Silviase/jgov_v1.4_full_ann huggingface dataset:
that job losses in routine occupations are responsible for jobless recoveries. These factors have been compounded by an erosion in implicit life-time employment guarantees and an increase in the fraction of workers with temporary jobs.

The final data observation that we wish to point to is rising wealth inequality. Piketty (2014) provides evidence that wealth inequality in advanced economies has risen sharply and suggests that taxes on capital should be increased. Ohtake et al. (2013) confirm this result for Japan. According to their analysis, wealth inequality increased between 1984 and 1989, decreased between 1989 and 1994 and increased again between 1994 and 2004. Lise et al. (2013) find a small but steady decline in the share of wealth held by the bottom 10 percent from approximately 0 in 1984 to -1 percent in 2004. The share of wealth held by the top 10 percent, in contrast, rises from 40 to 45 percent over the same period.

Our paper has two objectives. First, we use a model to show that economic stagnation, increasing wealth inequality, and wage polarization could all be related. The research described above suggests that rising earnings inequality is concentrated in medium-skilled routine occupations. Our model reproduces this observation by endowing households with unskilled labor and skilled labor. We assume that the latter form of labor is special because it can be augmented via investment but is subject to household specific non-diversifiable risk. An increase in the variance of this risk produces earnings polarization.

Earnings polarization depresses output because households are risk averse. When the risk of investing in human capital increases, they choose to invest more of their savings in physical capital and less in human capital. The resulting misallocation of investment acts to depress aggregate output and consumption.

In our model, an increase in uninsured earnings risk also increases wealth inequality. Households of all income levels choose to save the same fixed proportion of their earnings. Thus, an increase in the risk of investing in human capital not only produces higher earnings inequality and but also higher wealth inequality. Taken together, these results imply that an increase in earnings polarization can result in both economic stagnation and higher wealth inequality.

Our second objective is to analyze how fiscal policy can ameliorate the negative effects of higher earnings risk on economic activity, inequality and welfare. We compare policies that stimulate output with policies that provide more social insurance to the poor. One way to deal with wage polarization is to reduce tax rates. Lower
tax rates stimulate economic activity. However, if other forms of government spending are held fixed, as we assume, then government transfers to the poor will also decline. An alternative way to deal with wage polarization is for the government to help poor households cope with higher earnings risk by increasing public transfers to the poor. Higher public transfers, however, require additional tax revenue and increasing tax rates depress economic activity. Using a version of our model that is calibrated to Japan, we find that households prefer lower tax rates and in particular a lower capital tax rate even though this means that households receive less social insurance.

The fact that households prefer lower capital taxes is noteworthy. Carey and Tchilinguirian (2000) update and revise the effective tax rate data of Mendoza, Razin and Tesar (1994) find that Japan has some of the highest overall tax rates on capital income among advanced economies $ ^{1} $ . Based largely on concerns from businesses, the Abe administration has recently legislated reductions in the overall tax rate on capital by reducing corporate profits taxes. Corporate profits taxes are gradually being reduced from 37 percent in 2013 to 29.74 percent in 2018. Although reducing the tax rate on capital, improves the incentives for firms to undertake investment, this policy runs counter to the advice of Piketty (2014) who argues that the capital tax rate should be increased to improve equity.

This leads us to consider what if any tradeoffs arise between efficiency and equity in our model. We find that lowering the overall tax rate on capital in Japan from its current level of about 63 percent, to a level as low as 45 percent increases welfare and reduces wealth inequality.

A lower tax rate on capital increases the ability of agents to self-insure against low earnings because the after tax return on their savings increases. Poverty among the elderly declines, and thereby reducing overall wealth inequality.

The nature of the tradeoff between equity and efficiency depends on the specific instrument. For instance, if the consumption tax rate is reduced, instead wealth inequality increases. We find that the most effective ways to deal with the higher earnings risk faced by Japanese workers are to reduce tax rates on labor and/or capital. Both policies stimulate economic activity, reduce wealth inequality, and increase welfare $ ^{2} $ .
Our model builds on previous research by Krebs (2003), Angeletos and Panousi (2009), Toda (2014), Gottardi, Kajii, and Nakajima (2015), and Nirei and Aoki (2016). To capture job polarization we assume that households are endowed with two types of labor: skilled and unskilled. Unskilled labor is not risky but earns a low wage rate. Skilled labor can be enhanced over time by investing in human capital; however, these investments are risky and subject to idiosyncratic uninsurable risk. Households can self-insure against their risky investments in human capital by acquiring physical capital. The economy faces no aggregate risk. However, the amount of resources that households are willing to allocate to investing in risky human capital varies with the extent of uninsured risk. If the level of uninsured risk increases, households respond by reducing their investment in human capital and increasing their investment in physical capital; this has a depressing effect on output and consumption. To analyze the effects of alternative fiscal policies, we model proportionate taxes on labor income (skilled and unskilled), capital income, and consumption. The government uses this income to finance government purchases, fund public transfer programs, and to pay interest on government debt.

The remainder of the paper is organized as follows. Section 2 describes the model. Section 3 contains our main results, and Section 4 presents our concluding remarks.

### 2. The model economy

## Households

We consider a continuum of households of measure one. Each household dies with a constant probability; as in research by Blanchard (1985), we assume that actuarially fair life insurance is available for everyone. At each instant of time, new households are born so that the population size of each type is constant (unity). There are no aggregate shocks.

At every point in time, each household supplies two types of labor: skilled and unskilled. The amount of unskilled labor is constant and denoted by L>0. The amount of skilled labor (in efficiency units) changes over time, which we call “human capital.” The accumulation of human capital is a risky activity, and subject to idiosyncratic, uninsurable risks.

Households may hold two types of assets: a risk-free (financial) asset, b, and human capital, h. A new born household is endowed with $ h_{0}>0 $ units of human capital but no
financial assets. Consider a household that is born at time s (call it household $ (i, s) $ ). Her lifetime expected utility is given by

$$ E_{s}\int\limits_{s}^{\infty}e^{-(\rho+p)(t-s)}\ln c(i,s,t)d t, $$

where $ \rho>0 $ is the time discount rate; p>0 is the death rate; and $ c(i,s,t) $ is the amount of consumption at time t of household $ (i,s) $ .

Let $ b(i, s, t) $ and $ h(i, s, t) $ be the amount of the risk-free asset and human capital that household $ (i, s) $ holds at time t respectively. Each household is subject to different forms of taxes. Let $ \tau_{c}(t) $ denote the consumption tax at time t, $ \tau_{r}(t) $ the tax rate on interest income, and $ \tau_{w}(t) $ the tax rate on labor income (common across skilled and unskilled labor). Furthermore, there are also lump sum public transfers $ \tau(t) $ , which are identical across (living) households.

Define $ r(t) $ to be the before-tax interest rate, $ w_{h}(t) $ the before-tax wage rate for skilled labor; and $ w_{l}(t) $ the before-tax wage rate of unskilled labor. The after-tax rates of these variables are then

$$ \begin{aligned}\hat{r}(t)\quad&\equiv\quad r(t)(1-\tau_{r}(t)),\\\hat{w}_{h}(t)\quad&\equiv\quad w_{h}(t)(1-\tau_{w}(t)),\\\hat{w}_{l}(t)\quad&\equiv\quad w_{l}(t)(1-\tau_{w}(t)).\end{aligned} $$

The flow budget constraint at time t is expressed as

$$ \begin{align*}(3)\quad&(1+\tau_{c}(t))c(i,s,t)dt+dh(i,s,t)+db(i,s,t)\\=&\big[(\hat{r}(t)+p)b(i,s,t)+\hat{w}_{l}(t)L+\tau(t)+(\hat{w}_{h}(t)-\delta_{h})h(i,s,t)\big]dt+\sigma h(i,s,t)dz(i,s,t),\end{align*} $$

where $ \delta_{h}>0 $ is the (constant) depreciation rate of human capital. The riskiness of investments in human capital is reflected in the stochastic term $ \sigma h(i,s,t)dz(i,s,t) $ , where $ z(i,s,t) $ is the standard Brownian motion and $ \sigma>0 $ is the constant that measures the amount of the risk. The risk is idiosyncratic so that $ z(i,s,t) $ is independent across households.

Define $ T(t) $ by

(4)

$$ T(t)\equiv\int\limits_{t}^{\infty}\exp\Bigl\{-\int\limits_{t}^{s}\bigl[\hat{r}(u)+p\bigr]d u\Bigr\}\bigl[\hat{w}_{l}(s)L+\tau(s)\bigr]d s, $$

so that

(5)

$$ T^{\prime}(t)=-\big[\widehat{w}_{l}(t)L+\tau(t)\big]+\big[\widehat{r}(t)+p\big]T(t). $$

Now let $ a(i, s, t) $ be the total wealth held by household $ (i, s) $ at time t: (6)

$$ a(i,s,t)\equiv b(i,s,t)+T(t)+h(i,s,t), $$
and let $ \eta(i,s,t) $ be the share of human capital:

$$ \eta(i,s,t)\equiv\frac{h(i,s,t)}{a(i,s,t)}. $$

Then the flow budget constraint becomes

$$ \begin{align*}(1+\tau_{c}(t))c(i,s,t)dt+da(i,s,t)\\=&\left[(\hat{r}(t)+p)(1-\eta(i,s,t))+(\hat{w}_{h}(t)-\delta_{h})\eta(i,s,t)\right]a(i,s,t)dt\\&+\sigma\eta(i,s,t)a(i,s,t)dz(i,s,t).\end{align*} $$

Note that the amount of “initial wealth” held by a newborn household at time s is $ a(i, s, s) = a_{0} \equiv h_{0} + T(s) $ .

Since there are no aggregate stochastic shocks in the economy, the real interest rate $ \tilde{r}(t) $ and the wage rate $ \widehat{w}_{h}(t) $ are deterministic processes that depend only on time t. Thus, the value function for each household is expressed as a function $ J(a,t) $ , which depends on the value of total wealth a and time t. Let $ \eta_{c}=(1+\tau_{c})c/a $ . The Bellman equation is

$$ \begin{align*}0=&\max_{\eta c,\eta}\ln\Big(\frac{\eta_{c} a}{1+\tau_{c}(t)}\Big)-(\rho+p)J(a,t)+J_{t}(a,t)\\&+J_{a}(a,t)\big[(\hat{r}(t)+p)(1-\eta)+(\hat{w}_{h}(t)-\delta_{h})\eta-\eta_{c}\big]a+\frac{1}{2}J_{aa}(a,t)\sigma^{2}\eta^{2}a^{2}.\end{align*} $$

Let $ A(s, t) $ be the average amount of wealth held at time t by households born at time s. Its evolution is given as

$$ \frac{d A(s,t)}{d t}=\left[(\hat{r}(t)+p)(1-\upeta(t))+(\hat{w}_{h}(t)-\updelta_{h})\upeta(t)-(\uprho+p)\right]A(s,t). $$

Let $ A(t) $ be the aggregate wealth for the economy at time t. Since each household dies at rate p, aggregate wealth is given by

$$ A(t){=}\intop_{-\infty}A(s,t)p e^{p(s{-}t)}d s. $$

Firms

We assume perfectly competitive firms produce a single good with a constant-returns-to-scale technology:

$$ F(k,h,l)=z k^{\alpha}h^{\beta}l^{1-\alpha-\beta}, $$

where z is a constant, k is the input of physical capital, h is the input of skilled labor (human capital); l is the input of unskilled labor; and $ \alpha $ , $ \beta $ are constant share parameters with $ \alpha $ , $ \beta > 0 $ and $ \alpha + \beta < 1 $ . The aggregate output at time t is

$$ Y(t)=z K(t)^{\alpha}H(t)^{\beta}L^{1-\alpha-\beta}, $$
where $ K(t) $ and $ H(t) $ are aggregate stocks of physical and human capital, respectively. The factor market equilibrium requires

$$ \begin{align*}\hat{r}(t)\quad&=\quad\alpha z K(t)^{\alpha-1}H(t)^{\beta}L^{1-\alpha-\beta}-\delta_{k},\\\hat{w}_{h}(t)\quad&=\quad\beta z K(t)^{\alpha}H(t)^{\beta-1}L^{1-\alpha-\beta},\\\hat{w}_{l}(t)\quad&=\quad(1-\alpha-\beta)z K(t)^{\alpha}H(t)^{\beta}L^{-\alpha-\beta}.\end{align*} $$

## Government

The government issues debt $ D(t) $ , consumes goods $ G(t) $ , makes lump sum public transfers $ \tau(t) $ , and imposes taxes on consumption, labor, and capital income at the rates $ \tau_{c}(t) $ , $ \tau_{w}(t) $ , and $ \tau_{r}(t) $ , respectively. It follows that the government's flow budget constraint is

$$ \begin{align*}\dot{D}(t)+\tau_{r}(t)r(t)\big[K(t)+D(t)\big]+\tau_{w}(t)w_{h}(t)H(t)+\tau_{w}(t)w_{l}(t)L+\tau_{c}(t)C(t)\\=G(t)+r(t)D(t)+\tau(t).\end{align*} $$

where $ C(t) $ is the aggregate consumption of households.

## Equilibrium

Having completed the description of the economy, we can now define a competitive equilibrium.

## Definition 1 : Competitive Equilibrium

A competitive equilibrium consists of an allocation $ \{c(i,s,t),a(i,s,t),\eta(i,s,t)\} $ , a price system $ \{r(t),w_{h}(t),w_{l}(t),\tilde{r}(t),\tilde{w}_{h}(t),\tilde{w}_{l}(t)\} $ , and a fiscal policy $ \{G(t),D(t),\tau(t),\tau_{c}(t),\tau_{w}(t),\tau_{r}(t)\} $ such that (i) given the price system and fiscal policy, the allocation $ \{c(i,s,t),a(i,s,t),\eta(i,s,t)\} $ solves the utility maximization problem for each $ (i,s) $ ; (ii) firms maximize profits, that is, (14) holds for all t; (iii) the fiscal policy is feasible, that is, it satisfies the government's flow budget constraint (15) as well as the boundary condition: for each t,

$$ \lim_{u\to\infty}\exp\Bigl(-\int_{t}^{u}\hat{r}(s)ds\Bigr)D(u)=0\;; $$

and (iv) all markets clear.

In the analysis that follows, we will consider a particular type of competitive equilibrium in which government policy is constant.

## Definition 2 : Steady State Competitive Equilibrium
A steady state equilibrium is a competitive equilibrium in which (i) the fiscal policy is constant: $ G(t)=G $ , $ D(t)=D $ , $ \tau(t)=\tau $ , $ \tau_{c}(t)=\tau_{c} $ , $ \tau_{w}(t)=\tau_{w} $ and $ \tau_{r}(t)=\tau_{r} $ for all t; and (ii) the aggregate variables are constant: $ C(t)=C $ , $ \eta(t)=\eta $ , $ H(t)=H $ , $ K(t)=K $ , $ A(t)=A $ , $ Y(t)=Y $ for all $ t^{3} $ .

## Welfare

We will evaluate welfare from the perspective of a newborn household. In the steady state, the value function of an individual with wealth a is given by

$$ J(a){=}\frac{1}{\rho{+}p}\ln a{+}\psi, $$

where

$$ \psi{=}\frac{1}{\uprho+p}\left\{\ln{\left(\frac{\uprho+p}{1+\uptau_{c}}\right)}{-}1\right\}{+}\frac{1}{2(\uprho+p)^{2}}\left\{\left(\frac{\hat{w}_{h}{-}\updelta_{h}{-}\hat{r}{-}p}{\sigma}\right)^{2}{+}2(\hat{r}{+}p)\right\}. $$

Thus, the welfare of a new born household in the steady state, $ \bar{U} $ , is

$$ \begin{align*}\bar{U}=&\frac{1}{\rho+p}\ln(h_{0}+T)+\psi\\=&\frac{1}{\rho+p}\left\{\ln\Bigl(h_{0}+\frac{\tau+\hat{w}_{l} L}{\hat{r}+p}\Bigr)+\ln\Bigl(\frac{\rho+p}{1+\tau_{c}}\Bigr)-1\right\}\\&+\frac{1}{2(\rho+p)^{2}}\Bigl\{\Bigl(\frac{\hat{w}_{h}-\delta_{h}-\hat{r}-p}{\sigma}\Bigr)^{2}+2(\hat{r}+p)\Bigr\}.\end{align*} $$

## Wealth Distribution

We are interested in how the distribution of wealth is affected by increases in uninsured earnings risk as well as fiscal policy. The distribution of wealth, $ a(i, s, t) $ , can be derived using a result by Toda (2014). Using Ito's lemma, the evolution of $ \ln a(i, s, t) $ is expressed as

(19)

$$ d\ln a(i,s,t)=\upmu_{a}dt+\sigma_{a}dz(i,s,t), $$

where

$$ \begin{array}{r l r}{\upmu_{a}}&{\equiv}&{(\hat{r}+p)(l-\upeta)+(\hat{w}_{h}-\sigma_{h})\upeta-(\uprho+p)-\frac{1}{2}\sigma^{2}\upeta^{2},}\end{array} $$

$$ \sigma_{a}\ \equiv\ \sigma\eta. $$

Thus, the steady state log wealth distribution $ x=\ln a-\ln a_{0} $ has the density function given by $ (a_{0}\equiv1+T) $
$$ f(x){=}\frac{p}{\kappa\sigma_{a}}\exp\Bigl(-\frac{\kappa|x|}{\sigma_{a}}{+}\frac{\upmu_{a}x}{\sigma_{a}^{2}}\Bigr), $$

where

$$ \kappa\equiv\sqrt{2p+\frac{\mu_{a}^{2}}{\sigma_{a}^{2}}}. $$

Its mean and variance are given as follows:

$$ \begin{aligned}mean\quad&=\quad\frac{\mu_{a}}{p},\\ variance\quad&=\quad\left(\frac{\sigma_{a}^{2}}{\kappa\sigma_{a}-\mu_{a}}\right)^{2}+\left(\frac{\sigma_{a}^{2}}{\kappa\sigma_{a}+\mu_{a}}\right)^{2}.\end{aligned} $$

### 3. Results

Having described the model and derived its steady state equilibrium, we are now in a position to apply the model to investigate the role of uninsured earnings risk in accounting for economic stagnation. Recall from the introduction that recent empirical work in Japan and the U.S. has found that the earnings distribution of medium- and highly skilled occupations is diverging. Wages in medium-skilled routine occupations are declining, whereas wages in higher-skilled nonroutine occupations are increasing. Low-skilled jobs show no evidence of divergence. Wage polarization has been accompanied by a secular increase in wealth inequality. Various economic mechanisms can account for these phenomena in isolation, but we believe that we are the first to provide a comprehensive theory that can account both for this micro-evidence of high earnings and wealth inequality and link it to the macro-economic evidence of stagnation.

## Parameterization of the model

Although our model has a rich set of implications, this comes at a cost. We are unable to compute closed-form solutions for the steady state. Instead, we must resort to numerical methods. This, in turn, requires us to assign specific numerical values to each model parameter. The specific values of the parameters we use are reported in Table 1. We briefly summarize the rationale for our choices. The model period is one year, and thus, we thus set $ \rho $ the preference discount rate to $ \rho $ at 0.03 per year. This value is close to values used in the real business cycle literature. Hayashi and Prescott
Table 1: Model Parameter Values

A. Parameters that Govern Preferences and Technology

Parameter	$ \rho $	$ p $	$ \alpha $	$ \beta $	$ \delta,\delta_{h} $	$ \sigma $
Value	0.03	0.01195	0.3	0.45	0.085	0.246
B. Fiscal Policy Parameters
Parameter	G/GDP	Debt/GDP	$ \tau_{r} $	$ \tau_{w} $	$ \tau_{c} $
Value	0.21	1.5	0.63	0.32	0.08

(2002), for instance, use a value of 0.036 per year. Our choice is also close to values used in the New Keynesian literature. Christiano, Eichenbaum, and Evans (2005), for instance, also assume a value of 0.03 per year.

The World Health Organization reports that life expectancy in Japan was 83.7 years in 2013. This implies that p=0.01195. Estimates of $ \alpha $ , the capital share parameter in the production function, range from 0.25 to approximately 0.4. Our value of 0.3 lies in the middle of this interval. The human capital share $ \beta $ is set at 0.45. Thus, the overall labor share in the production function is 0.7. The depreciation rate for physical and human capital is set at 0.085. This value is close to the value of 0.089 that is reported by Hayashi and Prescott (2002). We assume that the relative endowments of initial human capital and unskilled labor are four to one and the initial amount of uninsured earnings risk in human capital accumulation $ \sigma $ is 0.246, which is the 1991 value of household earnings dispersion in Japan reported by Lise et al. (2014).

The fiscal policy parameters are chosen in the following way. The overall tax rate on capital $ \tau_{r} $ , is set at 0.63. This value lies between the value of 0.836 estimated by Caray and Tchilinguirian (2000) and the value of 0.544 estimated by Braun and Joines (2015). The overall labor tax rate $ \tau_{w} $ including SS and medical contributions is set at 0.32. This is the OECD estimate of the overall labor wedge in Japan for the year 2014. The consumption tax rate $ \tau_{c} $ , is set at its 2015 statutory value of 0.08. The government purchase share of output is set at 0.21. This is its value in the calendar year of 2014. Finally we set the debt-to-GDP ratio to 1.5. This value is an estimate of Japan's net debt-to-GDP ratio for 2012 constructed by Braun and Joines (2015).

## Simulation Results

We now analyze the effects of an increase in uninsured earnings on output, public transfers, utility and wealth inequality. The scenario we consider is based on results reported by Lise et al. (2014). They find that the standard deviation of log earnings
among Japanese households gradually increased from 0.246 in 1991 to 0.284 in 2008. If we project this trend, it implies a standard deviation of log earnings of 0.30 in 2015 or a total increase in earnings inequality of 5.4 log points.

Table 2 reports the effects of increasing the standard deviation of earnings from 0.246 to 0.3 in our model. An increase in earnings inequality of this magnitude depresses output by 2.52 percent $ ^{4} $ . This decline in output is due to a change in household savings plans. Accumulating human capital is now riskier, and households allocate a larger fraction of their savings to physical capital. This savings response allows the model to reproduce the increase in capital intensity that has been documented in research by Hayashi and Prescott (2002). The physical capital-output ratio in the model increases from 2.61 to 2.65.

Lower output reduces government tax revenue. Under our assumption, the resulting gap in the government budget constraint is remedied by reducing government transfers. Public transfers are a particularly valuable form of insurance to households who have low earnings and low wealth. It is thus not surprising to see that utility declines. Households are now exposed to more risk and have less government provided insurance.

Finally, notice that financial wealth inequality increases by 5 log points. The capital-output ratio has risen. In our model, all households save the same proportion of their earnings. Thus higher earnings inequality translates into higher wealth inequality.

Higher earnings inequality creates a potential conundrum for policy makers. Should the policy maker take actions to boost output, or alternatively increase public transfers

Table 2: Effects of an Increase in Uninsured Earnings Risk on Output, Public Transfers, Utility, and Wealth Inequality

$ \Delta \ln Y $	$ \Delta \ln \tau $	$ \Delta U $	$ \Delta \sigma_{a} $
High Earnings Risk Scenario	-0.0252	-0.0333	-0.2398	0.0501

$ ^{*} $ All values are relative to their baseline value and “ln” refers to the natural logarithm.

A negative value indicates that the magnitude of the variable is smaller in the baseline scenario as compared with the high earnings risk-scenario with $ \sigma=0.30 $ .