I. Introduction

Since 2010, financial markets have expressed recurrent concerns about risks to debt sustainability in a number of countries. One symptom of these developments is the observed pattern of eurozone members sovereign yields since 2010, as shown in Figure 1. Various bailouts and interventions have been proposed or been executed, with considerable controversy and mixed success¹. Of particular interest to this paper is the ECB President Mario Draghi’s attempt to restore confidence by pledging to do “whatever it takes” to preserve the euro zone. The ECB followed this speech with a more details and a program known as outright monetary transactions (OMT) in September 2012. The program was intended to reduce country-specific distress yields per potentially unlimited purchases of the short-term government bonds of that country. The plan was intended to lower borrowing costs in the euro zone, and avoid the dissolution of the monetary union. Yields subsequently declined, despite such purchases never taking place. While ECB Draghi stated that “OMT has been probably the most successful monetary policy measure undertaken in recent time”, it has been attacked at German constitutional court hearings in June 2013 as fiscal policy and outside the legal framework provided by the Maastricht treaty. It received a favorable ruling by the European Court of Justice on June 16th 2015, but the issue has now returned to the German constitutional court, with the latest round of hearings in February 2016. At the heart of the controversy is whether this ECB program represents monetary policy or whether it represents fiscal policy and a bailout, financed by reductions in seignorage revenue for other member countries or an inflation tax.

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10yr yield spread to Germany.

Citation: IMF Working Papers 2016, 136; 10.5089/9781475581027.001.A001

Source: Bloomberg.

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This paper is motivated by these developments. It seeks to understand the dynamics of sovereign default crisis and the potential role of a large, risk-neutral investor or agency in coordinating expectations on a “good equilibrium”, when sovereign debt markets might be prone to panics and run. The perspective proposed here can be understood as a benign version of the OMT program. In particular, we characterize the minimal actuarially fair intervention that restores the “good” equilibrium of Cole-Kehoe (2000), relying on the market to provide residual financing. “Fair value” here means that the resources provided by the bail-out fund earn the market return in expectation. We believe this is an important benchmark, shedding light on the OMT program of the ECB. The key issue in this benchmark is that the bail-out agency is able to restore the “good equilibrium” without endangering resources of tax payers in other countries, and it does so just by announcing that it is ready to step in and purchase debt at market prices. The main insight of the paper is not that the “good equilibrium” can be restored by this agency (to some, this may be fairly obvious), but rather to characterize the implications of the implementation of such a policy.

The analysis has implications beyond current events of the European debt crisis. The issue of belief coordination and the scope for policy intervention by large agencies such as the IMF or a coalition of partner countries is of generic interest. Our analysis of the dynamics of a sovereign debt crisis builds on and extends three branches of the literature in particular. First, Arellano (2008) has analyzed the dynamics of sovereign default under fluctuations in income, and shown that defaults are more likely when income is low². Second, Cole and Kehoe (1996,2000) have pointed out that debt crises may be self-fulfilling: the fear of a future default may trigger a current rise in default premia on sovereign debt and thereby raise the probability of a default in the first place. Both theories imply, however, that countries would have a strong incentive to avoid default-triggering scenarios in the first place. We therefore build on the political economy theories of the need for debt contraints in a monetary union of short-sighted fiscal policy makers as in to provide a rationale for a default-prone scenario, see e.g. Beetsma and Uhlig (1999) or Cooper, Kempf and Peled (2010).

We study a dynamic endogenous default model à la Eaton and Gersovitz (1981). This framework is commonly used for quantitative studies of sovereign debt and has been shown to generate a plausible behavior of sovereign debt and spread. The model environment consists of three agents: a single government, international lenders, and a bailout agency. The government finances its consumption with tax receipts and non-contingent long-duration bonds. Tax receipts are exogenous and stochastic. In the model, defaults can occur both from negative income shocks and coordination failures among international investors. If the government defaults on its debt obligations, it then pays an exogenous one-time utility cost of default³, it is temporarily excluded from debt markets, and it consumes its tax receipts until re-entry into debt markets. The utility cost of default is time-varying and it can be interpreted as an âĂIJembarrassmentâĂİ of default that changes from government to government. Re-entry into debt markets occurs with some exogenous probability.

We consider a bailout agency, modeled as a particularly large and infinitely lived investor and who is committed to rule out the sunspot-driven defaults of Cole-Kehoe (2000) per debt purchases, even if all other investors do not. We assume that this bailout agency seeks an actuarially fair return, and characterize the minimal intervention. The bailout agency will not prevent defaults due to fundamental reasons as in Arellano (2008) nor impose additional policy constraints such as conditionality as in e.g. Fink and Scholl (2014).

With the restoration of the fundamental equilibrium, the agency does not need to know a priori the price, at which it is prepared to buy the debt: it just needs to commit to buy at the prevailing market price, once that equilibrium is restored (and thus only needs to know that the latter has taken place). Essentially, the agency has to commit to buy only at secondary market prices eventually prevailing in equilibrium. This happens to be a central constraint on the ECB regarding sovereign bond purchases, as enshrined by the Maastricht treaty. We find that the agency needs to be willing to potentially purchase (nearly) the entire amount of newly issued debt, casting doubts on proposals that, say, seek to limit the amount the ECB can buy a priori. At that maximum, we find that a small worsening in fundamentals will make the bailout agency jump from the commitment to buy the entire amount of newly issued debt to buying no debt at all and letting the country default: the country is let-go when a future recession becomes more likely than it was. We find that the policy overall leads to higher debt levels and possibly rather small changes in the probability of default, as the probability of default for fundamental reasons is increased. Our numerical analysis shows, that changing the maturity of the debt may have little influence on default probabilities: the main change instead may be the level of debt. Our analysis is “positive”, not “normative”. The impatience of the government and its objectives may well be different from those of the population, which a social planner would take into account. On purpose, we therefore refrain from assessing the efficiency and welfare implications: these would require additional assumptions.

Our study is related to the recent literature on quantitative models of sovereign default that extended the approach developed by Eaton and Gersovitz (1981). Different aspects of sovereign debt dynamics and default have been analyzed in these quantitative studies. Aguiar and Gopinath (2006) find that shocks to the trend are important for emerging economies. Moreover, Hatchondo and Martinez (2009) and Chatterjee and Eyigungor (2012) show that long-term debt is essential for accounting for interest rate dynamics in the sovereign default framework. Hatchondo et al. (2015) study the effects of imposing a fiscal rule on debt dynamics and sovereign default risk. Also, Arellano and Ramanarayanan (2012) endogenize the maturity structure and analyze how it varies over the business cycle. Hatchondo and Martinez (2013) illustrate the time inconsistency problem in the choice of sovereign debt duration. Mendoza and Yue (2012) endogenize the output costs of defaulting. Furthermore, Benjamin and Wright (2009) introduce debt renegotiation to explain large delays observed during debt restructuring episodes. Bianchi et al (2014) illustrate the optimal accumulation of international reserves as a hedge against rollover risk. Pouzo and Presno (2014) characterize the optimal fiscal policy of the governemtn when it levies distortionary taxes and issues defaultable debt. However, these studies do not consider defaults driven by a buyers strike and the role of bailouts in eliminating self-fulfilling debt crises.

A few recent papers also analyzed the role of bailouts in models of strategic sovereign default. Boz (2011) introduces a third party that provides subsidized enforceable loans subject to conditionality in order to replicate the procyclical use of market debt but the countercyclical use of IMF loans. Fink and Scholl (2014) also include bailouts and conditionality to reproduce the observed frequency and duration of bailout programs. Juessen and Schabert (2013) include bailout loans at favorable interest rates but conditional to fiscal adjustments, and show that this could not result in lower default rates. However, these studies do not consider self-fulfilling debt crises. In a paper subsequent to ours, Kirsch and Ruhmkorf (2013) incorporate financial assistance to a multiple equilibrium default model. In contrast to our paper, they model bailouts differently: bailout loans are provided at a fixed price schedule, are senior to market debt, and are subject to conditionality. Furthermore, the scope for the bailout is not to resolve the coordination problem completely as in our paper and it does not feature the “political considerations” present in our paper. Uhlig (2013) study the interplay between banks, bank regulation, sovereign default risk and central bank guarantees in a monetary union. He shows that governments in risky countries get to borrow more cheaply, effectively shifting the risk of some of the potential sovereign default losses on the common central bank. An alternative explanation for the home bias is provided by Gaballo and Zetlin-Jones (2016), who emphasize that purchasing domestic bonds makes it harder for the domestic government to bail them out, and thus provides a commitment device to domestic banks ex ante.

Our paper is closely related to the literature on multiple equilibria in models of sovereign default, most notably Cole and Kehoe (1996, 2000), Calvo (1988), Aguiar et al (2013), Conesa and Kehoe (2013), Corsetti and Dedola (2014), and Broner et al (2014). While we share with these papers that crises can be triggered by a buyers strike, we differ in the focus of our analysis. Calvo (1988) shows that there could be multiple equilibria due to the government’s inability to commit to its inflation target. Cole and Kehoe (1996, 2000) provide a characterization of the crisis zone and optimal policy in a dynamic stochastic general equilibrium model. Aguiar et al (2013) analyze the effect of inflation credibility in determining the vulnerability to rollover risk. Conesa and Kehoe (2013) show that under certain conditions government may find optimal not to undertake fiscal adjustments, thus “gambling for redemption”. Corsetti and Dedola (2014) show that the government’s ability to debase debt with inflation does not eliminate self-fulfilling debt crises, when the government lacks credibility. In many ways, it may be the analysis most closely related to ours, however. Broner et al (2014) propose a model with creditor discrimination and crowding-out effects to show that an increase in domestic purchases of debt may lead to self-fulfilling crises.

For the Eurozone more specifically, Kriwoluzky et al. (2015) analyze the role of exit expectations in currency unions. Bocola and Dovis (2015) measure the importance of self-fulfilling crises in driving interest rate spreads during the euro-area sovereign debt crisis. Lorenzoni and Werning (2014) investigate a different type of multiplicity. They assume that the government first chooses the proceeds from debt issuances it needs, and the lenders later choose what interest rate they ask for to finance the governmentâĂŹs needs. Since higher debt levels imply more default risk and thus higher interest rates, the governmentâĂŹs needs can be financed in either a good, low-debt, low-rate equilibrium or a bad, high-debt, high-rate equilibrium. Bacchetta et al (2015) build on Lorenzoni and Werning (2014) to analyze the mechanisms by which either conventional or unconventional monetary policy can avoid defaults driven by self-fulfilling expectations.

As we do, Aguiar et al (2015) highlights that coordination failures are a significant factor in sovereign bond markets. Their model also features multiplicity of equilibria but it differs from ours by incorporating time varying probability of rollover crises and stochastic risk premium demanded by foreign investors, which seem important to account for interest rate and debt dynamics in the data. However, they do not discuss the role of a bailout agency in mitigating these coordination failures, which is the main point of our study. Moreover, an important variation with respect to the literature is our utility cost of default formulation, and the interpretation of the utility function as representing the preferences of the policy maker. These modifications allow to study sovereign defaults driven by “political considerations”, and also provide a free parameter to enhance the quantitative implications of the model.

The rest of the article proceeds as follows. Section 2 introduces the model without bailouts. Section 3 introduces and characterizes the bailout agency. Section 4 presents the numerical results. Section 5 concludes.

II. A model of sovereign default dynamics: no bailout agency

This section closely follows Cole-Kehoe (2000) and Arellano (2008). We assume that there is a single fiscal authority, which finances government consumption c_t ≥ 0 with tax receipts y_t ≥ 0 and assets B_t ∈ R (with positive values denoting debt), in order to maximize its utility

where β is the discount factor of the policy maker, u(·) is a strictly increasing, strictly concave and twice differentiable felicity function, χ_t is an exogenous one-time utility cost of default and δ_t ∈ {0,1} is the decision to default in period t. We assume that tax receipts y_t are exogenous, while consumption, the level of debt and the default decisions are endogenous and chosen by the government.

In Arellano (2008) as well as Cole and Kehoe (2000), this is the utility of the representative household, y_t is total output and c_t is the consumption of the household, i.e. the fiscal authority is assumed to maximize welfare. The structure assumed here is mathematically the same, and consistent with that interpretation. It is also consistent with our preferred interpretation, where the utility function represents the preferences of the policy maker. For example, given the uncertainty of re-election, a policy maker may discount the future more steeply than would the private sector. Spending may be on groups that are particularly effective in lobbying the government. Finally, y_t should then be viewed as tax receipts, not national income.

A more subtle difference is the cost of a default, modeled here as a one-time utility cost χ_t, while it is modelled as a fractional loss in output in Arellano (2008) with Cole and Kehoe (2000). Note, however, that c_t = y_t in default, and that at least for log-preferences, u(c_t) = log(c_t), a proportional decline in consumption each period following the default can equivalently be written as a one-time loss in utility. The stochastic utility cost formulation intends to capture the non-pecuniary costs of defaults such as reputation costs and the role of political factors in sovereign defaults episodes. For instance, Sturzenegger and Zettelmeyer (2006) argue that “a solvency crisis could be triggered by a shift in the parameters that govern the country’s willigness to make sacrifices in order to repay, because of changes in the domestic political economy (a revolution, a coup, an election, etc.)…”. The election of the Syriza government in Greece in January 2015 can be understood as electing a government that was more willing to risk a default than the previous one, and can be captured here by a change in χ_t. A similar utility cost formulation has been used in recent studies on personal bankruptcy and mortgage defaults⁴, and in the political economy literature ⁵. Technically, it provides a free parameter to fine-tune the quantitative implications of the baseline specification of the model: a feature that we exploit in the numerical analysis. We wish to emphasize, however, that introducing this political-taste feature and its stochastic variability may be quite important on economic grounds for understanding sovereign default.

In each period, the government enters with some debt level B_t and the tax receipts y_t as well as some other random variables are realized. Traders on financial markets are assumed to be risk neutral and discount future repayments of debt at some return R, and price new debt B_t+1 according to some market pricing schedule q_t(B_t+1). Given the pricing schedule, the government then first makes a decision whether or not to default on its existing debt. If so, it will experience the one-time exogenously given default utility loss χ_t, be excluded from debt markets until re-entry, and simply consume its output, c_t = y_t in this as well as all future periods, while excluded from debt markets. We assume that re-entry to the debt market happens with probability 0 ≤ α < 1, drawn iid each period, and that re-entry starts with a debt level of zero. If the government does not default, it will choose consumption and the new debt level according to the budget constraint

where 0 < θ ≤ 1 is a parameter, denoting the fraction of debt that currently needs to be repaid. The parameter θ allows to study the effect of altering the maturity structure: the lower θ, the longer the maturity of government debt. The remainder of the debt θB_t will be carried forward, with the government issuing the new debt B_t+1 − θB_t.

III. Bailouts

We now introduce the possibility for a bailout per a large and infinitely lived, risk neutral outside investor. More precisely, we envision an agency with sufficiently deep pockets, possibly backed by, say, governments other than the one under consideration here. In the specific context of the European debt crisis, one may wish to think of this agency as the ECB: given that current inflation levels are low and that a large loss may lead to recapitalization of the ECB by Eurozone member countries, an analysis in real rather than nominal terms appears to be jusified. The issue of fiscal support for the balance sheet of a central bank has recently been analyzed by del Negro and Sims (2015).

We assume that this agency aims at ensuring the selection of the “good” equilibrium, while earning the market rate of return in expectation on its bond holdings. I.e., we imagine that this bailout agency insists on actuarially fair pricing. It may well be that actual policy interventions amount to a subsidy or perhaps even a penalty. We view the actuarially fair “restoration-of-the-good-equilibrium” as an important benchmark. It might be interesting to consider other mechanisms, which are not actuarially fair, as well, and we do so in the appendix B. An alternative is to examine the conditionality of such bailouts, combining help with insistence on fiscal discipline, see Fink-Scholl (2011).

If the bailout agency buys the entire debt, then the solution is easy in principle. It should calculate the π = 0-equilibrium described above, price debt accordingly, and let the country choose the debt level it wants, given this pricing schedule. Since the bailout agency is always there, also in the future, to guarantee the “good” equilibrium, the pricing is actuarially fair.

There is generally no need to buy the entire debt, however, in order to assure the π = 0 equilibrium. We therefore assume a bailout of minimal size. I.e., we characterize the minimal level of debt $B_a^{\prime }\left(s\right)$ such a agency needs to guarantee buying at the π = 0 equilibrium price, so that markets must coordinate on this equilibrium. We assume that the agency buys at the π = 0 equilibrium price, even if the rest of the market does not buy at all: this is only relevant “off-equilibrium”. It is important in this construction, that the debt held by the agency is treated the same as the debt held by market participants⁸. The country is indifferent between purchasing this debt from the agency or from the market, and so is the market. The guarantee just needs to be there, in the (now hypothetical) case that the market coordinates on the default outcome.

To characterize the minimal guarantee level $B_a^{\prime }\left(s\right)$, we need to re-examine and slightly modify the value function of the government. We need an assumption about the continuation in the case that the market does not buy, and whether the buyers’ strike persists or not. In order to truly characterize the minimal intervention, we make the “optimistic” assumption that a potential buyer’s strike only lasts for one period, i.e., given the presence of the large investor, the continuation value following a no-default today shall be given by the value function valid for the π = 0 equilibrium.

This may be appear to be a strong assumption, at first blush. What, if the buyer strike continues longer than a single period? For that, we shall interpret the length of a period as the maximal time that such a buyer strike may last, provided there is a finite upper bound: this upper bound is then the essential assumption we are making here. With that a buyer strike then does not last more than one period by definition: changes to the interpretation of the length of a period “only” change the quantitative implications. In principle, one could conceive of a situation without such an upper bound. In that case, the bailout agency would be the ultimate long-term lender, and markets might no longer provide a guide to the appropriate terms. We exclude this extreme outcome by assumption.

Given the policy $B_a^{\prime }\left(s\right)$, define the no default value under assistance (and current buyers’ strike, except for the large investor) as

Note the second constraint, encapsulating the limit of the assistance. Let ε > 0 be a parameter and small number to break indifference. Given q^(π=0) and ν^(π=0), one can therefore solve for $B_a^{\prime }\left(s\right)$ “state by state” such that

where $\overline{B}\left(z\right)$ is the maximum level of current debt consistent with no default in the π = 0 equilibrium. For $B>\overline{B}\left(z\right)$, define $B_a^{\prime }\left(s\right)=0$, but do note, that q(B′; s) = 0 for any B′ > 0 per definition of $\overline{B}\left(z\right)$. In other words, the agency could also provide the (meaningless) guarantee of willing to buy any positive level of debt $B_a^{\prime }\left(s\right)$ at a zero price.

Proposition 1. Suppose $B_a^{\prime }\left(s\right)$ satisfies ((19)). Then, $\underset{\_}{B}(z)=\overline{B}(z)$, i.e, there will not be a default, unless debt exceeds $\overline{B}\left(z\right)$.

Proof. Suppose that $\underset{\_}{B}(z)\ne \overline{B}(z)$. Then, ((14)) and ((15)) imply that $\underset{\_}{B}(z)<\overline{B}(z)$. It follows that for every $B\in (\underset{\_}{B}(z),\overline{B}(z)),{\overline{v}}_{ND}(s=(B,0,z\left)\right)>v_D\left(z\right(s\left)\right)-\chi (s=(B,0,z\left)\right)$ and ${\underset{\_}{v}}_{ND;a}(s=(B,0,z\left)\right)<v_D\left(z\right(s\left)\right)-\chi (s=(B,0,z\left)\right)$. However, if $B_a^{\prime }\left(s\right)$ satisfies ((19)), then ${\underset{\_}{v}}_{ND;a}(s=(B,0,z\left)\right)>v_D\left(z\right(s\left)\right)-\chi (s=(B,0,z\left)\right)$ for all $0\le B\le \overline{B}\left(z\right)$, which is a contradiction.

In the iid case and with a constant embarrassment utility costs χ > 0 of defaulting, a bit more can be said. In that case, some constant value $\beta {\tilde{v}}_D$

is the continuation value from defaulting. Likewise, when receiving the full guarantee $B_a^{\prime }\left(s\right)$, the continuation value of not defaulting is$\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s\left)\right)$, given by

Criterion ((19)) becomes

comparing the current utility gain from defaulting to the utility continuation loss from defaulting, including the embarrassment cost χ.

Proposition 2. In the iid and constant-χ case, we have

1. For two states s₁, s₂, if B(s₁) > B(s₂), then $B_a^{\prime }\left(s_1\right)\ge B_a^{\prime }\left(s_2\right)$.

2. If B(s) > 0 and the default set is nonempty, then

   $q^{(\pi =0)}\left(B_a^{\prime }\right(s);s)\left(B_a^{\prime }\right(s)-\theta B(s\left)\right)<(1-\theta )B\left(s\right)$

3. For two states s₁, s₂, if y(s₁) > y(s₂), then $B_a^{\prime }\left(s_1\right)\le B_a^{\prime }\left(s_2\right)$.

4. For two states s₁, s₂, if χ(s₁) > χ(s₂), then $B_a^{\prime }\left(s_1\right)\le B_a^{\prime }\left(s_2\right)$.

Proof.

1. Suppose, to get a contradiction, that $B_a^{\prime }\left(s_1\right)<B_a^{\prime }\left(s_2\right)$. Denote the consumption level associated to $\left(B\right(s_1),B_a^{\prime }(s_1\left)\right),\left(B\right(s_2),B_a^{\prime }(s_2\left)\right)$, and $\left(B\right(s_2),B_a^{\prime }(s_1\left)\right)$ by c₁, c₂, and ${\tilde{c}}_2$ respectively. Criterion ((19)) becomes

   $u\left(c_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_2\left)\right)=v_D\left(z\right(s\left)\right)-\chi +\epsilon$
   Then, by definition of $B_a^{\prime }$, we have

   $u\left(c_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_2\left)\right)>u\left({\tilde{c}}_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)$
   Given that ${\tilde{c}}_2>c_1$, we have

   $u\left({\tilde{c}}_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)>u\left(c_1\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)$
   But, by definition of $B_a^{\prime }$, we have

   $u\left(c_1\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)=v_D\left(z\right(s\left)\right)-\chi +\epsilon$

   which is a contradiction.

2. From proposition 2 in Arellano (2008) it follows that there is no contract available {q^(π=0)(B′; s), B′} such that q^(π=0)(B′;s)(B′ − θB(s)) − (1 − θ)B(s) > 0. The definition of our minimal guarantee implies that $B_a^{\prime }\left(s\right)\le B^{\prime }$. Thus, the contract $\left\{q^{(\pi =0)}\right(B_a^{\prime }\left(s\right);s),B_a^{\prime }(s\left)\right\}$ is available to the economy and it must be the case that $q^{(\pi =0)}\left(B_a^{\prime }\right(s);s)\left(B_a^{\prime }\right(s)-\theta B(s\left)\right)<(1-\theta )B\left(s\right)$.

3. Suppose, to get a contradiction, that $B_a^{\prime }\left(s_1\right)>B_a^{\prime }\left(s_2\right)$. Denote the consumption level associated to $\left(y\right(s_1),B_a^{\prime }(s_1\left)\right),\left(y\right(s_2),B_a^{\prime }(s_2\left)\right),\left(y\right(s_2),B_a^{\prime }(s_1\left)\right)$, and $\left(y\right(s_1),B_a^{\prime }(s_2\left)\right)$ by c¹, $c_2,{\tilde{c}}_2$, and ${\tilde{c}}_1$ respectively. By definition of $B_a^{\prime },B_a^{\prime }\left(s_1\right)>B_a^{\prime }\left(s_2\right)$ implies

   $\begin{array}{l}u\left({\tilde{c}}_1\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_2\left)\right)<u\left(c_1\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)=u\left(y\right(s_1\left)\right)+\beta {\tilde{v}}_D-\chi +\epsilon \\ u\left({\tilde{c}}_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)>u\left(c_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_2\left)\right)=u\left(y\right(s_2\left)\right)+\beta {\tilde{v}}_D-\chi +\epsilon \end{array}$
   Also, by concavity of the utility function and part 2 of this proposition, we have

   $\begin{array}{c}u\left(y\right(s_2\left)\right)-u\left({\tilde{c}}_2\right)>u\left(y\right(s_1\left)\right)-u\left({\tilde{c}}_1\right)=\beta \left({\tilde{v}}_{ND}\right(B_a^{\prime }\left(s_1\right))-{\tilde{v}}_D)+\chi -\epsilon \end{array}$

   This implies that $u\left(y\right(s_2\left)\right)+\beta {\tilde{v}}_D-\chi +\epsilon >u\left({\tilde{c}}_2\right)+\beta {\tilde{v}}_{ND}\left(B_a^{\prime }\right(s_1\left)\right)$, which is a contradiction.

4. This follows from criterion ((19)). □