this matrix is calibrated from the values of five parameters, household composition data, and OG-Core's demographics. Similar to the transfers term $TR_{t}$, the UBI transfers will not be distortionary.
 
   Government pension systems are modeled explicitly and enter the budget through the $pension_{j,s,t}$ term. The rules and parameters of the available public pension rules and parameters are in the {ref}`SecGovPensions` section of chapter {ref}`Chap_UnbalGBC`. The term $tax_{j,s,t}$ is the total tax liability of the household in terms of the numeraire good. This term can include means tested benefit programs. In contrast to government transfers $tr_{j,s,t}$, tax liability can be a function of labor income $(x_{j,s,t}\equiv w_t e_{j,s}n_{j,s,t})$ and capital income $(y_{j,s,t}\equiv r_{p,t} b_{j,s,t})$ and wealth, $b_{j,s,t}$. The tax liability can, therefore, be a distortionary influence on household decisions. It becomes valuable to represent total tax liability as the sum of an effective income tax rate $\tau^{etr,xy}_{s,t}$ function multiplied by total income plus an effective tax rate on wealth $\tau^{etr,w}_{s,t}$ multiplied by wealth,
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   ```{math}
   :label: EqTaxCalcLiabETR
-    T_{j,s,t} = \tau^{etr,xy}_{s,t}(x_{j,s,t}, y_{j,s,t})\left(x_{j,s,t} + y_{j,s,t}\right) + \tau^{etr,w}_{s,t}\left(b_{j,s,t}\right)b_{j,s,t} \quad\forall j,s,t
+    tax_{j,s,t} = \tau^{etr,xy}_{s,t}(x_{j,s,t}, y_{j,s,t})\left(x_{j,s,t} + y_{j,s,t}\right) + \tau^{etr,w}_{s,t}\left(b_{j,s,t}\right)b_{j,s,t} \quad\forall j,s,t
   ```
 
   where the effective income tax rate on labor and capital income can be a function of both labor income and capital income, respectively, $\tau^{etr,xy}_{s,t}(x_{j,s,t},y_{j,s,t})$, and the effective tax rate on wealth is given by the functional form described in Section {ref}`SecGovWealthTax`. 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.
@@ -295,8 +295,9 @@ 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}} + rm_{j,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
+    &\quad (1 + r_{p,t})b_{j,s,t} + w_t e_{j,s} n_{j,s,t} ...\\
+    &\qquad +\: bq_{j,s,t} + rm_{j,s,t} + tr_{j,s,t} + ubi_{j,s,t} + pension_{j,s,t} - tax_{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
   ```
 
   The nonnegativity constraint on consumption does not bind in equilibrium because of the Inada condition $\lim_{c\rightarrow 0}u_1(c,n,b') = \infty$, which implies consumption is always strictly positive in equilibrium $c_{j,s,t}>0$ for all $j$, $s$, and $t$. The warm glow bequest motive in Equation {eq}`EqHHPerUtil` also has an Inada condition for savings at zero, so $b_{j,s,t}>0$ for all $j$, $s$, and $t$. This is an implicit borrowing constraint.[^constraint_note] And note that the discount factor $\beta_j$ has a $j$ subscript for lifetime income group. We use heterogeneous discount factors following {cite}`CarrollEtAl:2017`. And finally, as discussed in Section {ref}`SecHHellipUtil`, the elliptical disutility of labor supply functional form in Equation {eq}`EqHHPerUtil` imposes Inada conditions on both the upper and lower bounds of labor supply such that labor supply is strictly interior in equilibrium $n_{j,s,t}\in(0,\tilde{l})$ for all $j$, $s$, and $t$.