Literature Review:

Our paper is part of a growing literature that emphasizes the role of secondary markets in enforcing debts. This work has so far focused on how secondary markets restrict the actions of governments ex post, i.e. close to maturity or after defaults. Close to maturity, Broner et al. (2010) show that secondary markets both reduce the probability of default on foreigners and make it difficult for governments to discriminate among creditors. Guembel and Sussman (2009), Broner and Ventura (2010 and 2011), Brutti (2011), and Gennaioli et al. (forthcoming) show that this inability to discriminate increases the probability of both repayment to foreigners and default on domestic residents. After default, secondary markets raise the bargaining power of creditors and reduce inefficiencies. For instance, Lanau (2011) shows that renegotiations lead to smaller haircuts when debts can be sold to those agents that can extract more repayment. Pitchford and Wright (forthcoming) show that secondary markets can increase repayment by concentrating debts on the optimal number of creditors. Bai and Zhang (2012) show that secondary markets can reduce delay by providing information on creditors’reservation values. In all these cases, since governments face a time inconsistency problem, the constraints imposed on them ex post by secondary markets can be either beneficial or damaging from an ex ante point of view.

Here we focus instead on how secondary markets restrict the actions of governments ex ante, i.e. far from maturity. We model a situation in which secondary markets are open now but might fail to be open in the future, for example due to capital controls. This creates an expectation of discrimination that leads foreigners to sell their non-maturing debts to domestic residents. These purchases of government debt by domestic residents crowd out investment, reduce growth, and can increase the probability of default. In this case secondary markets also constraint the actions of governments. In particular, secondary markets make it difficult for governments to segment domestic and foreign markets when debts are issued and to control the retrading of non-maturing debts. If we assume that governments are benevolent, these ex-ante constraints are damaging to welfare.

Our paper is also related to a recent literature that analyzes how sovereign defaults affect private investment and growth. One set of papers, Aguiar et al. (2009) and Aguiar and Amador (2011), show that high levels of public debt can reduce private investment and growth by increasing governments’incentives to default and expropriate private capital. As in our model, in these papers investment is affected by the size of government debt. But the allocation of debt between foreign and domestic creditors and the potential for crowding out play no role in the mechanism. Another set of papers, Brutti (2011), Erce (2012), Mengus (2012), and Gennaioli et al. (forthcoming), show that sovereign defaults can reduce investment and growth due to their effects on private balance sheets. Unlike our model, in these papers growth is not reduced by the accumulation of debts in domestic hands as risk increases, but rather by the actual realization of defaults. So these papers’ implications for the timing of events is very different from ours.

Finally, our paper is related to the literature on self-fulfilling debt crises, notably Calvo (1988), Cole and Kehoe (2000), Corsetti and Dedola (2012), Aguiar et al. (2013), and Conesa and Kehoe (2013). Like us, these papers show that crises can be triggered by creditors becoming pessimistic and, thus, either refusing to buy the debt or demanding very high interest rates. However, the mechanism in these papers is different from ours. For example, a key message of this literature is that lengthening the maturity structure reduces countries’vulnerability to self-fulfilling debt crises. The reason is that the longer the maturity the smaller the payments countries must make if the debt cannot be refinanced, i.e. the smaller the potential “run” by foreigners. In our model, however, this is not the case because secondary markets allow foreigners to sell non-maturing debts to domestic residents. As a result, the potential run by foreigners is not reduced by a longer maturity. Instead, the degree of discrimination is what determines the potential crowding out and, thus, whether the pessimistic equilibrium exists.

2.1 The baseline model

Consider a country with a private sector that consists of generations that live for two periods. All generations have size one and contain a measure μ of patient findividuals that maximize expected consumption during old age, and a measure 1—μ of impatient findividuals that maximize consumption during youth. Thus, the patient save all their youth income and invest these savings so as to maximize their expected return. The impatient consume all their income during youth.

All generations receive one unit of labor when young, which they supply inelastically. They have access to a Cobb-Douglas technology to produce goods: $F(l_t,k_t)={l_t}^{1-\alpha }\cdot {k_t}^{\alpha };$ where l_t is employment and k_t is the capital stock and α∈ (0,1). The production of one unit of capital in period t + 1 requires the investment of one unit of the consumption good at time t. We assume that capital depreciates at a rate δ∈ (0,1), and is reversible. Factor markets are competitive and, as a result, all available factors are employed and paid their marginal products:

where w_t and r_t are the wage and the rental rate, respectively. Equations (2) and (3) already take into consideration that l_t = 1 in equilibrium.

There is a risk-neutral international financial market willing to borrow or lend at a (gross) expected return of ρ > 1. We refer to ρ as the interest rate. Here we introduce the first friction: the private sector can pledge to its creditors only a return of ϕ < ρ per unit of investment.¹⁷ As a result, it faces the following credit constraint:

where f_t is the financing or credit that the private sector recefives from the international financial market. Equation (4) says that this credit cannot exceed the net present value of pledgeable funds. Since these funds are known as of period t, the credit obtained by the private sector is riskless. If the credit constraint is not binding, the return to investment equals the rental rate plus the value of undepreciated capital, i.e. r_t+1 + 1—δ. If the credit constraint is binding, the return to investment is higher since each unit of capital can be leveraged to further expand borrowing and investment, i.e. $\frac{\rho }{\rho -\varphi }\cdot (r_{t+1}+1-\delta -\varphi )\mathrm{\cdot }$ Thus, for each unit of output invested $\frac{\rho }{\rho -\varphi }$ units of capital are produced, and each of these units of capital delivers the rental rate plus the undepreciated capital minus the financing costs.

The law of motion of the capital stock is given by:

where s ≡ μ ⋅ (1—α) is the economy’s gross saving rate and $k^{\ast }\equiv (\frac{\alpha }{\rho +\delta -1}{)}^{\frac{1}{1-\alpha }}$ is the unconstrained level of capital. Equation (5) has upward-sloping and horizontal sections, depicted as the solid line in Figure 9. The domestic private sector would like to invest until the return to investment equals the interest rate, i.e. until k_t+1 = k∗. But this investment might be unattainable if the credit constraint binds. In this case, the private sector invests as much as possible and $k_{t+1}=\frac{\rho }{\rho -\varphi }\cdot s\cdot {k_t}^{\alpha }\mathrm{\cdot }$¹⁸ Since the law of motion is globally concave, this economy has a single steady state to which it converges monotonically.

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2.2 Sovereign debt, default and crowding-out effects

We now introduce sovereign debt into the analysis. In particular, we consider a government that inherits an amount of debt d_t. The government can issue one-period bonds. Let R_t be the gross contractual rate of one-period bonds issued at t—1. Let x_t be the primary budget surplus, which equals the proceeds from consumption taxes minus (useless) government spending.¹⁹ With probability p_t ≤ 1, the institutions of the country succeed in forcing the government to pay the debt. When institutions fail, the government defaults. We follow much of the literature on sovereign debt and assume that a default, either full or partial, leads to a permanent inability to issue new debt. This means that the primary budget surplus is zero from then onwards.

Thus, we can write the law of motion of the debt conditional on not having defaulted before period t as follows:

Equation (6) shows that the evolution of the debt depends on two key variables: (i) the size of the fiscal adjustment or effort, as measured by the primary budget surplus x_t; and (ii) the quality of institutions or credibility of the government, as measured by the probability of repayment p_t. These variables play a key role in what follows.

The effect on sovereign debt on the economy depends crucially on who purchases the debt. And this, in turn, depends on what happens when the government defaults on its debt. Consider first the case in which the government defaults on foreign and domestic creditors alike. In this case, the contractual interest rate on government debt is given by:

Equation (7) says that the spread on sovereign debt must compensate for default risk. With this compensation, foreign and domestic creditors are indifferent between purchasing sovereign debt or lending in the international financial market. Interestingly, this implies that sovereign debt does not affect the law of motion in Equation (5), and, therefore, it does not affect investment and growth. If the return to investment exceeds ρ and the credit constraint is binding, the domestic private sector devotes all its resources to investment and the debt is purchased entirely by foreigners. If the return to investment equals ρ and the credit constraint is not binding, any purchase of sovereign debt by the domestic private sector replaces its lending to the rest of the world, and not investment. To be clear, the budget surplus and the probability of default determine the amount of debt that can be issued and welfare. But these variables have no impact on investment and growth.²⁰

Consider next the case in which the government defaults on the debt held by foreign creditors, but it repays the debt held by domestic creditors. This creates a wedge between the expected return to holding debt by domestic and foreign creditors. In particular, the expected return for domestic and foreign creditors is R_t+1 and R_t+1 ⋅ p_t+1, respectively. Thus, the contractual interest rate on sovereign debt depends on the identity of the marginal buyer. If this is a foreign creditor, the contractual interest rate of debt must be such that this creditor is indifferent between purchasing sovereign debt or lending. If the marginal buyer of debt is a domestic creditor, however, purchasing sovereign debt always dominates lending and the contractual interest rate must be such that this creditor is indifferent between purchasing sovereign debt and investing. This implies that the contractual interest rate is given as follows:

Equation (8) says that, if the return to investment exceeds the interest rate, the domestic private sector does not buy the debt and the marginal buyer is a foreign creditor. If instead the return to investment falls short of the contractual interest rate, the domestic private sector buys the debt and the marginal buyer is a domestic creditor.

This discussion suggests that the identity of the marginal buyer of debt, and hence the effects of debt on investment and growth, depends on the economy’s capital stock. Indeed, we can write the law of motion of the capital stock as follows:

where $\overline{k_t}$ is the capital stock at which the marginal buyer shifts from a foreign to a domestic creditor, and it is implicitly deined as follows:

Equation (9) shows the law of motion of the capital stock when default affects only foreign creditors, and is depicted as the dashed line in Figure 9. Recall that the solid line depicts the law of motion when default affects foreign and domestic creditors alike, and coincides with k₊₁(⋅; 1).

The most noticeable aspect of this law of motion is the convex region in which the solid and dashed lines do not coincide. We refer to this region as the ‘crowding-out region’, because inside it sovereign debt crowds out investment and lowers growth. If $k_t<\overline{k_t},$ this crowding-out effect is only partial, as some of the debt is held by foreign creditors. If instead $k_t\ge \overline{k_t},$ this crowding-out effect is full, as all the debt is held by domestic creditors. Figure 10 helps us build intuitions on how the size and shape of the crowding-out region depends on the stock of debt and the quality of institutions. The top panel shows the effects of changes in d_t for a fixed p_t+1, while the bottom panel shows the effects of changes in p_t+1 for a fixed d_t.

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Without the credit constraint, the crowding-out region would not exist. To see this formally, recall that as ϕ grows, the credit constraint is relaxed. Indeed, in the the limit $\varphi \to \rho$ the credit constraint becomes irrelevant. The reason is that each unit of credit, which requires payment of ρ units of goods tomorrow, allows the investors to purchase one unit of capital, which produces pledgeable funds of ρ units tomorrow. Thus, credit is unbounded. As we approach this limit, the law of motion in Equations (9)-(10) converges to:

Equation (11) simply says that, in the absence of a credit constraint, the patient young borrow enough to purchase the sovereign debt and invest until the return to investment equals the interest rate. Thus, sovereign debt does not affect investment and growth.²¹

2.3 Secondary markets and discrimination

When does the government default? When does the government discriminate between domestic and foreign creditors? There are three elements or assumptions that deine our view of sovereign default. The first one is that the institutional framework within which governments operate impedes default in normal times. This is particularly true in European countries where judicial and parliamentary systems are strong and use their power to ensure that governments do not break the law and honor the contracts they sign.

The second element is that, even in European countries, the institutional framework fails sometimes and governments can act opportunistically and default. In the previous section, this meant always defaulting on foreign creditors and we explored two alternative scenarios with and without default on domestic creditors. These scenarios are simple and clear cut but, naturally, the picture is more nuanced once we think more deeply about the implications of acting opportunistically.

When it comes to domestic creditors, the choice to repay or default revolves around the identity of domestic debt holders and taxpayers. After all, defaulting on domestic debt is equivalent to making a transfer from the former to the latter. Whether the government favors such a transfer depends on political factors that are beyond the scope of this paper. But it seems reasonable to think that these factors fluctuate over time and, as a result, so does the preference of governments towards domestic default.

When it comes to foreign creditors, the choice to repay or default depends on the magnitude of international sanctions. Defaulting on foreign debt consists of a transfer from foreign debtholders to taxpayers. The traditional approach in the sovereign debt literature is that the government dislikes this transfer and the only deterrent to it is the fear of international sanctions. Whether the government defaults on foreign creditors depends on the size of the foreign debt relative to the cost of international sanctions. Once again, it seems reasonable to think that these sanctions fluctuate over time and, as a result, so does the preference of governments towards foreign default.²²

The third element of our view is that the presence of secondary markets provides a link between the decisions to default on domestic and foreign creditors.²³ As argued above, these decisions depend on different factors and there is no ‘a priori’ reason to expect equal treatment. And yet, secondary markets make it difficult for governments to discriminate between domestic and foreign creditors.

The argument is based on the classic notion that markets transfer assets to those that value them most. And this value is derived not only from differences in risk or patience, but also from differences in the ability to be repaid. For example, imagine that the government announces a policy to repay domestic creditors and default on foreigners. Then the latter has incentives to go to the secondary market and sell their debts to the former. Competition among domestic creditors ensures that these debts are purchased at face value. When the government pays the debt, it is already in the hands of domestic creditors. And even though foreigners are not paid ‘de jure’they are paid ‘de facto.’ Thus, the government can only default on both groups or none.

Attempts at discrimination are more likely to succeed when secondary markets fail or work imperfectly. The presence of transaction costs and non-competitive behavior can reduce trade in secondary markets. Also, governments can impose capital controls to make it easier to discriminate. For these and other reasons, secondary markets sometimes fail to discipline governments and discrimination succeeds.

How should one model this rich and complex view of sovereign debt crises? We refer the reader to the papers cited in the introduction for a wide range of possible environments in which some these issues arise. Here the focus is less on the microfoundations of alternative default scenarios and more on their macroeconomic implications. Thus, we adopt the expedient device of defining the following set of repayment probabilities:

Foreign\Domestic	Pay	Default
Pay	p t+1	$p_{t+1}^F$
Default	$p_{t+1}^D$	$1-p_{t+1}-p_{t+1}^F-p_{t+1}^D$

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Foreign\Domestic	Pay	Default
Pay	p t+1	$p_{t+1}^F$
Default	$p_{t+1}^D$	$1-p_{t+1}-p_{t+1}^F-p_{t+1}^D$

There are four default scenarios. With probability p_t+1 both domestic and foreign creditors are repaid. This is the sum of the probability that institutions impose full repayment on the government and the probability that institutions fail but the government still repays in full. Also, $p_{t+1}^F$ and $p_{t+1}^D$ are the probabilities that institutions fail and the government repays, respectively, only foreign and only domestic creditors. Finally, $1-p_{t+1}-p_{t+1}^F-p_{t+1}^D$ is the probability that institutions fail and the government defaults on both domestic and foreign creditors.

A key assumption that we use throughout is that discrimination is more likely against foreign creditors than domestic ones:

This assumption is crucial in what follows since the crowding-out effects that we emphasize here are the result of a wedge between the expected return to holding debt by domestic and foreign creditors, and this wedge is in turn a consequence of discriminatory treatment against foreigners in the event of default. When this assumption fails, the domestic private sector prefers to lend abroad than to the government, and the crowding-out effects that we emphasize here vanish.

To generalize the model in the previous section to account for this richer set of probabilities, we need first to replace Equation (8) by the following one:

Equation (13) recognizes that defaults do not always discriminate in favor of domestic creditors, and that it is even possible that they sometimes discriminate in favor of foreign creditors. We also need to replace p_t in Equations (9)-(10) with the following variable:

where ${\tilde{p}}_{t+1}$ is the ratio of repayment probabilities to foreign and domestic creditors. Assumption (12) ensures that ${\tilde{p}}_{t+1}\le 1.$ These couple of simple adjustments generalize the model.

There are two additional aspects of discrimination that are worth mentioning. The first one is that discrimination might vary across different types of government debt. This might happen, for example, across currency denomination, jurisdiction of issuance, maturity, or issuing government agency. If the wedge between the domestic and foreign valuations varies across types of debt, the domestic private sector first purchases the type of debt with the highest wedge, then the one with the second-highest wedge, and so on. Ultimately, the size and shape of the crowding-out region depends on the valuation wedges and total stocks of the different classes of debt.²⁴

The second aspect is that discrimination need not happen only in the event of default. The government could offer a favorable tax treatment to domestic debtholders, or it could impose regulations on the portfolios of domestic residents that are fulfilled or relaxed by holding its debt. Whatever these additional advantages of holding domestic debt might be, they only reinforce our basic argument: that discrimination introduces a wedge between domestic and foreign valuations of debt, giving rise to crowding-out effects.

3.1 Dynamics

An interesting implication of the crowding-out region is that it gives rise to non-standard dynamics and the possibility of multiple steady states. We show this with the help of a sequence of examples in which (i) the government keeps the debt constant, i.e. x_t = (R_t–1)⋅ d_t; and (ii) the degree of discrimination also remains constant, i.e. ${\tilde{p}}_{t+1}=\tilde{p}\mathrm{\cdot }$ These assumptions are helpful because they make the law of motion time-invariant and the system can be analyzed with standard graphical tools.²⁵

Moving over the panels of Figure 11, we find a sequence of economies with progressfively higher degree of discrimination, i.e. lower $\tilde{p}\mathrm{.}$ Since increases in the probability of default raise the degree of discrimination, we use this sequence of examples to illustrate how changes in the probability of default affect the dynamics of the economy.

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Panel (a) shows an economy where the probability of default is zero. In this economy, there is no crowding-out region. There is a single and stable steady state, $k_H^{\ast }\mathrm{\cdot }$ As usual, the economy converges to this steady state monotonically.

Panel (b) shows an economy where the probability of default is small. This generates a small crowding-out region. The steady state does not change. But the crowding-out region slows down the rate of convergence towards it making the growth rate non-monotonic. Growth slows down when the country enters the crowding-out region and some of the savings that would have been invested are instead used to purchase sovereign debt. Only after all sovereign debt has been purchased by domestic creditors, additional savings are used for investment and growth picks up again. Eventually, the country exits the crowding-out region and reaches the steady state $k_H^{\ast }\mathrm{\cdot }$

Panel (c) shows an economy where the probability of default is intermediate, and so is the crowding-out region. An interesting novelty is that there are now two stable steady states $k_H^{\ast }$ and $k_L^{\ast }\mathrm{,}$ and an unstable one $k_M^{\ast }\mathrm{.}$ If the economy starts above $k_M^{\ast }\mathrm{,}$ the economy converges towards the high steady state $k_H^{\ast }$ outside the crowding-out region. Just as in the previous example, the convergence is slower and the growth rate non-monotonic. But these effects are only temporary, as the economy eventually exits the crowding-out region and reaches the high steady state $k_H^{\ast }\mathrm{.}$ If the economy starts below $k_M^{\ast }\mathrm{,}$ however, it converges to the low steady state $k_L^{\ast }$ inside the crowding-out region. As a result, crowding-out effects become permanent.

Panel (d) shows an economy with a high probability of default. The crowding-out region is so large that the high steady state no longer exists. The economy has a single steady state $k_L^{\ast }$ located inside the crowding-out region. Regardless of where the economy starts, crowding-out effects are always permanent as the economy converges towards the low steady state $k_L^{\ast }\mathrm{\cdot }$

This sequence of examples is suggestive of what might be happening in GIIPS. An increase in the probability of default has raised sovereign spreads and provided incentives for domestic creditors to purchase sovereign debt. This has created crowding-out regions. If these regions have not become too large, it might still be possible to outgrow them just by stabilizing debts, even without institutional reforms that lower the probability of default and sovereign spreads. It might take time for this to happen, of course, but GIIPS would eventually return to a high-output equilibrium. If these crowding-out regions have become too large, though, it might not be possible anymore to outgrow them without reducing debts and undertaking institutional reforms that lower the probability of default and sovereign spreads. Without these policy adjustments, GIIPS would remain permanently trapped in a low-output equilibrium.

3.2 Efficiency

Within the crowding-out region, the economy is inefficient in the traditional sense that a government that can transfer resources costlessly among domestic residents could implement a Pareto improvement. To see this, consider a given sequence of public debt ${\left\{d_s\right\}}_{s=t}^{\infty }\mathrm{\cdot }$ In any period t, the repayment of this debt is financed partly through taxation and partly through the issue of new debt, d_t+1. This new debt, in turn, can be either sold to domestic creditors, in which case we denote it by $d_{t+1}^D,$ or to foreign ones, in which case we denote it by $d_{t+1}^F\mathrm{.}$ Let y_t be the total domestic income. Then, we have that:

where

Equation (15) shows the expected total income at time t+1. This includes output and undepreciated capital, net of expected credit payments by the domestic private sector and of debt payments to foreign creditors by the government. Since we are considering what happens in the crowding-out region, the expression implicitly assumes that the private sector is credit constrained by setting its credit payments equal to ϕ ·k_t+1. It also takes into account that, in expectation, the unit cost of borrowing from foreign creditors equals ρ

Consider next the effects of replacing one unit of foreign borrowing with one unit of domestic one, while keeping the total stock of debt constant. The derivative of total expected income with respect to $d_{t+1}^D$ equals

which is negative whenever the private sector is credit constrained. Thus, increases in domestic debt reduce the economy’s expected income. The reason is very intuitive: each unit of debt sold to foreign creditors has an expected cost equal to the interest rate, while each unit of debt sold to domestic creditors reduces investment and therefore has a cost equal to the return to investment. Since the return to investment exceeds the interest rate, an increase in domestic borrowing reduces total income by replacing a cheap (external) source of funds with an expensfive (internal) one. Hence, the crowding-out effect imposes an inefficiency on the economy by reducing its total income.

Imagine that the government can adopt a policy that prevents domestic debt purchases altogether and gets rid of this inefficiency. It could do so, for instance, by eliminating the source of discrimination between domestic and foreign creditors. The gains from this policy are unevenly distributed, though. The patient young at time t lose because this policy does not affect their wage but lowers the return to their savings. The young at t + 1 gain from this policy, however, as they are born into an economy with more capital and higher wages. Because the economy’s total income increases, the gains of the young at t + 1 exceed the losses young at t. If the government could redistribute income from the latter to the former the policy could be made Pareto improving.

Outside the crowding-out region, there is no inefficiency associated with sovereign debt. If the capital stock is below the crowding-out region, the domestic private sector does not purchase any debt. If the capital stock is above the crowding-out region, the return to investment already equals the interest rate and there is no gain from reducing domestic debt holdings.

The model therefore suggests that repurchases of sovereign debt by the private sectors of GIIPS are inefficient, and policies that penalize these purchases would bring efficiency gains. The model also points at some of the political-economy problems that are likely to arise when policies of this sort are considered. The efficiency gains are likely to benefit future generations, while current generations experience loses. And the transfers that would ensure that everybody is made better of may or may not be politically feasible.

4.1 The model with endogenous defaults

Assume next that: (i) institutions succeed at time t + 1 and force the government to repay in full with probability π_t+1; (ii) if institutions fail, domestic creditors are always repaid. It follows then that ${\tilde{p}}_{t+1}=p_{t+1}\in [{\pi }_{t+1},1]\mathrm{\cdot }$ What determines the probability of (discriminatory) default 1 −p_t+1? The conventional answer is that governments repay in order to avoid costly international sanctions, such as the loss of reputation, trade embargoes or even military interventions. When these penalties are large relative to the payments owed to them, governments repay; otherwise, they default.²⁶

We introduce these considerations into the model by assuming that, for each unit of debt that is defaulted upon, the country suffers a loss equal to a fraction λ of the economy’s capital stock. This loss could be interpreted as the result of sanctions or penalties imposed by defaulted creditors, or as the resources that taxpayers have to provide in order for the government to defend itself against the legal actions of creditors. Whatever the interpretation, we assume that it is a deadweight loss that does not report any benefits to the creditors themselves.

Foreign creditors are defaulted on if the benefits of doing so exceed the costs. The benefit of a default is the reduction in expected net payments from domestic taxpayers to foreigners. If we let $d_t^F$ denote the stock of debt in the hands of foreigners at time t, that benefit is equal to $R_t\cdot d_t^F\mathrm{.}$ The costs of a default are equal to $\lambda \cdot k_t\cdot R_t\cdot d_t^F\mathrm{\cdot }$ Thus, a default on foreign creditors at time t is optimal only if

Equation (16) provides the default decision rule. Its key aspect is that default depends negatively on the capital stock and, thus, on past investment. This feature is crucial for the results to come.

To assess whether full repayment may be sustained in equilibrium, imagine that foreign creditors at time t are ‘optimistic’ and expect to be repaid in period t + 1, i.e. p_t+1 = 1. Hence, there is no crowding-out effect and capital accumulation is given by the law of motion in Equation (5), or Equation (9) for p_t+1 = 1:

To check whether these expectations are consistent with equilibrium, we just need to verify that the government repays foreign creditors at t + 1 even when institutions fail. According to Equation (16), this is the case as long as k₊₁(k_t, 1) ≥ λ⁻¹. Thus, if we define

there exists an optimistic equilibrium at time t as long as k_t ≥ k^O.

The intuition behind Equation (18) is clear. In the optimistic equilibrium the government repays foreigners because k_t+1 and, thus, the cost of default are so high that defaulting on foreigners is not worth its cost. In such an equilibrium, everyone expects the government to repay its foreign creditors: because of this, there is no crowding out effect and domestic investment is high, which raises the costs of default ex-post and validates expectations. Since k₊₁(⋅; 1) is weakly increasing, if the optimistic equilibrium exists for a given k_t it will also exist for higher levels of the capital stock. This is why there is a threshold level k^O such that the optimistic equilibrium exists if k_t ≥ k^O. If λ⁻¹ > k^∗, however, this equilibrium cannot exist: even if foreign creditors expect to be fully repaid, the cost of default is so low that government finds it optimal to default even if the unconstrained level of capital is achieved. The solid line of Figure 12 below illustrates the optimistic law of motion for the case in which λ^-1 < k*.

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The economy can also display a ‘pessimistic’ equilibrium. In this equilibrium the government repays foreigners at t +1 only when institutions succeed, i.e. p_t+1 = π_t+1. If foreigners expect to be repaid with probability π_t+1, they are less willing to buy government debt, there is more scope for crowding out, and capital accumulation is given by the law of motion of Equation (9) for p_t+1 = π_t+1:

To check whether this is an equilibrium, we need to verify that the government actually prefers to default on foreign creditors at t + 1 when institutions fail. According to Equation (16), this is the case as long as k₊₁(k_t,π_t+1) ≤ λ^-1. Thus, if we define,

there exists a pessimistic equilibrium at time t whenever k_t ≤ k^P.

The intuition behind Equation (20) is also clear. In the pessimistic equilibrium the government defaults on foreigners because k_t+1 and, thus, the cost of default are so low that repaying foreigners is not worth its cost. In such an equilibrium, everyone expects the government to default when institutions fail: because of this, the risk premium on debt is high, which leads to crowding out of domestic investment, and in turn lowers the costs of default and validates expectations. Since k₊₁(⋅, π_t+1) is weakly increasing, if the pessimistic equilibrium exists for a given k_t it will also exist for lower levels of the capital stock. That is why there is a threshold level k^P such that the pessimistic equilibrium exists if k_t ≤ k^P. The pessimistic law of motion is depicted by the dashed line in Figure 12. Because k₊₁(k, π_t+1) ≤ k₊₁(k, 1) for all k, it follows from Equations (18) and (20) that k^P ≥ k^O.

The law of motion has at least two distinct parts or sections. For low capital stocks, k_t ≤ k^O, only the pessimistic equilibrium exists. For high capital stocks k_t ≥ k^P, only the optimistic equilibrium exists. If k^P = k^O, the law of motion contains no other part or section. If k^P ≥ k^O, the law of motion contains a third additional part or section in which k^O < k_t < k^P. We refer to this section as the “crisis zone”. Within it, both the pessimistic and optimistic equilibria exist and outcomes depend on expectations. As usual, we model these expectations with a sunspot variable that takes two values, ‘optimism’ and ‘pessimism’.

The crisis zone is a subset of the crowding-out region for the simple reason that, outside this region, k^O = k^P and both equilibria cannot exist simultaneously. But the existence of a crowding-out region does not guarantee the existence of a crisis zone. If λ is either too small or too large, only the pessimistic or only the optimistic equilibria exist within the crowding-out region. For both equilibrium to coexist, λ must take intermediate values.²⁷

4.2 Dynamics with crisis zones

When the economy visits the crisis zone it might experience substantial volatility. This volatility is not caused by any fundamental shocks, but instead by fluctuations in investor sentiment. In optimistic periods, foreign creditors purchase the sovereign debt at zero spreads. The domestic private sector uses all its resources to invest. As a result, growth is high and the probability of default drops to zero. In pessimistic periods, foreign creditors are reluctant to purchase the debt and require large spreads. Thus, some (or all) of this debt is purchased by domestic creditors and this crowds out investment. As a result, growth is low and the probability of default is high. Alternance of optimistic and pessimistic periods generates volatility in spreads and the share of sovereign debt that is in the hand of domestic and foreign creditors. It also generates volatility in investment and growth.

An interesting question is whether visits to the crisis zone are permanent or temporary. And, if temporary, what determines whether the economy will exit from above and converge to a highoutput equilibrium, or instead exit from below and converge to a low-output equilibrium. We illustrate different possibilities in Figure 13 using four examples in which (i) the government keeps the debt constant, i.e. x_t = (R_t − 1) ⋅ d_t; and (ii) the quality of institutions remains constant, i.e. π_t = π. Under these assumptions the laws of motion are time invariant and the system can be analyzed with simple graphical tools.²⁸

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In all the examples of Figure 13, we depict economies that have two stable steady states, $k_L^{\ast }$ and $k_H^{\ast }\mathrm{.}$ In the high steady state, the probability of default is zero. In the low steady state, the probability of default is 1 − π. In all examples, we have chosen a value of λ that ensures that there is a crisis zone within the crowding-out region.

Panel (a) shows an example in which both steady states are within the crisis zone. Regardless of the initial condition, the economy always converges to the crisis zone. Once inside this zone, the economy never leaves it. The two steady states act as barriers that impede the economy to go below $k_L^{\ast }$ or above $k_H^{\ast }\mathrm{.}$ Thus, the economy wanders between these two values forever. In optimistic periods, the probability of default is low and investment and growth are high. In pessimistic periods, the probability of default is high and investment and growth are low.

Panels (b) and (c) show examples in which one steady state lies within the crisis zone and the other one outside. The only absorbing steady state is the one outside of the crisis zone. Even if the economy is arbitrarily close to the steady state inside the crisis zone, a sequence of investor sentiment shocks can take the economy close enough to the other steady state and outside of the crisis zone. Thus, the economy in panel (b) eventually will reach the high steady state with zero probability of default, while the economy in panel (c) will reach the low steady state with a high probability of default. Convergence towards the absorbing steady state might take time and, while in the crisis zone, the economy might be subject to an arbitrarily large number of cycles of optimism and pessimism.

Panel (d) shows an example in which both steady states lie outside the crisis zone. Thus, both of them are absorbing states. If the economy starts above the crisis zone, it converges to the high equilibrium. If the economy starts below the crisis zone, it converges to the low equilibrium. If the economy starts within the crisis zone, it will eventually exit it. But only luck determines how this happens. A long enough sequence of optimism would lead to an exit above and convergence to the high steady state. A long enough sequence of pessimism would lead to an exit below and convergence towards the low steady state.

These examples suggest another interpretation of what might be happening in GIIPS. Perhaps the deep recessions and the uncertainties that followed the financial crisis of 2007 had already placed these countries inside the crisis zone. Optimistic market expectations kept sovereign spreads low for a couple of years. The events in late 2009, and most notably Greece’s revelation that its deficits were larger than reported, caused market expectations to turn pessimistic. Looking forward, one might ask what it takes to ensure that GIIPS exit the crisis zone from above. If the high steady state is inside the zone, this might require a substantial reduction in their debts that shrinks the crowding-out region until the high steady state falls outside of the crisis zone. In the meantime, there is a role for policies that coordinate market expectations towards the optimistic equilibrium.

5.1 The model with an economic union

Assume next that the economy belongs to an economic union. For our purposes, the defining feature of the economic union is that residents of the union recefive preferential treatment in the event of default. Members of an economic union are also likely to have more integrated goods and factor markets. We abstract from these issues here, though, to keep the analysis manageable.²⁹

The government, like all other governments in the union, pays its debt to domestic creditors, to creditors in the rest of the union, and to creditors outside the union with probabilities 1, p_U,t+1, and p_t+1, respectively. Naturally, 1 ≥ p_U,t+1 ≥ p_t+1 ≥ 0. When p_U,t+1 = 1, creditors in the rest of the union are treated as domestic creditors. When p_U,t+1 = p_t+1, creditors in the rest of the union are treated as creditors outside the union. Although there are other ways to interpret this parametrization, in what follows we emphasize the probability of exiting the union. In particular, we think of 1 − p_U,t+1 as the probability that there is a default and the country exits the union, in which case creditors in the rest of the union are treated as foreign. Similarly, we think of p_U,t+1 − p_t+1 as the probability that there is a default and the country stays in the union, in which case creditors inside the union are treated as domestic. Thus, when we refer to an increase in the probability of exiting the union we are specifically referring to a decline in p_U,t+1, holding p_t+1 constant.³⁰

To determine the interest rate on the debt, we now must take into account that there are three potential marginal buyers: domestic creditors, creditors in the rest of the union, and creditors outside the union. Their expected returns to holding the sovereign debt are R_t+1, R_t+1 ⋅ p_U,t+1 and R_t+1 ⋅ p_t+1, respectively. Define ρ_U,t+1 as the lowest return to investment in the rest of the union. For further reference, we note that ρ_U,t+1 ≥ ρ since residents of the union never invest beyond the point in which the return to investment equals the interest rate. With this notation, we can write the contractual interest rate as follows:

Equation (21) shows how the interest rate varies with the marginal buyer and should be familiar at this point. The only novelty is that there is now a new set of creditors from the rest of the union. These creditors are willing to purchase the debt if R_t+1 ⋅ p_U,t+1 is as large as ρ_U,t+1.

Assume now that our economy is small relative to the union so that we can take ρ_U,t+1 as exogenous. We can then write the law of motion of the capital stock as follows:

where ${\overline{k}}_t$ is defined as

Equations (22)-Equation (23) show the law of motion of the capital stock in the presence of an economic union. A comparison of Equations (22)-(23) and (9)-(10) reveals that the economic union affects the law of motion through ρ_U,t+1. This return depends on the whole distribution of capital stocks and sovereign debts in the union.

Figure 14 plots the law of motion with and without the economic union for this case. The solid line shows the familiar case without a union which we have studied so far, while the dashed line shows the new case with a union. The figure shows that belonging to an economic union affects the law of motion of our economy in two regions.

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First, there is a range of relatively low capital stocks where the union shifts the law of motion up. This happens because the contractual rate on the debt is lower due to the lower probability of defaulting on other union members: p_U,t+1 > p_t+1 The return to investment is high in this range of capital stocks. Without the union, though, the domestic private sector would reduce investment to purchase all or part of the debt since the contractual interest rate is even higher. But the preferential treatment granted to creditors in the rest of the union lowers the contractual interest rate and reduces domestic purchases. In this range of capital stocks, our economy is effectively exporting crowding-out effects to the rest of the union. This raises investment and growth.

Second, there is a range of relatively high capital stocks where the union shifts the law of motion down. This happens because all members of the union are credit constrained and the lowest return to investment in the union exceeds the interest rate: ρ_U,t+1 > ρ. The return to investment is low in this range of capital stocks. Without the union, though, domestic creditors would use additional funds to invest since the interest rate is even lower. But the preferential treatment that domestic creditors recefive from other members of the union induces the private sector to stop investing and instead purchase debt from other union members. In this range of capital stocks, our economy is effectively importing crowding-out effects from the rest of the union. This lowers investment and growth.

The union therefore redistributes crowding-out effects from members that have low capital stocks and high debts, towards members that have high capital stocks and low debts. The importance of this re-distribution depends on how preferential the treatment of creditors within the union is. As this preferential treatment increases the crowding-out effect moves away from union members that issue the debt and towards the richer union members.

5.2 Dynamics of an economic union

The analysis so far has considered the cross-sectional implications of an economic union. To have a complete analysis of the dynamics of the union, we cannot take ρ_U,t+1 as given. As mentioned above, ρ_U,t+1 is the lowest return to investment in the union. Thus, this return must be determined endogenously together with the whole distribution of capital stocks in the union.

To show some of the implications of the theory, we consider a simple example in which the union has two sets or groups of countries with equal populations. North countries have high initial capital stocks. South countries have low initial capital stocks. Within each group there is no heterogeneity. As a result, each group can be treated as if it were a single country, and the whole union can be treated as a two-country union. We want to determine how debt affects convergence and growth within the union.

Figure 15 shows four simulations of the model. In all of them, the union starts in period t = 0 without debt. For a while, both South and North grow towards their common steady states. Since South starts out with a lower capital stock, it grows faster than North. In period t = 6, South issues an amount of debt d to inance consumption and then keeps this debt constant from then onwards. We choose parameter values such that both South and North are credit constrained even in the long run. The four panels in Figure 15 show the effects of debt on growth and convergence for different levels of debt d and the probability of a union breakup 1 − p_U,t+1.

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Panel (a) shows the dynamics of South and North when debt d is small and the probability of breakup 1 − p_U,t+1 is low. In this case, debt generates a mild recession. The recession affects North to a greater extent since South exports most of the crowding-out effects. As a result, growth slows down in the union but convergence accelerates. In the long run, South and North converge to a similar steady state with South’s debt evenly divided between South and North.

Panel (b) illustrates the case of small debt and high probability of a breakup. Since debt is small, the recession is mild as in the previous case. But the high probability of breakup implies that the crowding-out effects are stronger in the country that issued the debt. Thus, the recession affects South more than North and convergence slows down. In the long run, South converges to a steady state that is lower than North’s as most of the debt remains in South.

Panel (c) shows the case in which debt is large and the probability of breakup is low. The large debt generates a much larger recession than in the previous two cases. But the low probability of breakup means that the recession is more severe in North and convergence speeds up. The large debt also leads to a much lower steady state in both South and North. But since the probability of breakup is low the debt is held in both South and North and their steady states are similar.

Finally, panel (d) shows the case of large debt and high probability of breakup. The recession is severe and, due to the high probability of breakup, its effects are concentrated in South. Convergence falls both in the short and in the long run, as the crowding-out effects remain concentrated in South. This is the case that resembles the most what we have observed in Europe since the start of the crisis. However, it does not capture the potential role of official creditors, who can lean against these forces driving private investment decisions and affect growth outcomes. We turn to this next.

5.3 A role for transfers

If there are crowding-out effects within the union, the equilibrium is inefficient in the same sense that we discussed in section 3, i.e., if governments could transfer resources costlessly among residents of the union they could implement a Pareto improvement. Imagine, for instance, that governments adopt a policy that prevents debt purchases within the union, and get rid of this inefficiency. This policy necessarily raises the income of the union as a whole because it allows governments to borrow from foreigners at a unit cost of ρ and frees up union resources for productive investment, which has a higher return than ρ. The gains from this policy are unevenly distributed within the union, though. Countries that issue debt gain, because now they can do so at a lower interest rate. Countries that purchase debt lose despite being the ones that suffer the crowding-out effect of this debt. The reason is that now they recefive a lower interest rate on their savings. Because the union’s total income increases, though, it is always possible to redistribute income so as to attain a Pareto improvement. The interesting twist is that this requires redistribution across countries, which might be hard to coordinate among national governments.

Even if the union’s private sectors cannot be prevented from purchasing debt, they might be ‘discouraged’ from doing so through government intervention. This can be done insofar as the quality of institutions varies across union members. To see this, consider again the case of a union with two groups of countries, South and North. In South, the government: (i) keeps the debt constant and equal to d_S, and; (ii) repays domestic creditors, North creditors and creditors outside the union with probabilities 1, p_U and p_S, respectively, where p_U ≥ p_S. In North, there is no debt to begin with and the government repays all creditors with probability 1. In this case, the debt of South pays a risk premium and it tends to crowd out capital accumulation within the union, possibly in North.

Imagine that the government of North intervenes to outbid the union’s private sector and purchase the debt of South itself. It can finance this purchase by selling an amount d_N of debt to outside creditors.³¹ Because North always repays all creditors, Equation (21) says that the interest rate on its debt is R_N,t+1 = ρ. The key aspect of such a debt purchase by North is that it lowers the interest rate at which South borrows, R_S,t+1: by taking debt away from the union’s private sector, it fosters capital accumulation within the union and reduces ρ_U,t+1. Once d_N = d_S and all of South’s debt is held by the government of North, the union’s private sectors do not hold any debt and R_S,t+1 is determined by some bargaining process between both governments.³²

What are the effects of such a policy? Whenever d_N > 0, South is partly borrowing from outside creditors by using North as an intermediary. From the perspective of the union as a whole, this raises total expected income. The reason is, once again, the one we have outlined for the case of a single country: every time the government of North purchases a bond of South, it increases debt funding by creditors outside of the union (at an expected cost of ρ) at the expense of debt funding by the union’s private sectors (at a cost higher than ρ). To maximize the total resources of the union, North would therefore have to set d_N = d_S and completely eliminate the crowding-out effect.

The benefits of this policy might be unevenly distributed, though. South clearly benefits because it is able to borrow at a lower interest rate: from its perspective, d_N should be as large as possible. The effects on North, however, are ambiguous. To see this, let $d_S^N$ denote the debt of South that is held by the private sector of North. In this case, the total income of North at time t + 1 is given by

where

Equation (15) denotes the total amount of goods available to residents of North at time t + 1: it includes output and undepreciated capital, as well as debt payments received by the domestic private sector and proits made by the government of North as a consequence of its intermediation. The expression implicitly assumes that the private sector is credit constrained by setting its credit payments equal to ϕ ⋅ k_N,t+1.

By substituting Equation (25) into Equation (15), we can compute the derivative of North’s total income with respect to d_N as

Equation (26) says that an increase in d_N has opposing effects on the total income of North. The first term captures the proits made by North on each unit of debt that it intermediates, and it could be positive of negative depending on R_S,t+1. The second term captures the effect of d_N on R_S,t+1, which is always negative: as long as North is a creditor of South, the fall in R_S,t+1 reduces its income. The net effect of a transfer or debt-purchase scheme on the total income of North is thus unclear.

These results suggest that, if left to decide the size of such an intermediation scheme on its own, North might suboptimally set d_N < d_S or it might even choose not to set up a scheme at all. If North could be properly compensated for intermediating the entire stock of South’s debt, however, it would do so and fully eliminate the crowding-out effect. There is nothing in our union to prevent this compensation, though! South can always issue additional debt $d\prime$ and ‘gfive’ it to the government of North in exchange for running the optimal scheme. Because we know that the total income of the union is maximized when d_N = d_S, this compensation can always be tailored to raise the income of both countries.

The main message of this section is that an economic union maximizes its income by channeling its external borrowing through its most ‘credible’ members. To illustrate this, we have focused on the extreme case in which North always repays creditors outside of the union. This means that R_N,t+1 = ρ, so that an intermediation scheme like the one proposed can fully eliminate crowding-out effects within the union. The same logic would apply if North only repaid creditors from outside of the union with probability p_N < 1, though. The only difference is that the debt issued by North as part of the scheme would then pay an interest rate $R_{N,t+1}=\frac{\rho }{p_N}\mathrm{.}$. Although, at this interest rate, North might still be able to intermediate for South, it might not be able to fully eliminate the crowding-out effects of debt.