Households’ Savings, HIV/AIDS and Banking Stability in Developing Countries Patrick L. Leoni∗

Abstract We argue that the recent large drops in households’ savings in developing countries with high HIV/AIDS prevalence is associated with the spread of the disease. We also argue that the need to pay for individual treatments force large-scale withdrawals of households deposits, and that those large withdrawals put the banking industry at risk. In a standard demand-deposit model where the HIV prevalence among depositors is random, we show that 1- the probability of a large-scale banking failure without bank run increases as the odd of any prevalence level increases, and 2- it is always optimal to deposit, and thus to accept the risk of banking failure, to maintain long-term investments in place.

∗

University of Southern Denmark, Department of Business and Economics, Campusvej 55 DK-

5230 Odense M, Denmark. E-mail: [email protected], and EUROMED School of Management, Domaine de Luminy - BP 921, 13 288 Marseille cedex 9, France.

1

1

Introduction-last paragraph must be simplified

The spread of HIV/AIDS has been dramatic since the early 90s, in particular in SubSaharan countries. Roughly since this period, those countries experienced a sharp decline in households’ savings as documented later. Several reasons at national level have been given so far, ranging from high unemployment to high inflation (see South African Savings Institute [12] for South Africa). We argue that HIV/AIDS is another significant factor explaining the large drop in households’ savings in developing countries with high prevalence, and that in turn the lack of savings and early withdrawals caused by the disease put at significant risk the banking industry in those countries. In a theoretical framework related to that in Diamond and Dybvig [2], we show that the higher the likelihood of prevalence (in the sense of Firs-Order Stochastic Dominance) the higher the risk of widespread banking failure without bank run caused by the severe strain on reserves, and regardless of the odds it is always socially optimal to deposit in banks to maintain active long-term investments. The point is that most of the treatments costs in those countries are out-ofpocket health expenditures, and the cost is typically high (even for standards in developed countries, the annual cost of treatments in 2001 for an AIDS patient in the U.S. was between U.S.$12,000 and U.S.$20,000, Binswanger [1]). Despite foreign aid which amounted to 60% worldwide in 2005 (Lewis [10]) of the overall HIV/AIDS budgets, and the huge strain on public budget caused by the disease (up to 1.6 % of consolidated national expenditures in Nigeria in 2006 for instance, Hickey [6]), solely 15% of the infected population worldwide had access to public care in 2005 (Lampey et al. [8]. When experiencing significant symptoms requiring treatments, those patients will withdraw long-term investments through deposits for instance. In countries like South Africa where the prevalence can reach 35% of the adult population

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in some areas (see next section for aggregate prevalence), the resulting widespread withdrawals put at significant risk local banks that have invested in long-term plans with insufficient reserves to handle this problem. Even without reaching this extreme withdrawal threshold where banks can no longer be solvent, the fact that reserves are running out because of HIV/AIDS related withdrawals sends bad signals to outside investors about the soundness of the banking industry, with the possibility of widespread panic. The anticipation of this situation may force banks to call back previously issued loans, leading to an economic slowdown, and even to outside bankruptcies when those funds are be brought back to banks. In Section 3, we develop this point in a standard theoretical model, where we will show that widespread banking failure may occur solely from immediate need for cash because of AIDS morbidity, and independently of widespread panic effects as typically argued. Many studies have addressed the economic impact of HIV/AIDS in developing countries with high prevalence, although none has pointed that the disease puts a severe strain on the soundness of the domestic banking industry, in particular that in charge of handling deposits. Haacker [5] gives a broad view of the macroeconomic effects of the disease, whereas Johansson [7] studies more specifically the optimality of fiscal policies under high prevalence. Young [14] and SantaEulalia [11] are more closely related to our work. Young [14] argues that the epidemic has a detrimental impact on the accumulation of human capital, and that a large infection rate lowers fertility and increases the scarcity of labor. Even when considering low educational attainments, typical when more children become orphans and the number of teachers is lowered as argued earlier, Young finds that the fertility effects dominate and consumption per capita increases. On the other hand, SantaEulalia [11] argues that the lack of accumulation human capital defers the industrial transition that those developing countries need to catch up with developed ones, and the economic effects of the 3

disease are overall negative. In contrast, we argue that the need to maintain active long-term investments for future growth is so important that agents find it optimal to deposit regardless of the odds of banking failure caused by HIV/AIDS or any other economic conditions. Other effects of bank runs on economic growth have also been pointed out in Ennis and Kiester [?], although runs are triggered by sunspots in this model. In more details, we develop a theoretical framework to make our point, where standard contracts and long-term investments are modelled as in Diamond and Dybvig [2]. The fraction of infection population is random, and it is unknown at the time the rates are set by the banks (this is conceptually equivalent to setting up reserves). There are two types of equilibria: either every agent anticipates a bank failure and thus does not deposit, or every agent deposits and a bank failure occurs with strictly positive probability. This probability of failure is an increasing function, in the sense of first-order stochastic dominance, of the probability of infection. We also show that individuals’ welfare is strictly higher in the second class of equilibria, despite the risk of collapse of the banking system. The point is that the relative benefit of using the long-term productive technology is greater than the possibility of losing one’s income in case of failure. The insight of those results is that, when individuals decide to deposit, the risk in the banking industry stems from the possibility of a too high demand for withdrawals because of a sudden increase in infections. When the infection rate increases, more individuals need to withdraw early to pay for their treatments (they become infected), leading in turn to the collapse of the banking system. This possibility exists in every equilibrium in our fairly standard model, showing the importance of this phenomenon and the permanent threat to the banking system. The paper is organized as follows. In Section 2, we give the main stylized facts in developing countries with high prevalence and develop the implications for the banking industry; in Section 3, we develop our theoretical model and present our 4

main results; Section 4 contains some concluding remarks and the technical proofs are given in the Appendix.

2

Stylized facts and the consequences for banks

We now present the stylized facts about the decline of households’ savings and the spread of HIV/AIDS in South Africa since the early 90s. We focus on this country for two reasons: it has the largest HIV prevalence worldwide, and arguably the most reliable source of data of all Sub-Saharan countries. The economic and medical situations seem comparable in other Sub-Saharan countries. The first cases of HIV were detected in the early 80s in South Africa, but the pandemic spread fast and the prevalence became significant after the turn of the 90s. Fig. 1 give the basic facts about the spread of the virus after the 90s. The figure shows the rapid prevalence growth, which seems to have peaked after 2005. The adult population is the most severely affected, and it is this class of the population that typically saves the most for retirement purposes for instance. We next discuss the evolution of savings in South Africa over a longer horizon to make comparisons relevant with the spread of the disease. Fig. 2 gives the households’ savings as a fraction of disposable income, and the domestic savings as a fraction of GDP since the 60s. The striking aspect is that the large drop in households ’ savings is strongly correlated with the increase in HIV prevalence since the 90s, and so is the percentage of domestic savings as a fraction of GDP. Despite some slowdowns in households ’ savings prior to the occurrence of the pandemic, it seems at least visually that the three phenomena are related. A casual test for Granger-causality give some mild evidence that HIV has caused the drop in savings, but the results are not reliable because of the too-small sample. There are other macroeconomic factors that could have caused this drop, such as the high unemployment rate and the high inflation 5

(a) HIV Prevalence among adults (15-49)

(b) Population living with HIV

Figure 1: HIV prevalence in South Africa. Source: UNAIDS/WHO [13]. rate in South Africa, although the facts above strongly suggest that HIV/AIDS is part of the picture. The causal link between HIV and the stability of the banking industry may be explained as follows. When the disease strikes a household, previous savings must be withdrawn to compensate for an overall reduction in family income. The reduction income is two-fold, first most individuals will have to pay for their own treatment as explained earlier, and at late stage of the disease the morbidity is so high that an individual must abandon any professional activity. When a significantly high fraction of households are affected this way, the overall withdrawal level may exceed the reserve level held by banks and the need to forego long-term investments to cope with this has long-term negative consequences. With a roughly 20% prevalence among the adult population, thus the most productive and more prone to save, the causality effect of HIV/AIDS on savings and in turn on the fragility of the banking industry, is consistent with the facts above. 6

(a) Households’ savings as a fraction of dis- (b) Domestic savings as a fraction of GDP posable income

Figure 2: Savings in South Africa. Source: South African Savings Institute [12]. The consequences for banks are described next. Banks keep in reserve only a small fraction of deposits, and most of those deposited funds will be made available to investors for long-term investments and typically illiquid projects. By doing so, and therefore by using deposits for long-term and productive investments available in financial markets, economic growth is fostered. Large-scale withdrawals caused by HIV/AIDS force banks to call back some of those loans, when possible, to alleviate the severe strains on reserves and the potential insolvency when reserves run out. Whether the benefits of those long-term investments offset the possible bankruptcy losses resulting from an unexpectedly high level of withdrawals is discussed next. We consider a theoretical model similar to that in Diamond and Dybvig [2], in order to formalize this point. Every depositor is rational and capable of strategic behavior. We show that two possible types of equilibrium can occur: either every depositor anticipates bank insolvency and no one deposits in the bank, an outcome that may be regarded as the anticipation of a standard bank run by rational agents, or alternatively every agent deposits, no depositor panics and the bank fails when the number of

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withdrawals is too high (a typical case would be an underestimation of the AIDS incidence among the depositors when setting up bank reserves). Given the data on savings in South Africa above, the second type of equilibrium above is likely to occur. More and more agents either forfeit savings or withdraw prematurely, a behavior consistent with the observed decrease in savings. The prediction of our model is that when the fraction of withdrawals becomes too large because of AIDS, the whole banking industry will be at significant risk; or equivalently, the higher the prevalence the more at risk the banking industry.

3

A model of banking failure linked to HIV/AIDS

We now formalize the idea that an unexpectedly high increase in withdrawals of deposited funds, caused by a high incidence of AIDS in a given country, may cause the collapse of the banking system independently of a bank run. This theoretical framework will also allow us to evaluate the theoretical probability of banking failure as a function of the HIV prevalence, and to identify other triggering factors of failure such as widespread panic.

3.1

The model- add references on banking

Our model is taken from Leoni [9] Ch. 4, and it is very similar to the one presented in Diamond and Dybvig [2]. For sake of simplicity, we will focus on the case where banks cannot suspend payments when too many individuals attempt to withdraw. A similar result holds in the other case, as shown in Leoni [9] Section 4.4. The model has three periods (T = 0, 1, 2), and a single consumption good. In period 0, there is a continuum of non-infected individuals represented by the interval [0, 1]. Every individual is endowed with 1 unit of consumption good in period 0, which will be invested in various assets described later to smooth out consumption 8

and to hedge the risk of income loss if getting infected in period 1. Every individual can be of two distinct types, denoted by type 1 and type 2. A type 1 individual values consumption in period 1 only, this type of individual becomes impatient because of AIDS infection or any other economic reason justifying the immediate need for cash. In contrast, a type 2 individual values consumption in periods 1 and 2, which can be interpreted as an individual who did not get infected for instance. In period 1, nature draws a type for every individual that corresponds to a HIV/AIDS infection or not. A type 1 individual is infected, and thus has a lower life expectancy. We model this issue by assuming that this individual values consumption in period 1 only, and dies in period 2. In contrast, a type 2 individual is not infected and thus values consumption both in periods 1 and 2, which corresponds to a longer life-spam. A fraction t ∈ [0, 1] of the individuals will be of type 1; that is, a fraction t will become infected and will need immediate cash for period 1 only. The random number t is drawn by nature according to a probability distribution f over [0, 1]. We assume that f is continuous, with f (1) = 0. We assume that the associated cumulative distribution F is such that F (t) < 1, for all t ∈ [0, 1); this last assumption captures the idea that a high fraction of infected people is relatively unlikely even if possible. The individuals are equally likely to become infected in period 0, and the density of probability f is common knowledge among the individuals. In period 1, every individual privately learns whether infection occurred, or her type. To hedge against the risk of infection and thus to manage the need for consumption in a particular period, the individuals can use their initial endowment in three different manners. First, they can deposit in period 0 part of their endowment in a bank. The return of any deposit to the bank will be described later. Second, every individual can privately store a quantity of consumption good of her choice in every period, in order 9

to consume it in the next period. The storage is costless, and provides no return to the individuals. Finally, the individuals have access to a competitive complete market for claims on future goods, which is open in every period. With the deposit from the individuals, the bank uses a long-term investment technology exhibiting constant returns to scale. Formally, one unit of consumption good invested by the bank in period 0 yields R (R > 1) units of consumption good in period 2. If the investment is withdrawn in period 1, the salvage value will be exactly the value of the investment. In the financial market, it can be shown, as in Diamond and Dybvig [2], that the period 0 price of period 1 consumption is 1, and the period 0 price of period 2 consumption is R−1 to avoid arbitrage. Let c1 (resp. c2 ) denote individual consumption of an individual in period 1 (resp. period 2), and let Θ be the type of the individual. The utility derived from the consumption of the bundle (c1 , c2 ), as a function of her type, is   u(c ) if Θ = 1 1 U (c1 , c2 , Θ) =  u(c1 + c2 ) otherwise

(1)

where u : <+ → < is twice continuously differentiable, strictly increasing, strictly concave, and satisfies the Inada conditions u0 (0) = ∞, and u0 (∞) = 0. Similarly to Diamond and Dybvig [2], solely for technical reasons and without a significant loss of generality, we assume that −cu00 (c)/u0 (c) > 1 for c ≥ 1. We also normalize units so that u(0) = 0. Every individual is assumed to maximize the ex-ante expected utility E[U (c1 , c2 , Θ)], where the expectation represents the risk of being infected. We next describe the banking industry in more detail. There is a large number of banks that behave competitively. The banks offer demand deposit contracts to their depositors; that is, the banks offer the depositors a contract specifying a fixed claim of r1 per unit deposited to depositors withdrawing in period 1. The banks are mutually owned, and they are liquidated in period 2. This implies that period 2 10

withdrawer will equally share among themselves the remainder of the banks assets. Those contracts are the most commonly used in practice. The banks also satisfy a sequential service constraint; that is, the banks serve the individuals withdrawing in period 1 in the order that they arrive at the bank until the bank runs out of assets. This assumption is essential since it captures the possibility of a bank run or partial suspension of payments as an equilibrium. In order to formally capture the idea of sequential service constraint, denote by A the total amount of deposits in period 0, and consider an individual j ∈ [0, 1] willing to withdraw in period 1. Let fj denote the fraction of period 1 withdrawers arriving at the bank before individual j, and let V1 denote the period 1 payoff per unit deposit to this individual j. When implementing a demand deposit contract as above, the bank has in reserve an amount of funds r1−1 A. We thus have that   r if f < r−1 A 1 j 1 V1 (fj , r1 ) =  0 otherwise. Let now V2 denote the period 2 payoff per unit deposit not withdrawn in period 1, and let f be the number of demand deposits withdrawn in period 1. There are 2 possible cases: the withdrawn funds have exceeded the bank reserves and the bank is bankrupt, otherwise the period 2 claimants share the profits. We thus have that   R (A − r1 f ), 0 . V2 (f, r1 ) = max 1−f Denoting by wj the fraction of deposit withdrawn in period 1 by individual j (j ∈ [0, 1]), and assuming that individual j deposits all of her endowment in the bank, the overall payoff to individual j is then given by wj V1 (fj , r1 ) + (1 − wj )V2 (f, r1 ). The banks choose r ≥ 0 in order to maximize the ex-ante utility of the individuals; this behavior is a consequence of the competitive nature of the banking industry. Moreover, risk aversion will ensure that the optimal period 1 rate r is in [1, R]. 11

Individuals will individually choose whether to deposit at a bank, for all possible interest rates the banks offer. If they deposit nothing, it is straightforward to see that they will invest in the financial market. Also, for every possible interest rate, and deposit choice, an individual will choose either to withdraw in the first period or to wait, conditional on her type and others’ strategies. Formally, an individual chooses (d, w), where d is a function from [1, r] into {0, 1} representing the decision to deposit at the bank as a function of the offered rate r,1 and w is a function from [1, R] × {0, 1} × Θ into {0, 1} representing the decision of withdrawing in period 1 as a function of the rate offered by the bank, her previous decision to withdraw and her infection type. We make the convention that d(r) = 1 stands for the choice of depositing, and similarly w(r, d, Θ) = 1 means that she will withdraw in period 1. A bank failure occurs when the fraction of funds withdrawn in period 1 exceeds the banks reserves in this period. In order to evaluate different strategies, every individual needs to estimate the probability of arriving at the bank before it fails, conditional on an anticipated failure. Conditional on the fraction t of infected individuals in the population, this probability depends on the interest rate r offered by the bank, on the fraction s at which the bank suspends withdrawals (if allowed), and on the strategies chosen by the other individuals. We focus only on symmetric strategies; that is, we focus on common strategies that every identical agent will choose. It is thus enough for an individual to estimate the probability above in the three following cases: 1) when all individuals withdraw in period 1, 2) when only infected individuals withdraw in period 1, and 3) when she alone withdraws in period 1 in addition to all of the infected individuals — these functions will be denoted respectively by αa (r, s|t), αi (r, s|t), and α1 (r, s|t). We assume that for all y = a, i, 1, the function αy is continuous with respect to t, is 1

This decision is binary without loss of generality; that is, all of the endowment is invested or

nothing, and this can be easily justified by arbitrage considerations on the financial market.

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differentiable with respect to r and s with bounded partial derivatives, and that Z 1 αa (x|t) f (t)dt = 1. x 0 A Nash equilibrium is then r∗ for the bank, and (d∗ , w∗ for every individual such that w∗ (r, d, Θ) is optimal for all (r, d, Θ), d∗ (r) is optimal for all r, and r∗ is optimal taking as given individuals’ strategies. We can refine this equilibrium notion as they occur at macro-economic level. An autarkic equilibrium is a Nash equilibrium as above such that no individual decides to deposit in period 0, typically as a consequence of an anticipated bank failure.

3.2

Equilibrium behavior

We next characterize the symmetric equilibria of this game, and we study their properties. We first show that such equilibria can be of two distinct types: either individuals avoid depositing (an autarkic equilibrium resulting from the anticipation of a bank failure), or they all deposit and non-infected individuals wait until the last period. For sake of simplicity, we carry out the analysis with the assumption that every fraction of infected individuals can be drawn by nature with strictly positive probability. This assumption can be relaxed in order to get the same result (see Leoni [9] Section 4.6 for the details). Proposition 1 There are two classes of symmetric equilibrium: 1. Autarkic equilibria. No individual deposits; i.e., d∗ (r∗ ) = 0. 2. Non-autarkic equilibria. Every individual deposits, every non-infected individual waits, and the interest offered by the bank strictly exceeds 1; i.e., r∗ > 1, d∗ (r∗ ) = 1, and w∗ (r∗ , d∗ (r∗ ), 2) = 0.

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Proposition 1 shows that, in every equilibrium, either every in every agent anticipate a failure and thus do not deposit, or otherwise in every non-autarkic equilibrium, every agent deposits and the equilibrium rate is greater than 1. Since the interest rate is greater than 1, a bank failure may occur with strictly positive probability; despite the risk of a bank failure, everyone finds optimal to deposit in period 0 their endowment in the bank to have access to the long-term productive technology. We also have the following corollary, which shows that non-autarkic are more likely to occur if individuals are capable of coordinating themselves on the most socially beneficial equilibria. Corollary 2 In every non-autarkic equilibrium, the individuals’ ex-ante expected utility is strictly greater than in any autarkic equilibrium. Furthermore, in every nonautarkic equilibrium, the banks fail in the first period with positive probability given by 1 − F (1/r∗ ). Corollary 2 shows that individuals are better off depositing rather than staying in autarky, regardless of the risk of the failure as described here. Another interesting result is that the risk of banking failure is a direct function of F , the cumulative probability of having a fraction of the population infected. It is also straightforward to derive that the higher the likelihood of infection of a given fraction of the population, the greater the odds of banking failure. Goldstein and Pauzner [4] find a similar result in a different setting. Both Proposition 1 and Corollary 2 will be proven in the Appendix. More severe penalties in case of bankruptcy than the one used here would lead to the similar conclusion that the long-term benefits are greater than short-term risk of failure in equilibrium. In this same spirit, the conclusion that non-autarkic equilibria yield a higher social welfare than autarkic equilibria is natural because the long-term investment technology is used despite the risk of failure, something that is impossible 14

in autarkic equilibria where the banking industry is anticipated as too risky and therefore ignored by the individuals.

4

Conclusions

We have argued that the spread of HIV/AIDS in developing countries with high prevalence has a causal effect on the sharp drop household’ s savings, and in turn that the disease puts a severe strain on the stability of the banking industry. From a purely theoretical viewpoint, we have shown that two types of equilibrium behavior can occur. Either every individual anticipates insolvency and no individual consequently deposits, or every individual deposits with the following action: no individual panics (or in other words, no individual withdraws earlier than when its type demands) and a bank failure occurs with positive probability without occurrence of a bank run. No bank run has yet been seen in Sub-Saharan countries, regardless of any signs of withdrawals, and this fact is also consistent with our model because we show that savings may drop without always triggering banking failure. Bank failures are thus shown here to possibly stem from an uncontrolled spread of AIDS, and in particular from the financial pressure that the morbidity of the disease puts on households and the resulting need to call in any previous savings. Since banks’ reserves may not be significant enough to cope with large amounts of withdrawals, banks always face the risk of distress as the disease grows in magnitude. It turns out that accepting this risk is socially optimal under our assumptions, since the alternative prospect of forgoing long-term investments is worse. Further works may involve the same theoretical analysis under the optimal deposit contract, instead of the standard deposit contracts. Adding partial suspension of payments, as often seen in practice, does not change the qualitative results.

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A

Proofs

We now prove Proposition 1 and its corollary. Since every infected individual will withdraw in period 1, we have only four equilibrium behavior in a symmetric equilibrium: 1. No individual deposits, and non-infected individuals withdraw in period 1, 2. No individual deposits, and non-infected individuals withdraw in period 2, 3. Every individual deposits, and non-infected individuals withdraw in period 1, 4. Every individual deposits, and non-infected individuals withdraw in period 2. It is easy to see that both Cases 1 and 2 above form an equilibrium for any possible interest rate offered by the bank – these two cases correspond to the autarkic equilibria in Proposition 1. Case 3 cannot be an equilibrium; this is true because any individual would prefer not to deposit in period 0 should all of the others follow the strategy described in this case. Thus, in order to complete the proof of Proposition 1, we show that there exists an equilibrium in which non-infected individuals wait, and that all of the equilibria of this kind have the properties described in Corollary 2. We first construct an equilibrium in which every individual deposits, and non-infected individuals wait to withdraw in period 2. Assume now that the bank offers a rate r, every individual deposits and non-infected individuals wait for the second period to withdraw. The expected 
R utility for a non-infected individual is given by [0,1] u max R 1−tr , 0 f (t)dt. If 1−t one non-infected individual decides to withdraw in period 1 when all of the other non-infected individuals withdraw in period 2, then her expected utility is given by R u(r)α1 (r|t)f (t)dt. [0,1]

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Thus, letting W =

n

r ∈ [1, R] :

R

o  1−tr
R , u max R , 0 f (t)dt ≥ u(r)α (r|t)f (t)dt 1 1−t [0,1] [0,1]

we have that, as long as r ∈ W , Case 4 is a candidate for an equilibrium because there is no incentive for unilateral deviation in this case. From the above, in Case 
4 a non-infected individual receives a utility of u max R 1−tr , 0 , and an infected 1−t individual receives a utility of αi (r|t)u(r), when the fraction of infected individuals is t. Thus, the ex-ante expected utility of any individual in this type of equilibrium is     Z 1 − tr ,0 f (t)dt. U (r) := tαi (r|t)u(r) + (1 − t)u max R 1−t [0,1] If an individual decides not to deposit, then her ex-ante expected utility is simply R1 [tu(1) + (1 − t)u(R)] f (t)dt. 0 n o R1 Hence, letting D = r ∈ [1, R] : U (r) ≥ 0 [tu(1) + (1 − t)u(R)] f (t)dt , we see that any individual will choose to deposit provided that r belongs to D, all other individuals deposit, and non-infected individuals wait until period 2 to withdraw. Finally, if the bank offers r∗ which maximizes U in the set D∩W , we can construct ∗ a symmetric equilibrium in the following way: the bank   offers r , and the individuals  1 if r ∈ D ∩ W  0 if r ∈ D ∩ W and d = 1 choose d∗ (r) = , w∗ (r, d, 2) = ,  0 otherwise,  1 otherwise, and w∗ (r, d, 1) = 1 for all (r, d). In order to construct the above strategy, we first

need to show that such a r∗ exists. This is done in the following lemma. Lemma. The function U has a maximizer in D ∩ W . Proof. Note that the set D ∩ W is compact, and non-empty, since r = 1 belongs to D ∩ W . The function U is a continuous function of r. Hence, there exists r∗ that maximizes U in D ∩ W . The proof is now complete.

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We have shown so far that there are two types of equilibria: in one type nobody deposits, and in the other type everyone deposits, and non-infected individuals withdraw in period 2. In order to complete the proof of Proposition 1, we are left to show that in every equilibrium of the second type, we have r∗ > 1. This result will establish that failure occurs when the fraction of withdrawing individuals is greater than 1/r∗ . Indeed, the bank has to pay r∗ t in the first period when the fraction of withdrawers is t, and thus the critical value of t above which the bank is no longer solvent is 1/r∗ . Notice also that r∗ > 1 implies a strictly positive probability of a bank failure, since the probability of a bank failure is exactly the probability that the fraction of individuals withdrawing in period 1 is at least 1/r∗ ; this probability is simply given by 1 − F (1/r∗ ) > 0 as previously claimed. Lemma. In every symmetric equilibrium such that every individual deposits, and all non-infected individuals wait, we have r∗ > 1. Proof. It suffices to show that there exists r˜ > 1 such that r˜ belongs to D ∩ W , and U (˜ r) > U (1), which in turn implies that r∗ > 1. Let W L(r) (resp. W R(r)) denote the left-hand (resp. right-hand) side of the inequality defining the set W . Since W L(1) = u(R) > u(1) = W R(1), we conclude that there exists a ball B around 1 contained in W . Note that D = {r ∈ [1, R] : U (r) ≥ U (1)}. Therefore, to prove existence of r˜ > 1 such that r˜ ∈ D ∩ W , and U (˜ r) > U (1), it is enough to show that U (r) − U (1) > 0. r&1 r−1

(*)

lim

This is so, because if (*) holds, then it cannot be the case that

U (r)−U (1) r−1

≤ 0 for

all r > 1 in the ball B around 1. Thus, there exist r˜ > 1 in B ⊆ W such that U (˜ r)−U (1) r−1

> 0; this, of course, implies that U (˜ r) > U (1), and r˜ ∈ D.

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Let g(r) = R(1 − tr)/(1 − t). We have that

= +

U (r) − U (1) r−1  Z 1/r  u(r) − u(1) u ◦ g(r) − u ◦ g(1) t + (1 − t) f (t)dt r−1 r−1 0 Z 1 1 [t(αi (r|t)u(r) − u(1)) − (1 − t)u(R)]f (t)dt. r − 1 1/r

Since the following holds Z 1 1 [t(αi (r|t)u(r) − u(1)) − (1 − t)u(R)]f (t)dt r − 1 1/r Z 1 1 ≥ −u(R) f (t)dt r − 1 1/r F (1) − F (1/r) → −f (1)u(R) = 0, = −u(R) r−1 we are left to show that  Z 1/r  u(r) − u(1) u ◦ g(r) − u ◦ g(1) lim t + (1 − t) f (t)dt > 0. r&1 0 r−1 r−1 Note that g 0 (r)u0 (g(r)) = −u0 (R)Rt/(1 − t). Defining  h i  t u(r)−u(1) + (1 − t) u◦g(r)−u◦g(1) f (t) r−1 r−1 hr (t) =  0

if t ∈ [0, 1/r] otherwise,

we see that limr→1 hr (t) = [u0 (1) − Ru0 (R)]tf (t). Thus, by the Lebesgue dominated convergence Theorem, we obtain  Z 1/r  u(r) − u(1) u ◦ g(r) − u ◦ g(1) lim t + (1 − t) f (t)dt r&1 0 r−1 r−1 Z Z 0 0 = lim hr (t)dt = [u (1) − Ru (R)] tf (t)dt > 0, r&1

[0,1]

[0,1]

since u0 (1) > Ru0 (R) (see Diamond and Dybvig [2], Footnote 2) and since 0. The proof is now complete. 19

R [0,1]

tf (t)dt >

References [1] Binswanger, H. (2001) Public health: HIV treatment for millions. Science 292, 221-223. [2] Diamond, D. and P. Dybvig (1983) Bank runs, deposit insurance, and liquidity. Journal of Political Economy. 91, 401-419. [3] Ennis, H. and T. Kiester (2003) Economic growth, liquidity, and bank runs. Journal of Economic Theory 109, 220245. [4] Goldstein, I. and A. Pauzner (2005) Demand-deposit contracts and the probability of bank runs. Journal of Finance 60, 12931327 [5] Haacker, M. (Ed.) (2004) The Macroeconomics of HIV/AIDS. Washington, D.C.: International Monetary Fund. [6] Hickey A. (2004) New allocations for ARV treatment: an analysis of 2004/5 national budget from an HIV/AIDS perspective. The Institute for Democracy in South Africa, AIDS Budget Unit. [7] Johansson, L. (2007) Fiscal implications of AIDS in South Africa. European Economic Review 51, 1614-1640. [8] Lamptey, P., Johnson, J. and M. Khan (2006) The global challenge of HIV and AIDS. Population Bulletin 6, 1-24. University of stellenbosch. [9] Leoni, P. (2003) Beliefs, Learning and Economic Behavior. Ph.D. Thesis, University of Minnesota. [10] Lewis, M. (2005) Addressing the challenge of HIV/AIDS: Macroeconomic, fiscal and institutional issues. Center for Global Development Working Paper 58. 20

[11] Santaeulalia-LLopis, R. (2007) Aggregate effects of AIDS on development. Working paper, University of Pennsylvania. [12] South African Savings Institute (2007) Savings presentation 31 July 2007. Available: http://www.savingsinstitute.co.za/rp.html [13] UNAIDS/WHO (2008) Epidemiological fact Sheet on HIV and AIDS: Core data on epidemiology and response in South Africa. Geneva. [14] Young, A. (2005) The gift of the dying: The tragedy of AIDS and the welfare of future African generations. Quarterly Journal of Economics 120, 423-466.

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