3.1. Model economy

The model economy (which will serve as a laboratory for various quantitative analyses) extends the model developed by Aiyagari (Reference Aiyagari1994) to the endogenous labor supply.

Households: A continuum (measure one) of workers (households) with identical preferences exists and faces an idiosyncratic productivity shock $x$ , which evolves over time according to the Markov process with a transition probability distribution function $\pi _{x}(x'|x) = \mathrm{Pr}(x_{t+1} \leq x'|x_t = x)$ . When a worker with labor productivity $x_t$ chooses to work $h_t$ hours, its labor income is $w_t x_t h_t$ , where $w_t$ is the wage rate per efficiency unit of labor. The workers hold assets, $a_t$ , which yield the real rate of return, $r_t$ . The total (labor and capital) income, $\mathbf{y}_t = w_t x_t h_t + r_t a_t$ , is subject to a constant tax rate $\tau$ , and the workers receive $T(\mathbf{y}_t)$ , which is a transfer from the government. A household maximizes its lifetime utility given as follows:

\begin{equation*} \max _{\{c_t, h_t\}_{t = 0}^{\infty }} \mathbb {E}_0 \sum _{t=0}^{\infty } \beta ^t \left \{ \frac {{c_t }^{1 - \sigma } - 1 }{ 1 - \sigma } - B \frac {{h_t}^{1 + 1/ \gamma } } {1 + 1/ \gamma } \right \} \end{equation*}

subject to

\begin{equation*} \begin {aligned} c_t + a_{t+1} - a_t &= (1-\tau) ( w_t x_t h_t + r_t a_t) + T(\mathbf {y}_t), \\ a_{t+1} &\ge \underline {a}, \end {aligned} \end{equation*}

where $c_t$ represents consumption. Three parameters govern the preferences: $\sigma$ represents the relative risk aversion, $B$ denotes the relative weights on the disutility from working, and $\gamma$ denotes the (Frisch) labor supply elasticity.

Capital markets are incomplete in two senses. (i) Physical capital is the only available asset for households to insure against idiosyncratic productivity shocks. (ii) Households face an exogenous borrowing constraint $\underline{a}$ : $a_{t+1} \ge \underline{a}$ for all $t$ . The households differ ex post with respect to their productivity $x_t$ and asset holdings $a_t$ , whose cross-sectional joint distribution is characterized by the probability measure $\mu _t(a_t,x_t)$ .

Firms: The representative firm produces output according to a constant-returns-to-scale Cobb-Douglas production function in capital $K_t$ and effective units of labor $L_t = \int x_t h_t d\mu$ . Capital depreciates at rate $\delta$ in each period:

\begin{equation*} Y_t = F(L_t, K_t) = {L_t}^{\alpha }{K_t}^{1 - \alpha }. \end{equation*}

Government: The government operates a tax/transfer system. The government runs a balanced budget in each period:

\begin{equation*} \tau \int \big ( w_t x_t h(a_t, x_t) + r_t a_t \big) \, d\mu (a_t, x_t) = \int T(\mathbf {y}(a_t, x_t)) \, d\mu (a_t, x_t). \end{equation*}

Recursive representation: It would be useful to consider a recursive representation. Let $V(a,x)$ denote the value function of a household with asset holdings $a$ and productivity $x$ . Then, $V$ can be expressed as follows:

\begin{equation*} V(a, x) = \max _{c, h} \left \{\frac {c^{1-\sigma }-1}{1-\sigma } - B \frac {h^{1+1/\gamma }}{1+1/\gamma } + \beta \mathbb {E}\big [V(a', x')|x\big ] \right \} \end{equation*}

subject to

\begin{equation*} \begin {aligned} c + a' &= a + (1-\tau) ( w x h + r a) + T(\mathbf {y}(a,x)), \\ a' &\ge \underline {a}, \\ \mu '\left (a,x\right)&=\mathbf {T}\left (\mu \left (a,x\right)\right). \end {aligned} \end{equation*}

Equilibrium: A stationary equilibrium consists of a value function, $V(a,x)$ ; a set of decision rules for consumption, asset holdings, and the labor supply, that is, $c(a,x)$ , $a'(a,x)$ , and $h(a,x)$ , respectively; aggregate input $K$ and $L$ ; and the invariant distribution of the households $\mu (a,x)$ , such that:

1. 1. Individual households optimize: Given $w$ and $r$ , the individual decision rules $c(a,x)$ , $a'(a,x)$ , $h(a,x)$ , and $V(a,x)$ solve the Bellman equation.

2. 2. The representative firm maximizes its profits:

   \begin{align*} w & = \alpha (K/L)^{1 - \alpha } \\ r &= (1 - \alpha) (K/L)^{-\alpha } - \delta. \end{align*}

3. 3. The goods market clears:

   \begin{align*} \int \big \{ a'(a,x) + c(a,x) \big \} d\mu \left (a,x\right) = F(L,K) + (1 - \delta) K. \end{align*}

4. 4. The factor markets clear:

   \begin{align*} L & = \int x h(a,x) d\mu \left (a,x\right) \\ K & = \int a d\mu \left (a,x\right). \end{align*}

5. 5. The government balances the budget:

   \begin{align*} \tau \int \big ( w x h(a, x) + r a \big) \, d\mu (a, x) = \int T(\delta (a,x)) \, d\mu (a, x). \end{align*}

6. 6. Individual and aggregate behaviors are consistent: For all $A^{0}\subset {A}$ and $X^{0}\subset {X}$ ,

   \begin{align*} \mu (A^{0}, X^{0}) = \int _{A^0,X^0} \left \{ \int_{{A},{X}} \mathbf{1}_{a^{\prime} = a^{\prime}(a,x) } d\pi _x (x' |x) d\mu \right \} da' dx'. \end{align*}

3.2. Calibration

As the Aiyagari-type model is widely used in various macroeconomic analyses, we adopt the standard values for most of the parameters. The time unit is 1 year, the labor income share $\alpha$ is 0.64, and the annual depreciation rate of capital $\delta$ is 10%. The workers are not allowed to borrow, so $\underline{a}=0$ . As is common in the macroeconomic literature, we assume the relative risk aversion $\sigma$ to be one (i.e. the log utility in consumption) to be consistent with the balanced growth path.Footnote ²

We set the Frisch elasticity of the labor supply $\gamma$ to 0.5, close to the estimates in the literature [e.g. Chetty et al. (Reference Chetty, Guren, Manoli and Weber2011), Keane (Reference Keane2011)]. We assume that individual productivity $x$ follows an AR(1) process: $\ln{x'}=\rho \ln{x}+\varepsilon _x$ , where $\varepsilon _x \sim N(0,\sigma _x^2)$ . We assume that $\rho =0.91$ , which is a commonly used value. The chosen value for $\sigma _x$ is 0.21, which is an estimate provided by Floden and Lindé (Reference Floden and Lindé2001) based on the Panel Study of Income Dynamics (PSID). In the benchmark economy, there are no taxes and transfers. Finally, we set the time discount factor $\beta$ so that the real interest rate is 4%, which is the average real rate of return to capital in the United States in the post-World War II period. We also choose the disutility from working $B$ to generate the average number of hours worked of 0.33 in the steady state. Table 1 summarizes the parameter values of the benchmark model economy.

Table 1. Parameters of the benchmark economy

Table 2 reports the income and wealth distributions of the benchmark economy. The Gini coefficients of income and wealth are 0.33 and 0.64, respectively. Although the model exhibits significant dispersion in income and wealth, they are considerably smaller than those in the data (e.g. 0.53 and 0.76 according to the PSID and 0.63 and 0.78 according to the Survey of Consumer Finances). This outcome is inevitable given that the model is driven by a single source of heterogeneity (i.e. an idiosyncratic productivity shock). It is well known [see, Diaz-Gimenez et al. (Reference Diaz-Gimenez, Quadrini and Ríos-Rull1997)] that this type of model cannot account for the concentration found in the upper 1% of the income or wealth distribution. However, accounting for the likes of Bill Gates is not our primary concern so we do not focus on the extreme upper part of the wealth distribution. We compare the UBI and NIT in a simple heterogeneous agent model. In the United States, the median and mean household incomes (as of 2018) are $\$ $ 61,937, and $\$ $ 87,864, respectively. In our benchmark economy, the median and mean incomes are 0.346 and 0.435, respectively. Thus, the mean–median income ratio in the model (1.26) is not far from that in the data (1.42).

Table 2. Distributions of income and wealth

3.3. Comparison between UBI and NIT

In this subsection, we consider the UBI and NIT programs that are similar to Mankiw (Reference Mankiw2021) where the income tax rate and phase-out rate are 20%: $\tau = \hat{\tau } = \phi =0.2$ . We also assume that the households with zero income receive 20% of the pivotal income $T_0 = 0.2 \times \hat{y}$ in the NIT. To guarantee the balanced budget, we set the pivotal income in the NIT to the average income $\hat{y} = \bar{y}$ . Table 3 reports the steady states of the model economy.

Table 3. Steady states

Column (1) is the benchmark economy where no tax and subsidy exist. Columns (2) and (3) are the steady states under the Mankiw-like program described above. By construction, the UBI and NIT provide the same incentives to the households, resulting in an identical allocation. The aggregate output decreases by 13% (from 0.585 to 0.511) in the long run, capital and labor decrease by 22% and 9%, respectively. Two forces can explain the decrease of factor inputs. First, the introduction of a welfare program reduces the motive for precautionary savings and the labor supply. Second, the imposition of an income tax reduces the incentive to work and save. As capital decreases more than labor, the capital-labor ratio falls, and the equilibrium interest rate rises to 5.48% (and the wage rate falls by 6%), which slightly increases the Gini coefficients of wealth (from 0.640 to 0.657) and before-tax income (from 0.324 to 0.341). Owing to the redistribution policy, the after-tax income Gini drops to 0.273. As the aggregate output decreases significantly, the average income also falls. The UBI program requires 15.4% of the GDP, whereas the tax-to-GDP ratio under the NIT is only 3.8%. Overall, the average welfare slightly decreases (−0.56% in consumption-equivalence units).Footnote ³ In this particular case (of 20% effective marginal tax rate on income), the cost of distortion from taxes outweigh the insurance benefit of the subsidy.

3.4. Optimal UBI and NIT

In the above analysis, we assume that the tax rate is fixed at $\tau =0.2$ and the transfer amount for the households with zero income ( $\bar{T}$ or $T_0$ ) is 20% of the average income. We drop these assumptions and look for an optimal UBI amount $\bar{T}$ (still under a linear income tax schedule) along with a tax rate that balances the government budget.

Figure 1 plots the average welfare gains (in consumption-equivalence units) for a range of tax rates: $\tau \in [0.02, \, 0.2]$ . We compute the welfare gains relative to the benchmark economy with no welfare program ( $\tau =0$ ). The figure clearly illustrates the trade-off (between the insurance benefit and efficiency costs) in the UBI programs. As the size of the UBI increases, the inefficiency from high taxes starts to dominate the insurance benefit. According to our model, the highest social welfare (0.41% in consumption-equivalence units) is achieved at a tax rate of 8% ( $\tau =0.08$ ). Under this tax rate, everyone in this economy receives 7.9% ( $=\frac{0.033}{0.419}$ ) of the economy-wide average income as the basic income. The detailed statistics of the steady state under this optimal UBI tax rate $\tau =0.08$ are reported in column (4) of Table 3.

Figure 1. Welfare gains of UBI under various tax rates.

In Section 3.3 above, we assume that the pivotal income in the NIT is the same as the economy-wide average income ( $\hat{y} = \bar{y}$ ), and the phase-out rate is identical to the tax rate ( $\phi = \tau$ ). While this assumption easily guarantees a balanced budget, in practice, the government has a more leeway to choose from various combinations of these parameters. Now we allow the government can choose one of the two parameters ( $\hat{y}$ or $\phi$ ), and the other parameter is set to maintain the balanced budget of the NIT program. For a direct comparison with the optimal UBI of $\tau =0.08$ (which we just derived), we assume that the tax rate above the pivotal income remains at 8%.

The first panel of Figure 2 shows the combinations of pivotal income $\hat{y}$ (ranging from 0.24 to 0.56 in $x$ -axis) and the phase-out rate $\phi$ (decreasing from 49% to 2.4% in $y$ -axis) which achieve the balanced budget of the NIT. Note that the income tax rate above the pivotal income remains the same at $\tau =0.08$ . Since the phase-out rate is not necessarily the same as the income tax rate, the effective marginal income tax schedule is potentially piece-wise linear. The second and third panels show the subsidy amount for a zero-income household ( $T_0$ ) and tax-to-GDP ratio, respectively.

Figure 2. Welfare gains of NIT under various pivotal income.

Looking at the last panel, we obtain the highest welfare (CEV of 0.47%) when the pivotal income is 0.308—marked by red vertical line—which is the 45th percentile of the income distribution. (We also mark the NIT program that replicates the optimal UBI (of $\tau =\phi =0.08$ ) by the dotted vertical line in the figure for comparison). The phase-out rate under the optimal NIT (that achieves the balanced budget) is 0.249. Thus, the phase-out rate is much steeper than the income tax rate of 8% above the pivotal income. This optimal NIT features a subsidy that is generous to those with very little income and phases out quickly, providing a strong insurance to the very poor. Column (5) in Table 3 reports the detailed statistics of the steady state of this optimal NIT.

The households with zero income receive more than twice the subsidy than under the optimal UBI: $T_0 = 0.08$ in Column (5) versus 0.033 in Column (4) of Table 3. This results in a lower aggregate capital and labor in the long run (via a weaker motive for precautionary savings). While the aggregate hours fall by 7%, the efficiency units of labor decrease by 4.4% only because the decrease in the labor supply stems largely from the low-productivity workers. As the government collects taxes only from those above the 45th percentile, the tax-to-GDP ratio is a mere 2.3% in Column (5) (compared to 6% under the optimal UBI in Column (4)). Thus, the NIT can perform nearly as well as (actually slightly better than) the UBI with a much smaller budget.

3.5. Transition dynamics

So far, we have focused on the steady states only. In this subsection, we consider the transition dynamics between the steady states. Suppose that the economy is in the steady state without any welfare program (i.e. Column (1) of Table 3) in period 0. We assume that a redistribution program is introduced in year 1. During the transition to a new steady state, we assume that the government chooses the tax rates to achieve a period-by-period budget balance. For example, the government introduces a fixed-amount UBI and chooses a sequence of income tax rates ( $\tau _t$ ’s) to achieve a balanced budget each year. In the case of the NIT, the government announces a fixed phase-out rate and pivotal income in period 1 (along with a sequence of tax rates in the future to balance the budget in each period until the economy reaches the new steady state).

Figure 3 plots the transition dynamics of the economy for 40 years for three cases: Columns (2), (4), and (5) of Table 3 which we label as Models (2), (4), and (5), respectively. In all three scenarios, the model economy (nearly) converge to new steady states in 30 years. The half-life of convergence is about 10 years. In all three cases, there are periods of overshooting (i.e. temporary increase in wages and decrease in interest rates) because the economy’s current capital stock will be much higher than its new steady state when the government just introduced the welfare program.

Figure 3. Transition dynamics.

3.6. Political support for redistribution program

Finally, we compute the approval rate for each welfare program. Figure 4 illustrates the winners (in blue) and losers (in red) in each of the three scenarios: Columns (2), (4), and (5) of Table 3. The horizontal axis represents the asset holdings, and the vertical axis denotes the average labor income ( $wx\bar{h}$ ). For illustrative purposes, we convert the model units into dollars so the average income in the benchmark model coincides with the per capita GDP in 2018 ( $\$ $ 62,805).Footnote ⁴ All three policies easily gain the majority, which is not surprising, because the benchmark economy ( $\tau =0$ ) does not provide any social insurance at all. In Model (2), 70.7% of the workers are in favor of introducing a redistribution policy (either the UBI or NIT) with an income tax rate of 20%. The approval rate increases to nearly 81.9% for an optimal UBI (Model (4)). The optimal NIT obtains 74.4% (Model (5)). Compared with the UBI, the approval rate falls among the lower-middle asset and income households (where the population density is high given the skewed income and asset distributions), because the subsidy phases out quickly under this NIT.

Figure 4. Political support for introducing redistribution program.

In general, the workers with low productivity and low assets are in favor of introducing a redistribution policy. According to Model (2), among the households with asset holdings below $\$ $ 853,000, those with annual labor income less than $\$ $ 50,000 are better off from the introduction of a redistribution program whereas those with the labor income higher than $\$ $ 50,000 are against the UBI. As the asset holdings increase, the fraction of workers in favor of the redistribution policy diminishes. When the asset holdings reach $\$ $ 3 million, no one is in favor of an introduction of the UBI financed by a 20% income tax rate. However, in Models (4) and (5), quite a few rich households are in favor of a redistribution program, owing to so-called pecuniary externalities [e.g. Dávila et al. (Reference Dávila, Hong, Krusell and Ríos-Rull2012)]. On the one hand, redistribution is bad for wealthy households. On the other hand, redistribution raises the returns to capital in equilibrium because of the decreased supply of the aggregate capital (owing to a weak motive for precautionary savings).