The representative individual’s utility from consumption c_t is as follows

$E_0{\displaystyle \underset{t=0}{\overset{\infty }{\sum }}{\beta }^t\frac{c_t^{1-\gamma }-1}{1-\gamma }}$(1)

with the parameters γ and β being relative risk aversion and the discount factor, respectively parameter and β the discount factor. Henceforth we will suppress time notation, indicating future quantities with ꞌ (e.g. c = c_t and cꞌ = c_t+1).

Production y is assumed to be of the Cobb-Douglas form

$y=a{k}^{\alpha }$(2)

with y, k, $a$ denoting output, physical capital and total factor productivity (TFP), respectively.

TFP is stochastic (and the source of uncertainty in the model) and follows the AR (1) process:

$a^{\prime }=\mu +\rho a+ϵ$ (3)

where $a^{\prime }$ denotes with $ϵ~N\left(0,{\sigma }_{ϵ}^2\right)$.

We assume that a country defaulting on its debt is excluded from the international credit market for a period of time. A further assumption is that during this exclusion period the country’s TFP is below the level it otherwise would have been. This assumption is common in the previous literature and captures the stylized facts that output during default tends to be lower ³ .

With TFP during default denoted by $a^{def}$, production during the exclusion period is

$y^{def}=a^{def}{k}^{\alpha }$ (4)

While the boundaries of TFP under non-exclusion are $a\in \left(\underset{\_}{a},\bar{a}\right)$, the boundaries of TFP under exclusion are $a^{def}\in \left(\underset{\_}{a},{\bar{a}}^{def}\right)$, where ${\bar{a}}^{def}\le \bar{a}$.

If the country honors its debt, it has access to international borrowing and lending. We denote by b the amount lent. Consequently, if b is negative, the country borrows. Bonds have one-period maturity. If the country pays back −b, it can enter a new loan contract −bꞌ. The funds raised from borrowing is q (−bꞌ), with q being the bond price. This bond price in absence of default possibilities would just have been 1/(1 + r), and given by the international interest rate r. In the case of default possibilities, risk neutral lenders seek to infer the next period’s default probability. Since k is unobservable, q will depend on bꞌ and $a$ only.

The resource constraint is then

$c=y+b-q\left(b^{\prime },a\right)b^{\prime }+\left(1-\delta \right)k-k^{\prime }-\Phi \left(k,k^{\prime }\right)$ (5)

with δ being physical capital’s depreciation rate, and $\Phi \left(k^{\prime },k\right)$ the capital adjustment cost.

Under the exclusion period, the economy only has access to one asset, physical capital, implying the resource constraint

$c=y^{def}+\left(1-\delta \right)k-k^{\prime }-\Phi \left(k,k^{\prime }\right)$ (6)

where kꞌ denotes next period’s physical capital, and δ the depreciation rate. We also assume that there is a capital adjustment cost $\Phi \left(k^{\prime },k\right)$.

At any point in time, the borrowing country decides whether to honor the debt contract (repayment) or to default. Denote the value function of repaying by $v_g^r\left(b,k,a\right)$ and the value function of default by $v_g^d\left(0,k,a\right)$. Then, the value function for this decision is

$v_g^0\left(b,k,a\right)=\underset{\left\{d,r\right\}}{\max }\left\{v_g^d\left(0,k,a\right),v_g^r\left(b,k,a\right)\right\}$ (7)

The value function in case of repayment is

$v_g^r\left(b,k,a\right)=\underset{\left\{b^{\prime },k^{\prime }\right\}}{\max }\left\{u\left(c\right)+\beta \mathbb{E}{v}_g^0\left(b^{\prime },k^{\prime },a^{\prime }\right)\right\}$ (8)

subject to

$c,k^{\prime },q\left(b^{\prime },a\right)\ge 0$ (9)

and Equation (5). Notice that $\mathbb{E}{v}_g^0\left(b^{\prime },k^{\prime },a^{\prime }\right)$ takes into account the option of default in the future (see Equation (7)).

Denoting by θ the (exogenous) probability of re-entry into the international financial market, the default value function is as follows

$v_g^d\left(0,k,a\right)=\underset{\left\{k^{\prime }\right\}}{\max }\left\{u\left(c\right)+\beta \mathbb{E}\left[\theta v_g^0\left(0,k^{\prime },a^{\prime }\right)+\left(1-\theta \right)v_g^d\left(0,k^{\prime },a^{\prime }\right)\right]\right\}$ (10)

subject to

$c,k^{\prime }\ge 0$ (11)

and Equation (6).

For tractability, we assume that the capital adjustment cost $\Phi \left(k^{\prime },k\right)$ is quadratic

$\Phi \left(k^{\prime },k\right)=\frac{\Phi }{2}{\left(\frac{k^{\prime }-k}{k}\right)}^{2}k$ (12)

Denote by Ψ the perceived (by the lender) default probability. In equilibrium, since physical capital unobservable, this perceived default probability is only a function of bꞌ and $a$. As is common in the literature, we assume that international lenders are risk neutral. This implies that they are indifferent between the gross return of 1 + r for certain and the expected return (1 − Ψ)/q from lending to the risky country. Consequently, the bond price schedule is given by

$q\left(b^{\prime },a\right)=\left[1-\Psi \left(b^{\prime },a\right)\right]/\left(1+r\right)$ (13)

In a model of this kind, where the source of uncertainty is a TFP shock, physical capital tends to its steady state level relatively quickly. As we shall see later (Section 4.2), after a TFP shock capital approaches the steady state just after a few periods (between 3 and 6 quarters), see Figure 5 and Figure 6 . Also, in the simulations, see Figure 7 , with shocks in virtually every period, capital is fluctuating around its steady state level (with no tendency to stay above or below). In absence of any other information, it is reasonable for the lender to act agnostically, as if capital was at its steady state level $k^{*}$ (“best guess”).

Consequently, we look at the equilibrium default probability of a borrower endowed with $k^{*}$. The borrower’s choice of default or repayment is the given by

$v_g^0\left(b,a;k^{*}\right)=\underset{\left\{d,r\right\}}{\max }\left\{v_g^d\left(0,a;k^{*}\right),v_g^r\left(b,a;k^{*}\right)\right\}$ (14)

The value function of repayment is

$v_g^r\left(b,a;k^{*}\right)=\underset{\left\{b^{\prime }\right\}}{\max }\left\{u\left(c\right)+\beta \mathbb{E}{v}_g^0\left(b^{\prime },a^{\prime };k^{*}\right)\right\}$ (15)

subject to

$c=a{k}^{*}{}^{\alpha }+b-q\left(b^{\prime },a\right)b^{\prime }-\delta k^{*}$ (16)

where $q\left(b^{\prime },a\right)$ is given by Equation (13).

The value function of default is

$v_g^d\left(0,k^{*},a\right)=u\left(c\right)+\beta \mathbb{E}\left[\theta v_g^0\left(0,k^{*},a^{\prime }\right)+\left(1-\theta \right)v_g^d\left(0,k^{*},a^{\prime }\right)\right]$ (17)

subject to

$c=a^{def}{k}^{*}{}^{\alpha }-\delta k^{*}$ (18)

Therefore, the sets of repayment and default decisions are given by

$R\left(b\right)=\left\{a\in A:v_g^r\left(b,k^{*},a\right)\ge v_g^d\left(0,k^{*},a\right)\right\}$ (19)

$D\left(b\right)=\left\{a\in A:v_g^r\left(b,k^{*},a\right)<v_g^d\left(0,k^{*},a\right)\right\}$ (20)

where $R\left(b\right)$ is the repayment set and $D\left(b\right)$ is the default set. The perceived default probability is

$\Psi \left(b^{\prime },a\right)={\displaystyle {\int }_{D\left(b^{\prime }\right)}f\left(a^{\prime },a\right)\text{d}{a}^{\prime }}\ge 0$ (21)

where $f\left(a^{\prime },a\right)$ is the stochastic process (equation 3) and 0 ≤ Ψ ≤ 1. When $D\left(b^{\prime }\right)$= ∅, the default probability is zero. Equation (21) is the perceived default probability from the viewpoint of the lender and gives the bond-price schedule (Equation (13)).

We present the properties of the data in Figure 1 and Table 1 below.

In Figure 1 , the top left panel plots the log of GDP and consumption against time, together with their respective linear trends. The top right panel shows the de-trended series (with mean zero). The bottom two panels plot the consumption-output ratio (c/y), the debt-output ratio (−b/y), and the risk premium. The consumption to output ratio of Argentina was 75% in 1980 Q1, and peaked at 90% in 2017 Q4. Before the late 1980s, the debt to output ratio of Argentina had never exceeded 50%. It reached its lowest value at 9.07% in 1981 Q1 and increased to a peak of 153.63% in 2002 Q3. The spikes are clustered around the period of financial crisis and recession in 1990 and 2003.

Table 1 below presents business cycle statistics for the Argentine economy between 1980 Q1 and 2017 Q4. GDP and consumption are in log and linearly

Source: Authors’ computations.

de-trended with mean zero (denoted $\tilde{y}$ and $\tilde{c}$ respectively). The ratios debt/GDP (−b/y) and consumption/GDP (c/y) are also presented in Table 1 .

The standard deviations for consumption and output are 0.0969 ( ${\sigma }_{\tilde{c}}$) and 0.0917 ( ${\sigma }_{\tilde{y}}$), respectively. Thus, consumption is 1.07 times more volatile than output ( ${\sigma }_{\tilde{c}}/{\sigma }_{\tilde{y}}$). Moreover, we observe a positive correlation between output and consumption equal to 0.9223 ( ${\rho }_{\tilde{c},\tilde{y}}$). Mean and standard deviation of the consumption/output ratio are 0.7963 (μ_c/y) and 0.0336 (σ_c/y) respectively.

The auto-correlation of output is 0.6757 ( ${\rho }_{\tilde{y}}$). The debt ratio (−b/y) has a mean of 47.67% (μ_(−b/y)) and a standard deviation of 0.2719 ( ${\sigma }_{\left(-b/y\right)}$). The correlation coefficient between output and debt ratio is negative at −0.6552 ( ${\rho }_{\left(-b/y,\tilde{y}\right)}$). This relationship has clearly been detected in 1990 and 2002, when the debt ratio increased to its peak at the same time as output experienced the largest drop for the Argentinian economy.

The debt ratio is 2.97 ( ${\sigma }_{\left(-b/y\right)}/{\sigma }_{\tilde{y}}$) times more volatile than output and 2.81 ( ${\sigma }_{\left(-b/y\right)}/{\sigma }_{\tilde{c}}$) times more than consumption. The mean and standard deviations of the risk premium are 0.1689 (μ_Ψ) and 0.1420 (σ_Ψ) respectively. The correlation coefficient between output and risk premium is negative at −0.4713 ( ${\rho }_{\Psi ,\tilde{y}}$) whilst debt ratio and risk premium have a positive correlation at 0.6953 (ρ_(−b/y,Ψ)).

Overall, the signs of the correlation coefficients for Argentina are consistent with the RBC literature and stylised facts from developing countries (see Aguiar and Gopinath, 2006 ; Yue, 2010 , among others).

We take some of the parameters from the existing literature and calibrate others. First, we discretize the continuous stochastic process for TFP to obtain a 31-state Markov chain finite state Markov chain using the quadrature method (see Tauchen, 1986 ; Tauchen and Hussey, 1991 ) ⁵ .

The possible range of values of foreign assets (the negative of debt) and physical capital in the grids are set to $\left\{\underset{\_}{b},\bar{b}\right\}=\left\{-4.5,0.5\right\}$ and $\left\{\underset{\_}{k},\bar{k}\right\}=\left\{2,13\right\}$, respectively, with the number of grids being 51 and 71, respectively. Accordingly, the foreign assets ratio lies between $\frac{\underset{\_}{b}}{\underset{\_}{y}}$ and $\frac{\bar{b}}{\bar{y}}$, whilst the capital ratio lies within $\frac{\underset{\_}{k}}{\underset{\_}{y}}$ and $\frac{\bar{k}}{\bar{y}}$, to cover all recorded values of assets and capital for Argentina, as well as avoiding corners. The 51 grid points for foreign assets are equally distributed between the interval $\left\{\underset{\_}{b},\bar{b}\right\}$, while the grid for physical capital is set to cluster around the expected steady state ⁶ . Table 2 presents the parameters taken from the existing literature on Argentina.

We calibrate the other parameters (shown in Table 3 ). The stochastic structure of TFP shock, which follows the AR (1) process, is specified by using the moment matching method to target the Argentine economy. We first target volatility of output ( ${\sigma }_{\tilde{y}}$) and consumption ( ${\sigma }_{\tilde{c}}$) and the correlation between them ( ${\rho }_{\tilde{c},\tilde{y}}$).

Secondary targets are the relative consumption and output volatility ( $\frac{{\sigma }_{\tilde{c}}}{{\sigma }_{\tilde{y}}}$), the

average consumption ratio (μ_(c/y)) and the debt ratio (μ_(−b/y)). Third target statistic is the frequency of default periods, 12.5%. The parameters in the TFP equation, ρ_a and σ_є, are found to be 0.982 and 0.014 respectively. For θ (probability of re-entry after default) in the range 1% - 20%, the value 5% fits the targets statistics well; a higher (lower) value causes a too high (low) default frequency. For the capital adjustment cost parameter, Φ, after trying several values between 0 and 5 taken from the existing literature, we found that 2.4 fits well the actual changes in physical capital.

Source: Various sources as per table.

Source: Authors’ computations.

In Section 4.3 below, we run 100,000 Monte Carlo simulations to compare the model generated statistics to the target statistics ( Table 4 ).

In this section we present the equilibrium bond price schedule, default probability, and value function ( Figure 2 ), as well as the policy functions for debt and capital ( Figure 3 ), and the default probability for different levels of capital ( Figure 4 ).

The bond-price schedules, at the steady-state level of capital, are shown in the top left panel of Figure 2 . Each schedule shows how the bond price (q) is a function of the amount being borrowed (as b′ < 0) and corresponds to a different level of TFP. These plots indicate a negative relationship between next period level of debt and the cost of borrowing (recall that debt is the negative of b and a low q corresponds to higher cost of borrowing). If a country were to borrow more, the risk premium increases in the international market and the borrower will be charged a high price (low q). The bond price schedules are plotted for different levels of TFP, a_higher, a_high, a_ss, a_low and a_lower, representing the percentage difference from its steady state level, namely 6%, 3%, 0%, −3% and −6%, respectively. The sovereign country can borrow more at a lower risk premium when TFP is higher. Consequently, a negative shock to TFP can lead to an instant increase in bond risk premium for the same amount of debt. Thereby, a large negative shock can instantly induce a sovereign country with large debt to default because of an expensive borrowing rate (higher risk premium). The default probability is shown as a heat map chart in the top right quadrant of Figure 2 .

The top right panel of Figure 2 shows the default probability (Ψ), in color (zero indicated by dark blue and one indicated by light yellow) against bond selection (b′) (with negative value being debt) and TFP (a). From this plot, we can notice that if a country intends to save with a bond (b′) above or equal to zero, there will be no default in the following period. With Ψ = 0, the country can borrow or save money at the risk-free rate (without a risk premium; see the dark blue shaded area). However, if the country intends to borrow, there will be a positive default probability (light blue to yellow) in the next period with respect to the level of TFP. With a negative shock (lowering TFP), the borrower will be charged a higher risk premium in the following period because of a higher default probability. For a default probability of 100%, the borrower will be unable to borrow on the international market.

Finally, the bottom left panel of Figure 2 shows the value functions (solid lines) with the horizontal segment left of the kink indicating default.

Figure 3 shows the policy (optimal choice) functions for debt and capital, for different levels of TFP.

The left and right-hand panels plot the optimal choice of b′ and k′ as a function of b and k, respectively. The steady state is where the policy function (at steady state TFP) crosses the 45^◦ line. The policy functions also indicate convergence to the steady state.

Figure 4 shows the three-dimensional heat map charts of the default probability as a function of the borrower’s selection of capital investment (k′) and borrowing (–b′). Choices within the yellow area result in a positive probability of default in the next period. The top layer corresponds to high capital investment, whilst the middle and bottom layers correspond to the steady state and low capital investments, respectively. From the sliced layers in the chart, we notice that the equilibrium default probability in the following period depends on the amount of next period’s capital (k′). An increase in physical capital will increase future consumption possibilities in case of a future recession. Therefore, an increase in capital investment significantly reduces the default probability in the following period, as shown in Figure 4 (the yellow area getting smaller). We should note that, since capital is unobservable, the bond price schedule is not affected.

The second part of this section analyses the impulse response functions. Figure 5 shows the response of foreign asset (b), net borrowing (−b′ + b), output (y), physical capital (k), capital investment (k′ − k), net asset (k + b), consumption (c) and net cash flow (−Δb − Δk), following a positive and a negative TFP shock of 3% (without persistence), when the economy is endowed with its steady state level of capital.

At the steady state, when a positive shock occurs in period 1, the country starts to borrow more for both consumption and investment; net borrowing becomes negative, while capital investment and consumption increase. After that, the country slightly decreases capital, repays its debt and reduces consumption until the TFP shock fades. Hence, we can deduce that a country tends to borrow for both investment and consumption during a period of positive TFP shock.

On the other hand, for a negative TFP shock in period 1, the country decides to repay with a sharp decrease in capital investment, while consumption is slightly reduced. Interestingly, Figure 5 shows that it is optimal to sell physical capital in response to a temporary negative technology shock, rather than to borrow. This result can be explained by the fact that borrowers will face a higher risk premium during a bad period (see Figure 2 ).

We also provide the impulse response functions at above and below the steady state level of capital in Figure 6 .

As shown in Figure 6(a) , the country begins with an initial level of capital above the steady state. When a positive shock occurs in period 1, the country will choose to borrow more for both investment and consumption. The country can maintain a higher level of debt while the physical capital is above its steady state. However, after period 1, the country starts to sell physical capital and repays its debt. As can be seen in Figure 6(a) , net assets continuously decline until they reach their steady-state level in period 7; the same happens to consumption. In comparison to the previous figure, consumption in Figure 6(a) can be maintained above the steady state for 6 periods. Thus, a country with capital above the steady state will sell assets for consumption.

Figure 6(b) shows opposite results for an initial capital below the steady state. With an increase in TFP in period 1, the country will start to borrow for both consumption and investment. However, as the initial capital is below the steady state, the borrower can borrow less than in the previous scenario. Hence, physical capital slightly increases after the positive TFP shock but still remains below its steady state level.

A Monte Carlo simulation over 1000 periods is shown in Figure 7 . All variables are plotted in real terms.

The simulation starts at the economy steady-state and targets the business cycle statistics of the Argentine economy, as shown in Table 1 . The overlay default band (grey area) corresponds to the sovereign default decision in the model; the dotted lines represent the steady-state level of capital and debt (foreign assets).

From Figure 7 , we can see that the government tends to default after facing a series of negative shocks or a large drop in output. After each default, the country will be excluded from the international bond market with a probability of re-entry (θ) equal to 5%. During the default periods output is lower. As can be seen in Figure 7 , debt is always zero during defaults. Furthermore, after re-entering the bond market, the government will start borrowing and consuming more for a few periods, moving towards the steady state. In addition, the government will choose to stay in the bond market (honoring the debt) in absence of significant negative TFP shocks or if the physical capital stock is large enough. During a slight drop in output, the sovereign borrower tends to sell physical capital or reduce consumption in order to repay its debt. In the presence of high debt ratio, bond price fluctuates after a negative TFP shock. Interestingly, in some periods with a series of small negative shocks, the borrower sells physical capital to repay slowly its debt. With small debt and sufficient capital, the country can remain in the international financial market during recession and wait for a positive shock.

We subsequently run 100,000 Monte Carlo simulations of 1000 periods each. We compare moments of the computed numerical results with the target moments from the actual data for the Argentine economy between 1980 Q1 and 2017 Q4 in Table 4 .

Source: Authors’ computations.

Output volatility ( ${\sigma }_{\tilde{y}}$), consumption volatility ( ${\sigma }_{\tilde{c}}$) and consumption output volatility ( ${\sigma }_{\tilde{c}}/{\sigma }_{\tilde{y}}$) are 0.0940, 0.0924 and 0.9830 respectively. The correlation coefficient between consumption and output ( ${\rho }_{\tilde{c},\tilde{y}}$) is 0.9314. Most of the results are consistent with Argentine data. In particular, the model can target well the average consumption and debt ratio ⁸ of Argentina. However, the fraction of default periods ⁹ found by our model simulation is 7.40%, while the actual data shows a value of 12.50%.

Finally, we compare other moments of the simulated data from 100,000 Monte Carlo simulations with the corresponding moments from the actual data in Table 5 .

The model generated volatilities of debt and risk premium are 0.3472 and 0.0083, respectively. The model autocorrelation of output and consumption are 0.9838 and 0.8569, respectively. Besides, the model shows a negative relationship between output and risk premium of −0.18806, whilst the relationship between consumption and risk premium is −0.2138. Model minimum and maximum values of the consumption and debt ratios are 71.28%, 95.21%, 0%, and 184.73%, respectively ¹⁰ . Overall, our model fits the actual data well.

Source: Authors’ computations.

In order to highlight our contribution by modelling unobservable capital, we make a comparison to Park (2017) , where physical capital is observable. In Park (2017) , there is an extra incentive to capital in order to increase the bond price q, encouraging over accumulation of physical capital. If capital grows large, the probability of default increases, because with large k, the cost of exclusion (autarky) is less, relatively. Consequently, there is the possibility of default when k is high. In our model this does not happen. For large k the probability of default is lowered (see Figure 4 ). In our model, as lenders estimate k, they will offer less favourable loan conditions (lower q) when actual k is high, than they would have done if k was observable, and more favourable when actual k is low. Consequently, the country’s incentive to borrow is higher (relative to a model with observable k) when actual k is low. Conversely, when actual k is high the county has an incentive of borrow less (relative to a model with observable k), meaning that debt would not be as high to move the country into the default region.