Silviase/jgov_v1.4_full_ann huggingface dataset:
liquidity in the interbank market, followed by a contraction of economic activity. The fiscal default forces the central bank to finance the deficit through money creation, which will trigger inflation and further harms the economy. The decline in bond prices also affects the balance sheet of banks holding government bonds, and forces those banks to improve their balances at the sacrifice of their lending to the private sector.

We now turn to the behavior of the government. Given the outstanding debt $ d_{t} $ , the government budget constraint is stasndard :

$$ d_{t+1}-d_{t}=R_{t+1}d_{t}-s_{t+1}, $$

where $ s_{t} $ is the primary surplus (tax revenue minus government expenditure). The government is committed to follow a fiscal reaction function :

$$ s_{t+1}=\upmu+f(d_{t})+\upvarepsilon_{t+1}, $$

where the parameter $ \mu $ captures the systematic determinants of the primary surplus balance other than the past debt, and $ f(\cdot) $ is the response of the primary surplus to the debt, which is a continuously differentiable and nondecreasing function. The stochastic variable $ \varepsilon_{t+1} $ follows the distribution function $ G(\varepsilon_{t+1}) $ , which is continuously differentiable with a positive density defined over the interval $ [\varepsilon^{min}, \varepsilon^{max}] $ . The stochastic primary surplus reflects uncertainty over the tax revenues and/or unexpected government expenditure. The government levies tax $ \tau_{t}W $ on the young.

We obtain the debt dynamics from (2) and (3) by

$$ \upmu+f\left(d_{t}\right)+\upvarepsilon_{t+1}=\left(1+R_{t+1}\right)d_{t}-d_{t+1}, $$

where the left-hand side (LHS) is the fiscal surplus and the right-hand side (RHS) is the “net” interest payment. To capture the idea of a fiscal crisis, the function has the property that there exists a debt $ d^{m} $ such that, for any $ d_{t} > d^{m} $ ,

$$ \upmu+f\left(d^{m}\right)+\upvarepsilon^{\min}<\left(1+R_{t+1}\right)d^{m}-d^{m}\mathrm{~a n d~}f^{\prime}(d^{m})
Table 2: Returns on government bonds and capital

Non-default state	Default state
Government bonds	$ 1 + R_{t+1} $	0
Capital	$ 1 + A $	$ (1 - \theta)(1 + A) $

The government aims to maximize investor welfare and there is no conflict of interest between them. Investors who are domestic citizens hold all government bonds, and thus, the government has no incentive for strategic default.

For the default state, we assume that once default has occurred, the government cannot issue government bonds, at least during this period. The young’s budget constraint in the default state is given by $ W = k_{t} $ . Table 2 summarizes the returns on the government bonds and capital.

### 3. The Model Features

We first characterize the fiscal default. In doing so, we consider the "debt limit," denoted $ \overline{d}_{t+1} $ , as the maximum level of debt under which investors are willing to buy the bonds. For the present, we proceed to the analysis by assuming that there is a debt limit and that the government can roll over debt until it reaches this limit. If the shock $ \varepsilon_{t+1} $ is so small that the debt that is necessary to cover the interest payment exceeds the debt limit, $ \overline{d}_{t+1} < d_{t+1} $ , the government cannot roll over the debt and must default. Given the debt limit $ \overline{d}_{t+1} $ , there exists a cutoff variable $ \bar{\varepsilon}_{t+1} $ below which the debt that the government must issue to avoid default exceeds the debt limit:

$$ \bar{\varepsilon}_{t+1}=(1+R_{t+1})d_{t}-\upmu-f(d_{t})-\bar{d}_{t+1}\equiv\bar{\varepsilon}(R_{t+1},d_{t},\bar{d}_{t+1}). $$

From (3) and (4), the default probability is defined by

$$ \pi_{t+1}=p r o b.[\varepsilon_{t+1}<\bar{\varepsilon}_{t+1}]=G(\bar{\varepsilon}(R_{t+1},d_{t},\overline{d}_{t+1})), $$

which has the following properties: the higher the bond yield, the larger the current debt and the smaller the debt limit, the higher the default probability.

Agents should be indifferent to holding capital and bonds. The no-arbitrage condition is written as
$$ (1-\pi_{t+1})(1+R_{t+1})=(1-\pi_{t+1})(1+A)+\pi_{t}(1-\theta)(1+A), $$

for $ d_{t}>0 $ . The LHS is the expected return on government bonds such that agents receive payment only if the government rolls over the debt. The RHS is the expected return on capital such that agents receive dividend $ 1+A $ if the government rolls over the debt and receive $ (1-\theta)(1+A) $ if fiscal default occurs. The distinguishing feature is that a fiscal default produces a negative externality on the return on capital.

We rearrange (6) to yield the equation for the determination of the bond yield:

$$ 1+R_{t+1}=(1+A)+\frac{\pi_{t+1}}{1-\pi_{t+1}}(1-\theta)(1+A). $$

The second term on the RHS represents the risk premium in terms of $ 1+A $ . As $ \theta $ increases, the risk premium tends to decrease for a given $ \pi_{t+1} $ and is less likely to reflect the default risk fully. This is because when a fiscal default occurs, the return on government bonds declines, as does the opportunity cost of holding bonds. At the extreme for $ \theta=1 $ , the bond yield does not react to the fiscal risk. At the other extreme for $ \theta=0 $ , (7) reduces to the standard expression, such that the bond yield fully reflects the risk.

In this model, there is no safe asset, so we define the real interest rate by the expected return on bonds, or equivalently, the expected return on capital. Letting $ r_{t+1}^{n} $ denote the real interest rate, it follows from (6) that

$$ (1-\pi_{t+1})(1+R_{t+1})=(1-\theta\pi_{t+1})(1+A)=1+r_{t+1}^{n}. $$

The fiscal risk captured by the increase in the default probability $ \pi_{t+1} $ negatively correlates with the real interest rate.

We now turn to the formal analysis. We use (5) and (6) to define the net return function from government bonds by

$$ \begin{align*}\Omega\big(R_{t+1},d_{t},\overline{d}_{t+1},\theta\big)=&(1+R_{t+1})\{1-G\big(\overline{\Xi}\big(R_{t+1},d_{t},\overline{d}_{t+1}\big)\big)\}\\&-(1+A)\{1-\theta G\big(\overline{\Xi}\big(R_{t+1},d_{t},\overline{d}_{t+1}\big)\big\},\end{align*} $$

for $ 0 \leq \theta < 1 $ , where the first term is the return from government bonds and the second is the return from capital $ ^{10} $ . The function has three properties: (i) with the bond yield, the function at first increases, peaks, and then decreases —— it has a maximum, suggesting that there may or may not be a bond yield at which agents are willing to
buy government bonds; (ii) the function decreases as the actual debt increases; and (iii) the function increases as the debt limit increases. When there exists any bond yield satisfying $ \Omega(\cdot)>0 $ , there are two bond yields that satisfy $ \Omega(\cdot)=0 $ , as illustrated in Figure 3 $ ^{11} $ . Among these two, the lower bond yield is chosen as the equilibrium, but the higher bond yield is not $ ^{12} $ . Given the actual debt $ d_{t} $ and the debt limit $ \overline{d}_{t+1} $ , there is an $ R_{t+1} $ that satisfies

$$ \Omega(R_{t+1},d_{t},\bar{d}_{t+1},\theta)=0, $$

as long as $ d_{t}<\overline{d}_{t} $ . Accordingly, the default probability is given by

$$ \pi_{t+1}=G\left\{\left(1+R_{t+1}\right)d_{t}-\upmu-f\left(d_{t}\right)-\overline{d}_{t+1}\right\}. $$

Conversely, when $ \Omega(\cdot)<0 $ , for any $ R_{t+1} $ , there is no bond yield at which agents are willing to lend to the government. As is familiar from the literature on equilibrium credit rationing (e.g., Stiglitz and Weiss, 1981, Williamson, 1986), agents are never willing to hold government bonds irrespective of the bond yield. We now establish the determination of the bond yield.

Fig.3: The net return function
Proposition 1 : (i) There exists a finite bond yield at which the government can issue debt for $ d_{t} \in (0, \bar{d}_{t}] $ , and (ii) There exists no finite bond yield at which the government can issue debt for $ d_{t} \in (\bar{d}_{t}, +\infty) $ .

The first property implies that if debt remains at or below the debt limit, agents are willing to lend to the government at a positive finite bond yield. The second property implies that if debt exceeds its limit, there is no finite bond yield that can compensate for the default risk to agents. As a result, the government faces a complete loss of access to the bond market.

We now turn to the determination of the debt limit $ \overline{d}_{t+1} $ . Consider the function $ \Omega(\overline{R}_{t+1}, d_t, \overline{d}_{t+1}, \theta) $ when $ d_t $ reaches $ \overline{d}_{t+1} $ . By construction, for the debt limit to exist, there are some large $ \overline{d}_{t+1} $ , above which $ \Omega(\overline{R}_{t+1}, \overline{d}_{t+1}, \overline{d}_{t+1}, \theta) $ is decreasing in $ \overline{d}_{t+1} $ , or equivalently,

$$ R_{t+1}-f^{\prime}(\overline{d}_{t+1})>0. $$

Condition (A) guarantees that the debt limit satisfies (B) for any $ \bar{d}_{t+1} > d^{m} $ . There exists a debt limit $ \bar{d}_{t+1} $ and a “bond yield limit” $ \overline{R}_{t+1} $ that satisfy the following two equations :

$$ \Omega(\overline{R}_{t+1},\overline{d}_{t+1},\overline{d}_{t+1},\theta)=0,\mathrm{a n d} $$

$$ \partial\Omega(\overline{{R}}_{t+1},\overline{{d}}_{t+1},\overline{{d}}_{t+1},\theta)/\partial R_{t+1}=0. $$

Under Condition (A), the $ \Omega(\cdot) $ function is decreasing in $ \overline{d}_{t+1} $ . As shown in Figure 4, at the debt limit $ \overline{d}_{t+1} $ , the bond yield limit is the unique yield at which agents are willing to

Fig.4 : Bond yield and debt limits
hold the bonds. These two variables are time invariant; hereafter, we denote $ \overline{d}_{t+1}=\overline{d} $ and $ \overline{R}_{t+1}=\overline{R} $ without the time subscript. The default probability at the debt limit, denoted as $ \overline{\pi}=G\{(1+\overline{R})\overline{d}-\mu-f(\overline{d})-\overline{d}\} $ , is generally less than unity so that if the debt were to increase beyond the limit $ \overline{d} $ , the default probability would suddenly jump to unity.

We define the temporary equilibrium in period t by a set of variables $ \{s_{t}, d_{t}, k_{t}, R_{t+1}, \pi_{t+1}\} $ that satisfies five equations, (1), (2), (3), (10), and (11), given $ \tau_{t} $ and the fiscal shock $ \varepsilon_{t} $ . At the beginning of period t, $ d_{t-1} $ and $ R_{t} $ have been already determined. Once the fiscal shock $ \varepsilon_{t} $ is revealed, default occurs if $ \varepsilon_{t} < \bar{\varepsilon}_{t} $ , otherwise default does not occur. When default occurs, the economy shifts to the regime without debt. When default does not occur, (2) and (3) determine the fiscal surplus $ s_{t} $ and the new debt $ d_{t} $ . Equation (10) determines the bond yield $ R_{t+1} $ . Equation (11) determines the default probability $ \pi_{t+1} $ . Once $ d_{t} $ and $ R_{t+1} $ are determined, (1) determines $ k_{t} $ .

We now investigate several features of the model. Consider first the effects of an increase in debt.

Proposition 2: When debt $ d_{t} $ increases, the bond yield $ R_{t+1} $ and the default probability $ \pi_{t+1} $ increase and the real interest rate $ r_{t+1}^{n} $ decreases for $ d_{t} \in (0, \bar{d}] $ .

Proof: See the Appendix.

The relation among debt, bond yield, and default probability is a standard one. Additionally, debt negatively relates to the real interest rate. The increase in debt followed by a high default probability reduces the expected return on capital due to the negative externality. We next investigate the effects of a change in the no-risk hedge measure $ \theta $ .

Proposition 3: As the no-risk hedge measure $ \theta $ increases, the bond yield $ R_{t+1} $ and the default probability $ \pi_{t+1} $ decrease for $ d_{t} \in (0, \bar{d}] $ and the debt limit $ \bar{d} $ increases.

As the no-risk hedge measure $ \theta $ increases, the bond yield tends to be lower given the debt level. This arises because when there is fiscal default, the return on the government bond declines as does the opportunity cost of holding these bonds.

This finding suggests that the dynamics of debt and bond yield can differ across
economies depending on the no-risk hedge measure $ \theta $ . This explains why Japan, even with the OECD's highest debt-to-GDP ratio, enjoys low bond yields and no experience to date of a fiscal crisis. It also explains why several euro countries have suffered from fiscal difficulties with much lower debt-to-GDP ratios than that of Japan. The parameter $ \theta $ for Japan is supposedly high in that many large bondholders are domestic financial intermediaries lacking the superior knowledge and skills needed to manage international portfolios. In contrast, $ \theta $ in the euro countries is supposedly low in that high-debt countries such as Italy, Spain, Portugal, and Greece have adopted the common euro currency, freeing their investors of the foreign exchange risk associated with buying the safe assets of other euro countries.

Figure 5 plots bond yields as debt increases for three different values of $ \theta(0,0.5, $ and 0.8). We use the uniform distribution function $ G(\varepsilon_{t})=(\varepsilon_{t}+0.5\varepsilon^{m})/\varepsilon^{m} $ over the interval $ [-0.5\varepsilon^{m},0.5\varepsilon^{m}] $ , with the mean $ E[\varepsilon_{t}]=0 $ , and the response function $ f(d_{t})=\rho d_{t} $ . The chosen parameters are $ \varepsilon^{m}=10 $ , A=0.02, $ \rho=0.01 $ , and $ \mu=5 $ . When $ \theta $ is zero, the yield starts to rise at a low debt level and then the debt reaches the debt limit $ \overline{d}_{\theta=0} $ . As $ \theta $ increases, the yield rises only after some large debt level has been reached and the debt limit becomes large. In this example, the debt limit for the high- $ \theta(\theta=0.8) $ economy is about four times larger than that for the zero- $ \theta $ economy. The absence of safe assets contributes to the high level of sustainability. The bond yield limits also

Fig.5: The bond yields for different $ \theta $ 's

Note) For calculation, we use the distribution function $ G(\varepsilon_{t}) = (\varepsilon_{t} + 0.5\varepsilon^{m}) / \varepsilon^{m} $ over the interval $ \left[-0.5\varepsilon^{m}, 0.5\varepsilon^{m}\right] $ , with the mean $ E[\varepsilon_{t}] = 0 $ , and the response function $ f(d) = \rho d_{t} $ . The chosen parameters are $ \varepsilon^{m} = 10 $ , A = 0.02, $ \rho = 0.01 $ , and $ \mu = 5 $ .
differ. In fact, the yield limit of 12.1 percent for the zero- $ \theta $ economy is nearly double that of the 6.5 percent for the high- $ \theta $ economy. In our simulation, a low yield limit accompanies a large debt limit. Fiscal default can then take place before any hike in bond yield.

The next concern is to understand the relation between bond yield and real interest rate. Figure 6 illustrates the relation among bond yield, real interest rate, and default probability when $ \theta=0.5 $ . As shown, the bond yield does not rise sufficiently as the default probability increases but the real interest rate declines. While the bond yield remains low, the reduction in the real interest rate absorbs the fiscal risk. As $ \theta $ increases, this tendency becomes even stronger such that the bond yield increases slightly and the real interest rate declines significantly.

Caballero and Farhi (2014) uses the notion of a shortage of safe assets backed by sound collateral to explain the low yields on US government bonds. In contrast, we treat government bonds as risky assets with default risk and explain the low yields on JGBs by relying on a mechanism in which the bond yield is insensitive to fiscal risk. Instead, the reduction in the real interest rate absorbs the fiscal risk in an economy without safe assets. Our finding relates to the study by Kitagawa (1994) in explaining the low real interest rate in a world of risky capital.

The finding that the real interest rate declines as fiscal risk increases suggests that the stock market return negatively correlates with the outstanding debt to the extent that bond yields are insensitive to the fiscal risk. We can define the stock price as

Fig.6: The relation among bond yield, real interest rate, and default probability

Note) The chosen parameters are the same as the case for $ \theta=0.5 $ in Figure 5.
$$ Q_{t}=(1-\theta\pi_{t+1})(1+A)/(1+r_{t+1}^{n}), $$

which is equal to unity, as in the one-good production framework. However, the observed “price dividend” ratio, given that fiscal default has not occurred, is $ Q_{t}/(1+A)=(1-\theta\pi_{t+1})/(1+r_{t+1}^{n}) $ , which makes investors believe that the stock price is undervalued. When we observe two economies with the same bond yields but different debt levels, observing stock market prices may reveal what is actually going on.

## Welfare

A low bond yield is desirable from the viewpoint of fiscal sustainability, but it reduces the real interest rate when the debt is large. We examine the welfare of a generation using the expected consumption in old age:

$$ (1-\tau_{t})(1+r_{t+1}^{n})W. $$

Here, welfare decreases as the debt $ d_{t} $ increases (Proposition 2) or the tax rate $ \tau_{t} $ increases. Fiscal reform can then have an ambiguous effect on welfare. The trade-off is stringent when the actual debt is close to the debt limit. An increase in tax can reduce after-tax labor income, which is a source of worsening welfare, while a rise in the real interest rate follows the debt reduction, which is a source of improving welfare. Figure 7 plots the effects of the tax reform on welfare when $ \theta=0.5 $ . The horizontal axis represents the debt level before the reform. The bold unbroken line plots the case.

Fig.7 : Tax reform and welfare

Note) The chosen parameters are the same as the case for $ \theta=0.5 $ in Figure 5.
before the reform $ (\tau_{t}=0) $ . As shown, welfare starts to decline at about $ d_{t}=14.94 $ and reaches a minimum at a debt limit of $ d_{t}=15.086 $ . The bold dashed line plots the case for a tax reform where $ \tau_{t}=0.01 $ . When debt is small, post-reform welfare is lower; but as the debt goes beyond $ d_{t}=14.98 $ , welfare is higher. The thin dashed line plots the case for reform where $ \tau_{t}=0.02 $ . As shown, the reform gain becomes larger as debt approaches its limit.

### 4. Policy Implications

Our analysis provides several implications for fiscal sustainability and the macro economy. In this section, we select expansionary fiscal policy, financial market reform, and growth-enhancing policies for consideration.

## Expansionary Fiscal Policy

Consider the effect of government expenditure financed by government debt. Assume that labor income increases as government expenditure $ g_{t} $ increases. This is motivated by the idea that government expenditure yields positive externalities on the aggregate demand and output. The young's budget constraint is rewritten as

$$ (1-\tau_{t})W(g_{t}){\text{=}}k_{t}+P_{t}d_{t}, $$

where $ W'(g_t)>0 $ . Fiscal policies influence capital through three channels. First, an increase in government expenditure decreases the surplus $ s_t $ and increases debt $ d_t $ , which in turn has a crowding-out effect on capital. Second, the increase in debt increases the default probability followed by a decrease in the bond price, which has the effect of mitigating this crowding-out. Finally, the externality on output positively affects stimulating capital. The overall effects of these three channels on capital are ambiguous. However, as Proposition 2 states, the increase in debt $ d_t $ will decrease the real interest rate.

Proposition 4: When there is an increase in government expenditure financed by government debt, capital can increase or decrease, the bond yield $ R_{t+1} $ increases, but the real interest rate $ r_{t+1}^{n} $ decreases.

In a world of no safe assets, a decrease in the real interest rate follows expansionary fiscal policy. This finding contrasts sharply with Hick’s mechanism in the famous
IS-LM analysis, in which fiscal policy increases the interest rate that crowds out investment.

## Financial Market Reform

In Japan, the government either owns or regulates several large holders of JGBs; hence, regulations and misguided governance may induce them to hold larger amounts of government bonds $ ^{13} $ . We describe this environment as one where there are barriers that prevent investors from holding capital. Assume that a proportion $ \eta $ of the return on capital is lost because of these barriers. In this environment, we can rewrite the no-arbitrage condition between bonds and capital as

$$ (1-\pi_{t+1})(1+R_{t+1})=(1-\theta\pi_{t+1})(1-\eta)(1+A). $$

We then describe financial market reform as a reduction in $ \eta $ . Several effects of the reform are similar to those of a reduction in $ \theta $ . For example, the bond yield and default probability are higher, and the debt limit is lower (Proposition 3). The higher bond yield reduces the bond price, which in turn allows capital to increase. The direction of the real interest rate is ambiguous; the higher debt probability tends to reduce the real interest rate, while the higher return on capital tends to increase the real interest rate. The direction of the effect on welfare is also ambiguous.

Financial market reform contributes to stimulating investment in capital but worsens fiscal sustainability. Welfare may be better or worse. Specifically, given that the debt is already close to the debt limit, the reform may make it difficult or impossible to sustain the debt, and this will make agents worse off. The government then must understand the potential trade-off between investment boom and fiscal difficulty in conducting financial market reform.

## Growth-Enhancing Policy

We describe the policy for enhancing economic growth as an increase in the return on capital A. The effects are similar to those of financial market reform. When the
government implements a growth-enhancing policy, capital, the bond yield, and the default probability increase and the debt limit declines. The direction of its effect on the real interest rate and welfare are ambiguous. When the policy increases anxiety about fiscal sustainability, the stock market may remain stagnant. The government should then propose a policy package that advances economic growth and fiscal sustainability at the same time.

### 5. Discussion

Several related issues require further discussion.

## The Role of Monetary Easing

We have thus far investigated one possible mechanism that explains the low yields of JGBs. It would be meaningful to discuss the role of monetary policy as one of other channels for explaining the low yields. As of 2016, the BoJ has been conducting QE that involves large-scale purchases of long-term government bonds. Since the beginning of this policy implementation, the nominal yield on 10-year government bonds has declined from 0.9 percent to 0.4 percent as of 2015 end. Certainly, the QE policy has contributed to lowering yields on JGBs.

The BoJ is in a special position relative to private investors in that it does not aim to earn profits from holding JGBs $ ^{14} $ . Assume that the central bank does not request positive yields and purchases bonds at a higher price than the market price. When the central bank holds a share $ \phi(0 < \phi < 1) $ of the outstanding debt, the government budget constraint is rewritten by

$$ d_{t+1}-d_{t}=(1-\phi)R_{t+1}d_{t}-s_{t+1}. $$

This expression states that the government can economize on the expense of bond yields to the extent that the central bank holds the bonds. Taking other things as given, as $ \phi $ increases, the bond yield and the default probability decline. QE can improve fiscal sustainability and play a complementary role in strengthening the effects of financial market reforms and growth-enhancing policy. However, this policy is only effective if people value the money issued in exchange of purchasing bonds.