2.1 Households

There is a representative household with a continuum of members of measure unity, with a fraction $1-f$ that are workers and a fraction f that are bankers. Workers supply labor and return wages to the family, while bankers own a financial intermediary and return dividends to their household. Households can save in form of deposits held by intermediaries. They supply funds to banks in the form of noncontingent short-term debt (deposits, denoted $D_t$ ) that pay a risk-free gross real return rate $R_t$ . We additionally assume households cannot buy government bonds directly.

Households choose consumption ( $C_t$ ), labor supply ( $L_t$ ), and riskless debt to maximize expected discounted utility. We assume preferences in logaritmic form that follow a GHH specification (Greenwood et al. (Reference Greenwood, Hercowitz and Huffman1988)) to avoid the wealth effects on labor supply. ^{Footnote 9}

(1) \begin{equation} E_t \sum_{i=0}^{\infty} \beta^{i} \log \left[ C_{t+i} -\frac{\psi}{1+\varphi} L_{t+i}^{1+\varphi} \right].\end{equation}

Households are subject to the following budget constraint:

(2) \begin{equation} (1+\tau^{c}_{t}) C_t + \left( D_{t+1} - D_t \right) = (1-\tau^{w}_{t}) W_t L_t + (R_t -1 ) (1-\tau^{d}_{t}) D_t + \Pi_t - TR_t.\end{equation}

where $W_t$ is the wage rate, $TR_t$ are lump-sum tranfers payed (received) to the government, and $\Pi_t$ are the dividends obtained from the ownership of nonfinancial firms and banks. Tax rates are also indexed to t. Taxation is composed by consumption taxes ( $\tau_{t}^{c}$ ) and income taxes of two forms: taxes on wages ( $\tau_{t}^{w}$ ) and taxes on (net) returns of savings ( $\tau_{t}^{d}$ ).

From the first-order conditions for consumption/saving, we get:

(3) \begin{equation} E_t \beta \Lambda_{t,t+1} \left[ (1 - \tau^{d}_{t+1} ) (R_{t+1}-1) + 1 \right] = 1,\end{equation}

where $\Lambda_{t,t+1}$ is the households’ stochastic discount factor:

\begin{equation*}\Lambda_{t,t+1} \equiv \frac{\varrho_{t+1}}{\varrho_t} \frac{(1+\tau^{c}_{t})}{\left(1+\tau^{c}_{t+1}\right)},\end{equation*}

and $\varrho_t$ is marginal utility of consumption,

\begin{equation*}\varrho_t \equiv \left( C_t - \frac{\psi}{1+\varphi} L_{t}^{1+\varphi} \right)^{-1}.\end{equation*}

The first-order condition for labor supply writes:

(4) \begin{equation} \psi L_{t}^{\varphi} = \frac{(1-\tau^{w}_{t}) W_t}{(1+\tau^{c}_{t})}.\end{equation}

In every period, there is a probability $(1-\theta)$ that a banker becomes a worker. In order to maintain the fraction in each occupation constant over time, in each period there is a random fraction $(1-\theta)f$ of workers that become bankers. Workers who become bankers receive a “start up” capital from the household to start business. Expected survival time of a bank is thus $1/(1-\theta)$ . This prevents bankers from accumulating enough wealth so as to overcome their financial constraints.

Households also own nonfinancial firms (capital and goods producers). However, they are not able to acquire capital directly or to provide funds to these firms. All financial intermediation for production must be made by a bank.

2.2 Goods producers

The representative firm in this sector produces output in a competitive market, using labor and capital in a Cobb–Douglas technology:

(5) \begin{equation} Y_t = K_t^{\alpha} L_{t}^{1-\alpha},\end{equation}

with $0 < \alpha < 1$ .

As usual, labor demand implies that the real wage rate equals the marginal product of labor:

(6) \begin{equation} W_t = (1-\alpha) \frac{Y_t}{L_t}.\end{equation}

In order to produce in period $t+1$ , firms need to buy the amount of capital $K_{t+1}$ at the end of period t from capital producers. In order to finance the acquisition of capital, firms issue securities $S_t$ and an arbitrage condition ensures the value of these securities equals the value of the capital to be bought. The intermediaries buy these securities. Denoting by $Q_t$ the price of one unit of capital, we have:

\begin{equation*}Q_t K_{t+1} = Q_t S_t.\end{equation*}

There are no frictions in this process. Intermediaries have perfect information about the firm and about future payoffs, so securities are state-contingent. Frictions exist within the process of banks obtaining resources from households.

In order to satisfy the zero profit condition in the competitive market, goods producers buy capital goods up to the point that gross profits per unit of capital $Z_t$ equal the marginal product of this input:

\begin{equation*}Z_{t} = \frac{Y_t - W_t L_t}{K_{t}} = \alpha \frac{Y_t}{K_t}.\end{equation*}

A firm that sells $S_t$ securities to acquire capital must return all its profits in the next period to the bank. Call $R_{kt}$ the gross return to capital in time t, the amount a bank obtains as a return over each unit of credit supplied in the form of acquired securities. The representative goods producer owes a bank an amount $Q_t S_t R_{kt+1}$ at the end of the period. This value equals the sum of profits $\Pi_{ft}$ obtained through capital utilization in production (gross of capital remuneration) and the market value of the effective non-depreciated capital, that could be sold back in the market after production has taken place.

\begin{equation*}Q_t S_t R_{kt+1} = \Pi_{ft+1} + (1-\delta) Q_{t+1} K_{t+1}.\end{equation*}

Substituting for $\Pi_{ft}$ and $S_t$ and dividing both sides by $K_{t+1}$ :

\begin{equation*}Q_t R_{kt+1} = \alpha \frac{Y_{t+1}}{K_{t+1}} + (1-\delta) Q_{t+1}.\end{equation*}

Hence, the gross return to capital in period $t+1$ is given by the ratio between the value generated by one unit of capital acquired by the firm in period t over the price at which it was bought.

(7) \begin{equation} R_{kt+1} = \frac{Z_{t+1} + (1-\delta) Q_{t+1} }{Q_{t}}.\end{equation}

2.3 Capital producers

The market for capital is competitive. At the end of each period, capital producers build new capital for the following period using the final output as an input in the production. Capital goods are then sold back to goods producers at price $Q_t$ . They are subject to convex adjustment costs in this process. ^{Footnote 10} A capital producer chooses investment $I_{t}$ to maximize discounted profits, taking the price of capital $Q_t$ as given.

Adjustment costs are a convex function of investment. The capital producers’ problem is given by

\begin{equation*}max \quad E_t \sum_{\tau=t}^{\infty} \beta^{\tau - t} \Lambda_{t,\tau} \left[ Q_{\tau} I_{\tau} - \left[ 1 + f \left( \frac{I_{\tau}}{I_{\tau-1}} \right) \right] I_{\tau} \right],\end{equation*}

with $f(1) = f'(1) = 0$ and $f''(1) > 0$ . Nonzero profits are possible when the economy is not in steady state, and profits are transfered to the household.

The first-order condition for investment is given by

(8) \begin{equation}Q_t = 1 + f \left( \frac{I_{t}}{I_{t-1}} \right) + \frac{I_{t}}{I_{t-1}} f' \left( \frac{I_{t}}{I_{t-1}} \right) - E_t \beta \Lambda_{t,t+1} \left( \frac{I_{t+1}}{I_{t}} \right) ^{2} f' \left( \frac{I_{t+1}}{I_{t}} \right).\end{equation}

This condition states that capital price will equal the marginal cost of investment.

The adjustment cost function assumes the form:

\begin{equation*}f(.) = \frac{\eta_i}{2} \left( \frac{I_{t}}{I_{t-1}} -1 \right) ^{2},\end{equation*}

where $\eta_i$ refers to the inverse elasticity of investment with respect to the price of capital.

2.4 Government

Government spending is given by $G_t$ . To finance itself, government taxes households, banks, and issues debt, which for simplicity is bought only by banks.

The government issues debt with the following maturity structure: every period a fraction $\mu$ of the outstanding debt stock comes due and $1 - \mu$ goes on to add to the next period’s debt pile. This is equivalent to assuming the government always issues debt with varying maturities, being $1-\mu$ the ratio between the amount of debt coming due in $t+1$ and the amount coming due in t.

The objective of the paper is to study fiscal policy in the aftermath of a sovereign default. To capture the debt restructuring in an easily tractable way, we assume the default is caused by an exogenous policy shock. We introduce this possibility in the model by assuming that repayment is given by a random variable $m_t \in [0,1]$ . The variable $m_t$ represents the actual fraction of debt coming due at t that is repaid, and it is given by

(9) \begin{equation} m_t = \min \{ \iota_{t}, 1 \},\end{equation}

where

(10) \begin{equation} \iota_t = \rho_{\iota} \iota_{t-1} + (1 - \rho_{\iota}) + \varepsilon_{\iota t},\end{equation}

where $\rho_{\iota}$ is a positive constant and $\epsilon_{\iota t}$ is a normally distributed error term with mean 0 and standard deviation $\sigma_{\iota}$ . In steady state, $\iota_t = 1$ , meaning that the government fully repays the amount of debt that comes due in t. However, the fraction to be repaid is subject to shocks.

A debt restructuring episode triggered by a policy decision is captured by a one-off negative shock on $\iota$ . The auto-regressive specification captures the fact that in case of debt restructuring, sovereigns tend to repudiate short-term debt and lengthen the debt repayment profile. So repayment drops in the first periods following a default episode but grow in time, tending again to a hundred percent of the maturing fraction as the shock vanishes completely. ^{Footnote 11} The following section shows that in fact such a pattern was present in the Greek Bond Exchange from 2012.

In each period, the government repays $\mu m_t$ of the debt. Denoting $\chi_t$ the price of debt, the government’s financing requirement for period $t+1$ , $\chi_t A_{t+1}$ , is the difference between the fraction of debt repaid ( $\mu B_t$ times the fraction effectively honoured $m_t$ ) and the amount of government spending that is not covered by taxes (primary deficit):

(11) \begin{equation} \chi_t A_{t+1} = \mu m_t B_t + G_t - T_t.\end{equation}

Total outstanding (nominal) debt in $t+1$ is given by

(12) \begin{equation} B_{t+1} = (1-\mu) B_t + A_{t+1}.\end{equation}

Total taxes are given by the sum of all sources of taxation: lump-sum transfers, consumption taxes, income taxes (wage income and capital taxes), and taxes on banks’ profits (to be presented in the next subsection).

(13) \begin{equation} T_t = TR_{t} + \tau_{t}^{c} C_t + \tau_{t}^{w}W_t L_t + \tau_{t}^{d} (R_t - 1) D_t + \tau_{t}^{b} {\pi}_{t}.\end{equation}

We now turn to the fiscal rules assumed to close the model. The government has a set of tax rates to manipulate and can also increase exogenous expenditures, with the objective of not letting real value of debt deviate largely from steady state. The speed of this adjustment is given by the parameter $\rho_b$ and is related to the convergence of debt to the steady state level. Tax rates also have an autoregressive component, $\rho_x$ , as in the following equation:

(14) \begin{equation} X_t = (1-\rho_x) \bar{X} + \rho_x X_{t-1} + \rho_b \left( \chi_t B_t - \bar{\chi} \bar{B} \right),\end{equation}

with

\begin{equation*} X_t = \left\{ \tau_{t}^{w}, \tau_{t}^{c}, \tau_{t}^{d}, \tau_{t}^{b}, TR_{t}, G_t \right\}. \end{equation*}

The rule implies that the government raises taxes when its debt rises (see, e.g., Bi (Reference Bi2012) for a discussion). The autoregressive parameter ensures that tax rates move slowly, in accordance to the data.

To complete this subsection, gross return on bonds is the ratio between the expected value to be payed back by the government in the next period plus the expected value of the remaining outstanding debt divided by the current price of debt:

(15) \begin{equation} R_{bt+1} = \frac{\mu E_t m_{t+1} + (1-\mu) E_t \chi_{t+1}}{\chi_{t}}.\end{equation}

A shock to $m_t$ affects not only the haircut in t but also the expected repayment in the following periods, which induces changes in bonds’ prices, directly influencing banks’ balance sheets and investment decisions. Steady-state price of government debt $\chi_t$ is

(16) \begin{equation} \bar{\chi} = \frac{\mu}{\bar{R_b} - (1-\mu)}.\end{equation}

In the limiting case where $\mu=1$ , price of debt is as standard the inverse of the bond yield.

2.5 Banks

Following Gertler and Karadi (Reference Gertler and Karadi2011), banks can raise funds from households in form of deposits or from retained earnings, accumulating net worth. They use the available funds to buy state-contingent securities from goods producers, but also to buy government bonds $B_{t+1}$ at price $\chi_t$ . In each period, banks pay a tax $\tau_{t}^{b}$ on their profits.

Banks are the only agents that buy sovereign debt in this economy. In reality, sovereign default also transfers resources from firms and households to the government and also entails a variety of costs. Since the focus of the paper is the sovereign-bank loop, we do not consider any other default cost. But then, we do not include the default on firms and households either. Accordingly, our calibration strategy will consider debt held by banks only.

A bank’s balance sheet is composed by the assets it holds (government bonds and private securities), liabilities (deposits), and net worth:

(17) \begin{equation} \chi_t B_{t+1} + Q_t K_{t+1} = N_t + D_{t+1}.\end{equation}

$D_t$ are deposits raised from households and we used $S_t = K_{t+1}$ .

We make use of the notation $N_t$ to denote post-tax net worth:

\begin{equation*}N_{t} = N_{t-1} + \pi_{t} (1-\tau_{t}^{b}), \end{equation*}

with

\begin{equation*} \pi_{t} = r^{k}_{t} Q_{t-1} K_{t} + r^{b}_{t} \chi_{t-1} B_{t} - r_{t} D_{t}. \end{equation*}

Net worth in $t+1$ is the gross payoff from assets funded at t net of returns to depositors. Profits are given by subtracting the flow of compensation to depositors from earnings on assets. Let $R_{kt+1}$ denote the gross rate of return on a unit of a bank’s private securities from t to $t+1$ . Net worth before taxes is then given by

(18) \begin{equation} \tilde{N}_{t+1} = R_{kt+1} Q_{t} K_{t+1} + R_{bt+1} \chi_{t} B_{t+1} - R_{t+1} D_{t+1},\end{equation}

with

\begin{equation*}R_{kt+1} = (1+r^{k}_{t+1}) = \frac{Z_{t+1} + (1-\delta) Q_{t+1}}{Q_{t}}.\end{equation*}

The objective of a banker is to maximize its future terminal value, given by the discounted value of (net) net worth, accounting for the probabilities that she might exit at each future period:

(19) \begin{equation} V_t = E_t \left[ \sum_{i=1}^{\infty} \left( 1-\theta \right) \theta^{i-1} \beta^{i} \Lambda_{t,t+i} N_{t+i} \right].\end{equation}

The bank’s ability to obtain funds is limited by a moral hazard constraint as in Gertler and Karadi (Reference Gertler and Karadi2011). At each period, a banker may choose to divert a fraction $\lambda$ of assets in the form of dividends to her family and hence defaults on part of debt. In this case, the remaining fraction $1-\lambda$ of her assests will be recovered by other depositors, leading the bank to bankruptcy. This fraction $\lambda$ is exogenous and constant. This constraint could also be interpreted as a leverage constraint imposed by official regulation, along the lines of the Basel Agreements. In equilibrium, leverage is pinned down by this constraint.

Anticipating the possibility of funds diversion, depositors will limit their lendings to ensure banks won’t divert funds. The bank’s value must be at least as large as its gain from deviating funds, so as to discourage diversion.

(20) \begin{equation} V_t \geq \lambda (\chi_t B_{t+1} + Q_t K_{t+1}).\end{equation}

The expressions in (17) and (18) yield the evolution of the bank’s net worth as a function of the state variables $K_{t}$ , $B_{t}$ , and $N_{t-1}$ :

(21) \begin{equation} N_t = N_{t-1} + \left[ (r^{k}_{t} - r_t) Q_{t-1} K_{t} + (r^{b}_{t} - r_t) \chi_{t-1} B_{t} + r_t N_{t-1} \right] (1-\tau_{t}^{b}).\end{equation}

Let $V_t(K_{t+1}, B_{t+1}, N_t)$ be the maximized value of the bank’s objective. It will satisfy the following Bellman equation.

(22) \begin{equation} V_t (K_{t+1}, B_{t+1}, N_t) = E_t \beta \Lambda_{t,t+1} \{ (1-\theta) N_{t+1} + \theta \max \left[ V_{t+1} (K_{t+2}, B_{t+2}, N_{t+1}) \right] \}.\end{equation}

In each period, the banker chooses a portfolio composition of capital and bonds, $K_{t+1}$ and $B_{t+1}$ , to maximize her value function subject to the incentive constraint and the law of motion for net worth, taking into account that she might exit with probability $(1-\theta)$ .

We conjecture the value function to be linear in the balance sheets’ components:

(23) \begin{equation} V_t(K_{t+1}, B_{t+1}, N_t) = \nu_{t} Q_t K_{t+1} + \zeta_{t} \chi_t B_{t+1} + \eta_t {N}_t.\end{equation}

In the Appendix, we show that this conjecture is true, as long as

(24) \begin{equation} \eta_t = E_t \beta \Lambda_{t,t+1} \tilde{\Omega}_{t+1} \left[ 1 + r_{t+1}(1-\tau^{b}_{t+1}) \right],\end{equation}

(25) \begin{equation} \nu_{t} = E_t \beta \Lambda_{t,t+1} \Omega_{t+1} \left(R_{kt+1} - R_{t+1} \right),\end{equation}

(26) \begin{equation} \zeta_{t} = E_t \beta \Lambda_{t,t+1} \Omega_{t+1} \left(R_{bt+1} - R_{t+1} \right),\end{equation}

with

(27) \begin{equation} \Omega_t = (1-\tau_{t}^{b}) \tilde{\Omega}_t,\end{equation}

and

(28) \begin{equation} \tilde{\Omega}_{t+1} = 1-\theta + \theta \left[\phi_{t+1} \zeta_{t+1} + \varpi_{t+1}(\nu_{t+1} - \zeta_{t+1})+ \eta_{t+1} \right].\end{equation}

The auxiliary variable $\varpi_t$ is the leverage only in terms of capital:

(29) \begin{equation} \varpi_t \equiv \frac{Q_t K_{t+1}}{N_t}.\end{equation}

Each component of the banks’ value function can be interpreted as follows: $\eta_t$ is saving in deposits’ costs from an additional unit of net worth. The variables $\nu_t$ and $\zeta_t$ are the marginal discounted gains of expanding, respectively, private securities’ holdings and government bonds’ holdings. Finally, $\tilde{\Omega}_t$ is the shadow marginal value of net worth and affects the banks’ intertemporal discount factor.

Optimization for banks will imply the following no-arbitrage condition:

(30) \begin{equation} E_t \beta \Lambda_{t,t+1} \Omega_{t+1} \left(R_{kt+1} - R_{t+1} \right) = E_t \beta \Lambda_{t,t+1} \Omega_{t+1} \left(R_{bt+1} - R_{t+1} \right).\end{equation}

Define $\phi_t$ as the leverage ratio, the maximum ratio of bank assets over equity:

\begin{equation*}\phi_t \equiv \frac{Q_t K_{t+1} + \chi_t B_{t+1}}{N_t}.\end{equation*}

The constraint in (20) can be rewritten as

\begin{equation*}\nu_{t} Q_t K_{t+1} + \zeta_{t} \chi_t B_{t+1} + \eta_t N_t \geq \lambda (\chi_t B_{t+1} + Q_t K_{t+1}).\end{equation*}

If this constraint binds, we get

\begin{equation*}\phi_t = \frac{\eta_t + \varpi_t (\nu_t - \zeta_t)}{\lambda - \zeta_{t}}.\end{equation*}

Which, using (30), simplifies to

(31) \begin{equation} \phi_t = \frac{\eta_t}{\lambda - \zeta_{t}}.\end{equation}

2.6 Evolution of bank’s net worth

The total net worth in the banking sector equals the sum of existing banks’ net worth $N_{e,t}$ and entering banks’ start-up capital $N_{n,t}$ provided by their families. The net worth of existing banks equals the net earnings from assets over liabilities from one period to another, that is, earnings from holding securities plus earnings from holding government bonds minus costs from deposits, net of taxes. This expression must be multiplied by the fraction $\theta$ of banks that survive between periods:

(32) \begin{equation} N_{e,t} = \theta N_{t-1} + \theta N_{t-1} (1-\tau_{t}^{b}) \left[ (R_{kt}-R_{bt}) \varpi_{t-1} + (R_{bt}-R_{t}) \phi_{t-1} + (R_t-1) \right].\end{equation}

Families transfer to each new banker a constant fraction $\omega/\left( 1 - \theta \right) $ of total assets from exiting bankers, given by $(1-\theta) [ Q_{t} K_{t} + \chi_{t} B_{t}]$ , also after taxes. Hence, entering bankers’ net worth will be

(33) \begin{equation} N_{n,t} = \omega (1-\tau_{t}^{b}) (Q_t K_{t} + \chi_t B_{t}).\end{equation}

Total net worth from banks in the economy is thus

\begin{equation*}N_t = N_{e,t} + N_{n,t}.\end{equation*}

2.7 Market Clearing

Output can be used for consumption, government spending, or investment (including adjustment costs). Aggregate demand is given by

(34) \begin{equation} Y_t = C_t + I_t \left[ 1 + f \left( \frac{I_t}{I_{t-1}} \right) \right] + G_t.\end{equation}

Market clearing in the goods market requires the expression for demand in (34) to equal supply, given by (5).

The banks’ balance sheet can be written as

\begin{equation*}Q_t K_{t+1} + \chi_t B_{t+1} = \phi_t N_t.\end{equation*}

Demand for securities and bonds is given by the balance sheet constraint, given by (31). The supply of securities by firms is given by the expression for capital accumulation:

\begin{equation*}K_{t+1} = (1-\delta) K_t + I_t.\end{equation*}

Finally, market clearing for deposits is obtained from balance sheet identity. Total deposits supplied by families must equal the difference between banks’ assets and net worth.

\begin{equation*}D_t = Q_t K_{t+1} + \chi_t B_{t+1} - N_t.\end{equation*}

3.1 The greek debt restructuring

The 2012 Bond Exchange in Greece was the outcome of a public budget deterioration that became evident with the countercyclical policies following the 2007–2008 Great Recession and revealed itself much worse once previous unreliable fiscal data was revised. This led to a deep confidence crisis and a sharp increase in spreads for Greek debt. A first proposal for a bond exchange in 2011 (one year after a rescue package had been agreed upon with the IMF and the EU) and the following fiscal consolidation effort by the Greek government were shown to be insufficient, since the deep recession and the postponement of structural reforms precluded a deeper adjustment. After further negotiations a huge bond exchange program was agreed upon for March 2012, with major private sector involvement, a feature already present in the 2011 proposal. ^{Footnote 12}

The Greek Restructuring consisted of a lengthening of the average maturity and large debt relief. Restructuring implied an average residual maturity increase for Greek securities from 7 years in 2011 to more than 12 years in 2012, although at the aggregate level the repayment profile for bonds shifted into the future was largely compensated by short term repayment of EFSF (European Financial Stability Facility) notes (official loans). ^{Footnote 13}

Figure 1 plots the evolution of debt in aggregated terms and interest payments, both as fractions of GDP. One can observe a drop in general government consolidated debt as a share of GDP from 2011 to 2012 which, in terms of debt relief, resulted in a face value reduction of around 52 $\%$ , or a reduction of 12 percentage points in Debt/GDP ratio. Notwithstanding this immediate relief, debt-to-GDP ratios started to recover fast, returning to pre-default levels after a few years. The evolution of interest payments in terms of GDP is also interesting: it drops by almost half from 2011Q4 to 2012Q1, after the bond exchange was conducted. Average haircut was 65 $\%$ , with higher losses for short- term investors.

Figure 1. Greece: Debt-to-GDP.

3.2 The counterfactual scenario

The paper aims at evaluating how fiscal policy affects the sovereign-bank negative loop. For this reason, in our quantitative analysis, we (i) compare a situation with debt restructuring with a counterfactual scenario with normal debt servicing and (ii) we focus only on default on bonds held by banks.

The Greek Debt Restructuring was accompanied by some degree of fiscal contraction: total taxes were raised and expenditures were cut. Nevertheless, had the restructuring not happened, the fiscal effort required for servicing debt would have been larger than observed.

In our laboratory economy, we consider a counterfactual steady state that aims at capturing a scenario with no Bond Exchange in Greece in 2012. Since the absence of default would have substantially increased debt servicing costs, the debt and tax levels in this counterfactual steady state are higher than their actual values observed in Greece before default. We then simulate a shock that resembles the Greek Bond Exchange from 2012.

The debt restructuring is thus accompanied by expansionary fiscal policy. Importantly, that is relative not to reality in 2012, but to the counterfactual steady state with higher debt and taxes. The fiscal room open right after the restructuring shock is exactly the difference between the counterfactual and the actual debt servicing levels.

In the following periods, fiscal policy will react to deviations of debt from steady state such that in the medium run debt returns to pre-default levels. After a sizeable relief on impact, debt recovers and returns to pre-default levels after some years. This assumption is in agreement with empirical evidence suggesting that debt-to-GDP ratios tend to return, on average, to pre-default levels some years after restructuring. ^{Footnote 14} The quantitative analysis captures the idea that default opens some fiscal space in the short run, but the long run debt burden does not seem to be significantly reduced.

Second, we calculate the hypothetical increase in taxes that would have been necessary in the absence of debt restructuring considering only bonds held by banks. Sovereign default also creates some fiscal room by subtracting resources from firms, households, and foreign agents; the fiscal policy response affects how the economy reacts. However, our focus here is the sovereign-bank loop, so we abstract from default on firms and households and all resulting transfers from the government. ^{Footnote 15}

Denoting debt as the sum of securities and loans to the domestic government and assets as the sum of all domestic assets (to roughly reflect our closed economy), we get that financial institutions held $59.1$ billion Euros of government debt in 2011, out of a total amount of $356.3$ billion Euros. Hence, banks owned around $16.6\%$ of sovereign bonds issued by the Greek government (a figure similar to the one in Gennaioli et al. (Reference Gennaioli, Martin and Rossi2018)). Our quantitative analysis focuses on this $16.6\%$ share.

We then need to calculate the additional debt service in the absence of default. It comes from two sources: from the haircut itself, that is, from the face value reduction of debt; and from the decrease in interest payments over debt.

We estimate the hypothetical debt in the absence of default in 2012 as the debt observed at the end of 2011 times the average year-over-year debt growth between 2008 and 2011. This leads to $393.4$ billion Euros of debt. Subtracting the observed level of debt in 2012 ( $305.1$ billion Euros) from its counterfactual level yields a haircut of $88.3$ billion Euros. The average interest rate from 2007 to 2011 is $4.2\%$ . Multiplying it by the $88.3$ billion Euros, we get that the haircut itself reduced debt service in 2012 by $3.71$ billion Euros.

Second, the average interest rate paid in 2012 by Greece was roughly $3.2\%$ ( $9.7$ billion Euros in debt service devided by $305.1$ billion Euros of outstanding debt). The amount saved owing to the decrease in interest payments over debt was thus $(4.2\%-3.2\%) \times 305.1 = 3.05$ billion Euros in 2012. Hence, the estimated additional debt service in the absence of default is $3.71 + 3.05 = 6.76$ billion Euros.

Recall we focus only on the share of debt held by banks. Multiplying the $16.6\%$ share by $6.76$ billion Euros yields our final estimation: owing to debt restructuring, the amount paid by the Greek government to banks was $1.125$ billion Euros less than it would have been otherwise in 2012. This is the increase in taxes (or reduction in government spending) that would have been necessary in the absence of debt restructuring, as compared to their observed levels on 2012.

3.3 Calibration

Sovereign debt restructuring affects an economy in a variety of ways. Our model and calibration aim at capturing the sovereign-bank channel only. Hence, default on debt in the hands of households or foreign agents is not considered in our simulations. Our calibration matches key features of the Greek economy surrounding the March 2012’s Bond Exchange. The model’s steady state portrays Counterfactual Greece in 2012 and the default shock captures the main characteristics of the observed debt restructuring.

3.3.1 Fiscal variables

Figure 2 presents revenues per type of tax as a share of total taxes in 2012, calculated using data from Eurostat on tax items and aggregating by categories. The model is calibrated to match this tax composition. ‘Other’ taxes are bundled as lump-sum taxes.

Figure 2. Share of total tax collection. Source: Authors’ calculations using data from Eurostat

The estimated $1.125$ billion Euros not paid by the government to banks owing to debt restructuring implies that total tax revenues in case of default were 2.4 $\%$ lower than in Counterfactual Greece (considering bank’s debt only). We target this amount of tax reduction when default hits the economy.

In all our simulations, one tax rate is cut at a time, and the rule parameter is calibrated to achieve the desired size of decrease on impact for each tax. This allows us to compare the response of the economy when different fiscal instruments are chosen, which is our aim.

The parameter $\rho_b$ in (14) is calibrated to target desired response of taxes after the shock (it assumes negative values in case of government expenditures) that match both the initial desired drop in taxes and the time until debt converges again to steady state in line with the already mentioned empirical evidence on default. Across the exercises, we compare scenarios for which the government uses one tax instrument at a time, leaving the other tax rates fixed at steady state values. The autoregressive component of the tax rule, $\rho_x$ , is set to 0.9 and is also subject to variations in sensitivity analyses.

3.3.2 Parameters and steady state

In order to match the main characteristics of the Greek macroeconomy in 2012, government consumption is set to 21.3 $\%$ of GDP, gross capital formation is 12 $\%$ of GDP, and consumption comprises the remaining fraction (66.7 $\%$ ), given that we consider a closed economy. These values are almost exactly the ratios found in data for Greece in 2012 (Eurostat), with deviations corresponding to net exports, that are not accounted for in the model.

For the “real” sector, we set desired risk premium that, together with depreciation rate and capital share in output, determine the capital/labor ratio for the economy. In order to match those ratios we set depreciation rate ( $\delta$ ) to 8.5 $\%$ per year, capital share in production ( $\alpha$ ) to 0.33 and the intertemporal discount rate ( $\beta$ ) to 0.98. The steady-state spread of capital return over the risk-free rate, $R_k - R$ , that in the model also represents spread from bond return over risk free $R_b - R$ , is set to 5.8 $\%$ per year. This value is close to the annualized spread of 10-year Greek bonds over German bonds observed in 2010. ^{Footnote 16} Inverse elasticity of investment to capital price, $\eta_i$ , follows the conventional value from papers that add adjustment costs to investment, such as Gertler and Karadi (Reference Gertler and Karadi2011).

Parameters of the GHH preferences follow conventional calibration in the literature. Labor disutility equals 1 and the Frisch inverse elasticity of labor supply, $\varphi$ , is set to 3, implying a labor supply elasticity equal to 1/3. In case of labor income taxes, these values imply that the Laffer curve peaks when the labor tax rate is around 75%. ^{Footnote 17} Table 1 summarizes the parameters in our calibration.

Table 1. Parameters

The other set of aggregates regards debt and banks (the “financial side”). We target three key ratios: debt as a fraction of total assets, the leverage ratio, and the amount collected in taxes on banks as a proportion of total taxes. ^{Footnote 18}

As mentioned before, we focus on the share of Greek bonds owned by banks ( $16.6\%$ ). Hence, our counterfactual debt-GDP ratio is $33.5\%$ (and not around 200% as it would have been if we had considered the total amount of debt). Using consolidated data from the Bank of Greece on Balance Sheet of Credit Institutions, we arrive at a value of $16.2\%$ for bank debt/assets (in line with evidence presented by Gennaioli et al. (Reference Gennaioli, Martin and Rossi2018)).

Leverage ratio is set to 6.5, in line with Bankscope data on Greek banks, most of them important creditors of their own government. ^{Footnote 19} Last, the amount collected in bank taxes is $1.8\%$ of total taxes.

We then choose $\mu$ accordingly to reach a debt’s term to maturity equal to 7.4 years, the value observed in data before the restructuring. As already mentioned, we only focus on the specific channel of default through balance sheets of banks. Besides, we do not account for important movements in debt’s composition after default that, as shown in the previous section, moved toward higher participation of official lenders’ loans in expense of privately held securities, the share of total debt directly affected by the bond exchange.

Concerning the fiscal block of the model, steady-state government consumption and debt service imply taxes are 26% of GDP, a value slightly higher than the counterfactual one presented in the previous section. Table 2 shows model steady-state aggregates and ratios and comparison to counterfactual Greece. Details on the time series used and assumptions are in appendix.

Table 2. Model steady states and comparison to data*

*Data sources and details in appendix.

^†Values in parentheses refer to the counterfactuals calculated from data.

*Debt/GDP in the model: $\chi B / 4 Y$ . **Average 2010–2012. ***2010. ****Average 2001–2008.

3.3.3 The debt restructuring exercise

The bond exchange in Greece had two important features: a substantial lengthening of the repayment profile and a considerably high average haircut, leading to a present value debt relief of almost 50 $\%$ in terms of GDP (Zettelmeyer et al. (Reference Zettelmeyer, Trebesch and Gulati2013)). To match those two features, we set persistence of the shock to 0.93 and the standard deviation to 0.8, which imply the shock does not fade away during the simulation horizon. Those values refer to, respectively, $\rho_{\iota}$ and $\sigma_{\iota}$ . As showed in Figure 3, these two parameters from our shock specification match pretty closely actual haircuts according to debt maturity, as calculated in Zettelmeyer et al. (Reference Zettelmeyer, Trebesch and Gulati2013).

Figure 3. Actual (left panel) and Simulated (right panel) Haircuts. Source of actual haircuts: Zettelmeyer et al. (Reference Zettelmeyer, Trebesch and Gulati2013)

The baseline tax response targets the difference between taxes in the (counterfactual) steady state and the one observed in data for 2012. We target a reduction of 2.4 $\%$ on impact in aggregate taxes and calibrate $\rho_b$ for each tax rate to match this reduction. We compare the response of the economy by letting one tax component react according to the fiscal rule at a time, leaving fixed the other tax rates.

As for the face value reduction, endogenous effects might produce an initial impact that might be slightly different from the targeted 50 $\%$ drop due to endogenous response of other model’s variables when taxation is distortionary, what might affect debt price.