# Universal Basic Income (UBI)#

[TODO: This section is far along but needs to be updated.]

We have included the modeling of a universal basic income (UBI) policy directly in the theory and code for [OG-Core] on which dependency the OG-USA is based. UBI shows up in the household budget constraint (17), and is described in the Budget Constraint section of the Households chapter of the OG-Core documentation. We calculate the time series of a UBI matrix $$ubi_{j,s,t}$$ representing the UBI transfer to every household with head of household age $$s$$, lifetime income group $$j$$, in period $$t$$. We calculate the time series of this matrix from five parameters and some household composition data that we impose upon the existing demographics of OG-USA.

## Calculating UBI#

We calculate the time series of UBI household transfers in model units $$ubi_{j,s,t)}$$ and the time series of total UBI expenditures in model units $$UBI_t$$ from five parameters described in the ogusa_default_parameters.json file (ubi_growthadj, ubi_nom_017, ubi_nom_1864, ubi_nom_65p, and ubi_nom_max) interfaced with the OG-USA demographic dynamics over lifetime income groups $$j$$ and ages $$s$$, and multiplied by household composition matrices from the Calibrate class of the OG-USA/ogusa/calibrate.py module in the repository.

From the OG-USA repository, we have four $$S\times J$$ matrices ubi_num_017_mat$$_{j,s}$$, ubi_num_1864_mat$$_{j,s}$$, and ubi_num_65p_mat$$_{j,s}$$ representing the number of children under age 0-17, number of adults ages 18-64, and the number of seniors age 65 and over, respectively, by lifetime ability group $$j$$ and age $$s$$ of head of household. Because our demographic age data match up well with head-of-household data from other datasets, we do not have to adjust the values in these matrices.[1]

Now we can solve for the dollar-valued (as opposed to model-unit-valued) UBI transfer to each household in the first period $$ubi^{\}_{j,s,t=0}$$ in the following way. Let the parameter ubi_nom_017 be the dollar value of the UBI transfer to each household per dependent child age 17 and under. Let the parameter ubi_nom_1864 be the dollar value of the UBI transfer to each household per adult between the ages of 18 and 64. Let ubi_nom_65p be the dollar value of UBI transfer to each household per senior 65 and over. And let ubi_nom_max be the maximum UBI benefit per household.

(29)#$\begin{split} \begin{split} ubi^{\}_{j,s,t=0} = \min\Bigl(&\texttt{ubi_nom_max}, \\ &\texttt{ubi_nom_017} * \texttt{ubi_num_017_mat}_{j,s} + \\ &\texttt{ubi_nom_1864} * \texttt{ubi_num_1864_mat}_{j,s} + \\ &\texttt{ubi_nom_65p} * \texttt{ubi_num_65p_mat}_{j,s}\Bigr) \quad\forall j,s \end{split}\end{split}$

The rest of the time periods of the household UBI transfer and the respective steady-states are determined by whether the UBI is growth adjusted or not as given in the ubi_growthadj Boolean parameter. The following two sections cover these two cases.

## UBI specification not adjusted for economic growth#

A non-growth adjusted UBI (ubi_growthadj = False) is one in which the initial nonstationary dollar-valued $$t=0$$ UBI matrix $$ubi^{\}_{j,s,t=0}$$ does not grow, while the economy’s long-run growth rate is $$g_y$$ for the most common parameterization where the long-run growth rate is positive $$g_y>0$$.

(30)#$ubi^{\}_{j,s,t} = ubi^{\}_{j,s,t=0} \quad\forall j,s,t$

As described in the OG-Core chapter on stationarization, the stationarized UBI transfer to each household $$\hat{ubi}_{j,s,t}$$ is the nonstationary transfer divided by the growth rate since the initial period. When the long-run economic growth rate is positive $$g_y>0$$ and the UBI specification is not growth-adjusted the steady-state stationary UBI household transfer is zero $$\overline{ubi}_{j,s}=0$$ for all lifetime income groups $$j$$ and ages $$s$$ as time periods $$t$$ go to infinity. However, to simplify, we assume in this case that the stationarized steady-state UBI transfer matrix to households is the stationarized value of that matrix in period $$T$$.

(31)#$\overline{ubi}_{j,s} = ubi_{j,s,t=T} \quad\forall j,s$

Note that in non-growth-adjusted case, if $$g_y<0$$, then the stationary value of $$\hat{ubi}_{j,s,t}$$ is going to infinity as $$t$$ goes to infinity. Therefore, a UBI specification must be growth adjusted for any assumed negative long run growth $$g_y<0$$.[2]

## UBI specification adjusted for economic growth#

Put description of growth-adjusted specification here.