Investor Risk Aversion and Financial Fragility in Emerging Economies

Rovelli, Riccardo Working Paper Investor Risk Aversion and Financial Fragility in Emerging Economies Quaderni - Working Paper DSE, No. 380 Provided in Cooperation with: University of Bologna, Department of Economics Suggested Citation: Rovelli, Riccardo (2000) : Investor Risk Aversion and Financial Fragility in Emerging Economies, Quaderni - Working Paper DSE, No. 380, Alma Mater Studiorum - Università di Bologna, Dipartimento di Scienze Economiche (DSE), Bologna, https://doi.org/10.6092/unibo/amsacta/4929 This Version is available at: https://hdl.handle.net/10419/159221 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/3.0/ INVESTOR RISK AVERSION AND FINANCIAL FRAGILITY IN EMERGING ECONOMIES Jeffrey H. Nilsen Studienzentrum Gerzensee and University of Bern Tel. +41.31.780.3108; Fax +41.31.780.3100 e-mail: [email protected] Riccardo Rovelli University of Bologna Tel. +39.051.209.601; Fax +39.051.209.664 e-mail: [email protected] This version: July 10, 2000 Abstract Bank intermediated short-term capital inflows play a crucial role in the financial structure of many emerging economies. Yet since these funds are subject to the risk of early withdrawal, an excessive reliance on this financing is often associated with a financial or currency crisis. We model a situation where withdrawals are motivated by a change in either the domestic or foreign fundamentals. We show that, for a given change in fundamentals, a sudden reversal in the capital flows, and hence a financial crisis, is more likely the more risk averse are the foreign investors. We also show that a policy to tax early withdrawals may discourage the inflows more likely to cause fundamental runs, as it prevents the relatively more risk averse investors from investing. However, the policy must be fine-tuned to avoid discouraging all capital inflows. Keywords: Short-term capital inflows, Financial fragility, Chilean tax. JEL codes: G21, G28, O16, 023. Acknowledgments:This research was undertaken with support from the European Union Phare-ACE Programme 1996, Project No. P96-6114-R. Both authors thank the Studienzentrum Gerzensee for providing logistic support during this research. We thank an anonymous referee, Sudipto Bhattacharya, Elena Carletti, Gianluca Femminis, Luca Lambertini, and several participants at the October 1998 ASSET Conference in Bologna and at the July 2000 ESSET Conference at Gerzensee for useful comments and suggestions. 2 1. Introduction In the last six years, emerging market economies have been affected by a long series of financial crises, starting with Mexico in 1994-95, followed by Asia in 1997, Russia in 1998 and Brazil in 1999. Each crisis had its own particular characteristics and determinants. However, they also shared common factors: each demonstrated the “potential for sharp changes in investor sentiment”, often triggered by a combination of unsustainable external imbalance, overvalued exchange rate, financial fragility, poorly monitored bank loans, unwise investments, and unsustainable fiscal policy. In this paper we concentrate on one crucial step in the development of such crises: the interplay between financial fragility and deteriorating economic fundamentals. Financial fragility arises when financing capital accumulation via bank-intermediated short-term inflows since they are subject to the risk of early withdrawals. These withdrawals become more likely as the outlook of economic fundamentals gets bleaker and may bring about a sudden reversal of capital flows, precipitating a financial crisis. We propose a simple unified theoretical framework to evaluate these issues. While simple, our model is useful to analyze the origin of some financial crises, and also to evaluate the effectiveness of a tax on short-term inflows (we call it a Chilean tax as it is equivalent to a requirement on short-term deposits) in reducing the fragility of the financial structure. In particular we show that a Chilean tax works not simply by deterring short-term capital inflows, but more precisely by helping to select less risk averse lenders. In fact we find that, ceteris paribus, a financial crisis is more likely the more risk averse are the investors entering an emerging economy: by selecting out such investors, the tax renders a crisis less probable for any given configuration of the “fundamental” economic variables. The paper is organized as follows. In section 2 we review the literature and motivate our analysis. In section 3 we set up the basic model to examine the interaction between foreign investors and the financial and industrial sector of an emerging economy. In section 4 we examine under which conditions foreign investment will take place, and in section 5 we characterize under which conditions a crisis will take place. In section 6 we evaluate the role of a Chilean tax in preventing a crisis. Section 7 concludes. 3 2. Short-term capital inflows and financial crises The lack of domestic savings is a constraining factor in the growth of most emerging economies. Inflows of foreign capital may help sustain a phase of intensive capital accumulation. Several factors may however deter foreign investments: • An underdeveloped financial environment. Investing in emerging market economies may be quite difficult for domestic and especially for foreign investors. Financial markets may lack liquidity and transparency while regulations protecting investors’ rights are weak or difficult to enforce. • Demand for liquidity. While acknowledging the need for long-term investments, investors considering investment in an emerging economy may want to keep their options open in the case of either a sudden worsening of conditions in the host economy or a sudden improvement in the financial outlook elsewhere in the world. Thus demand for liquidity is often an integral part of foreign investors' requirements. • Risk of devaluation. Once an investment has been made, will the foreign investor be able to repatriate her investment into her own currency? Will a sudden devaluation reduce the value of the investment in terms of her consumption basket? Or might the repayment be rescheduled or partially written-off due to balance of payments problems? The first two issues often direct foreign investors to local banks since they provide ready solutions to both the monitoring and liquidity problems. As for the foreign exchange risk, an investor might to some extent hedge her currency exposure in forward markets, although it is much more difficult to purchase insurance against rescheduling and write-offs. Based on these considerations, our analysis models foreign investments as channeled through bank deposits into an emerging economy. We do not dispute the importance of other financial instruments, but rather seek a simple framework to highlight what we perceive to be the single most critical aspect of foreign investments into these economies: funds are committed for a shorter period than required for investment to mature and which must be repaid in a foreign currency acquired only after the investment succeeds. That this is a relevant issue is now widely acknowledged. As Radelet and Sachs (1998) observe: "Much of the economic activity supported by the capital inflows was highly 4 productive, and the loss of economic activity resulting from the sudden and enormous reversal in capital flows has been enormous". Several authors have argued that there has been a strong correlation between bank-intermediated short-term inflows and financial crises. For instance, a recent OECD report states that: “ensuring that banks are reasonably sound may not be sufficient to prevent self-fulfilling crises: it is also important to ensure that banks are not exposed to liquidity crises. ... Short-term maturity debt is risky because it increases the potential magnitude of capital outflows. ... It is the ratio of short-term debt to international reserves that matters” (OECD, 1999, pp.83-87). In the same vein, Furman and Stiglitz (1998) observe that, out of a sample of 48 developing nations, 11 had short-term debt ratios to central bank reserves higher than one; of these, only three escaped a currency crisis. Moreover they estimated that the effect of 1% rise in the ratio of bank debt/total debt may have increased the probability of a crisis by 6%. Also Rodrik and Velasco (1999) observe that the ratio of shortterm bank debt to central bank reserves is a robust predictor of financial crises. They estimate that the probability of a “sharp reversal” in capital flows increases by 3% to 5% with a 1% increase in the ratio. Parallel to the empirical literature, several authors have examined in theory the relationship between banking arrangements and financial fragility. While a survey of the bank run literature is beyond our scope, we note that several authors have examined the role of the information about fundamentals in precipitating bank runs following the seminal contributions of Bryant (1980) and Diamond and Dybvig (1983). In particular, Jacklin and Bhattacharya (1988) observe that bank runs become a problem only if long-term assets are insufficiently liquid. In their model, moreover, bad news does not necessarily lead to runs if depositors lack alternative stores of value. Accordingly, we also develop below the idea that investors may be induced to run by the presence of attractive outside opportunities.1 Papers in the Bryant and Diamond-Dybvig tradition generally examine banking allocations in the context of a closed economy. Recently, several authors have incorporated features of the Diamond-Dybvig model (henceforth: DD) into models of the financial crisis of an emerging economy. In a series of papers, Chang and Velasco (1998; 2000) model domestic residents of an emerging economy as risk averse DD investors who may additionally borrow abroad. A 1 Several other papers have examined information-based bank runs. Some authors have proposed the notion of an efficient bank run, i.e. a bank may draft an optimal deposit contract while recognizing the possibility of bank runs (See e.g. Alonso, 1996). However, if one considers the possibility that non- 5 term structure of interest rates on the foreign liabilities of domestic banks emerges that depends on the relative default risk premia. These authors also find that a flexible exchange rate arrangement with a lender-of-the-last-resort facility is socially optimal. Goldfajn and Valdés (1997) also model a situation where intermediaries may help attract capital inflows although they only consider risk neutral DD investors. In their model, patient investors may start a run if they observe bad news. Turning to capital controls, several authors favor imposing restrictions or controls on shortterm inflows. The previously quoted OECD report suggests that “in some instances a case can be made for limiting short-term capital inflows through taxes on capital imports, foreign deposit reserve requirements, or similar measures, but global financial integration is driven by forces that cannot easily be reined in and that generally carry many benefits” (OECD, 1999, pp.83-87). In a recent survey, Edwards (1999) observes that capital controls: (i) did not reduce aggregate capital inflows, but instead reduced the ratio of short-term to longer-term debt. (See also Montiel and Reinhart, 1999) (ii) did not affect real exchange rates (iii) did not affect the volatility of short-term interest rates (iv) did significantly affect the level of short-term interest rates (See also De Gregorio, Edwards and Valdés, 1998). These effects (or lack thereof) are entirely consistent with the predictions of our model (see Section 6 below). It is interesting to ask through which mechanism do capital controls operate. In some sense the answer is rather obvious: pecuniary disincentives to invest in shortversus longer-term debt mitigate liquidity problems (provided they do not deter investment altogether). However, the (positive) effects of controls are not limited to this case. For instance, Cordella (1998) argues that a tax on short-term inflows may prevent bank runs and also increase the amount of funds invested. In his model, "rational herding" arises when, after having invested in a long-term project, some investors' early withdrawal engenders anticipated investment liquidation, thus reducing the returns available to the remaining investors. And he finds that imposing a suitably high tax on short-term inflows will effectively discourage herding. All the above-mentioned papers chose to neglect some crucial aspect of the DD model. In this paper, we model foreign investors as full-fledged DD investors. In particular, our model includes both their aversion to risk and demand for liquidity as essential ingredients. We characterize the banking system of an emerging economy as offering an optimal term structure informed investors may run prior to the arrival of precise information (used by informed investors), then the notion of efficient runs evaporates (See Chen 1999). 6 of deposit rates explicitly tailored to the foreign investors’ risk aversion. Higher risk aversion, however, increases the likelihood of a financial crisis for a given dose of bad news. We then find that the imposition of a tax on short-term inflows brings about a steeper term structure of bank offered rates and thus reduces financial fragility, in particular by eliminating the more risk-averse investors, who are the more likely to precipitate a run2. 3. The model As the analysis will soon become inherently complicated, we design the model in a suitably simple way. We assume only three agents in the world economy: the productive sector in the emerging economy ("the firm"); the consolidated central bank and commercial banking sector of this economy ("the bank") and foreign investors. The central bank holds foreign reserves as a counterpart to the monetary base (bank reserves), but does not exercise any independent monetary policy, as the structure of domestic rates is a function of technology and investors’ preferences. The size and composition of the current and capital account depend on the capital accumulation decisions of domestic firms. There is no fiscal policy and no foreign debt of the government. While these omissions obviously reduce the overall realism of the model, the main point of the analysis stands up more clearly in this simplified setting. The overall setup and discussion of the model takes places through the following sequence: a) The firm requests a two period loan from the bank to finance its investment b) To finance the loan, the bank collects deposit from foreign investors c) The deposit contract is of the DD type, set for two periods with an option to withdraw after one period d) Foreign investors also have the outside option to invest in the foreign market e) New information on fundamentals of the domestic economy arrives in period 1. Under the new circumstances, some foreign investors who would not otherwise have done so decide to seek liquidity, hence to withdraw in period 1 f) A “run” is equivalent to the decision to withdraw in period 1 by depositors who would otherwise have stayed in for 2 periods g) A run causes a financial crisis, as the bank is subsequently unable to face its obligation towards foreign investors 2 We develop this notion precisely in Section 6 of the paper. 7 Having examined these issues, we conclude by examining whether, for a given change in fundamentals as in point (e) above, the imposition of a suitably defined “Chilean tax” would reduce the likelihood of a financial crisis. We examine points (a-c) in this section, point (d) in section 4, points (e-g) in section 5, and we discuss the effects of a "Chilean tax" in section 6. The firm. The firm begins with no initial resources and so requires initial finance (time = 0) to purchase imported inputs. The production process is long (2 periods) and partially illiquid in the intermediate (time = 1) period. Each unit of investment at time 0 yields a total return R > 1 at time 2. Thus the periodic rate of return is 1R −. In Section 5.1 we discuss the case when R is random. Output produced at time 2 is exported, the proceeds from which repays the initial foreign investment. Foreign inflows are the only possible source of funds for financing domestic production. While an extreme assumption, it helps us to concentrate on the main ingredient in our story. We would be able to provide no further insight by introducing a parallel domestic source of funds. Moreover, it is a quite realistic assumption: most emerging economies need extensive investment to (re)build their capital stock, an amount which exceeds domestic savings. Foreign investors. Foreign investment is made in the form of deposits with the domestic bank. Deposits inflow at time 0 and may be maintained for one or two periods with no possibility of pre-commitment. A proportion 0 < ε < 1 of depositors seek to receive at time 1 the pre-agreed rate r1 in domestic currency per unit of deposit. The remaining depositors expect to receive at time 2 the contract rate (r2)2 per unit of initial deposit (see below for greater detail on the depositors). The bank. The bank at time = 0 receives foreign currency from foreign depositors. This sum is in part converted into local currency and loaned to the firm, which reconverts it into foreign currency to pay for imported inputs. Those initial deposits not loaned to the firm are kept as (foreign currency) reserves in the consolidated banking system. At time = 1, the bank uses reserves to repay depositors wishing to withdraw early 3. The role of the central bank in the domestic economy is thus most concisely seen as a currency board.4 3 Strictly speaking, eq. (1.2) below should in general be written as a weak inequality. However in the optimum it will always hold as an equality, and for simplicity we use this fact from the beginning. 8 The consolidated balance sheet of the bank is defined by the following three equations (with inflows on the left, outflows on the right): (1.1) 0000 LFD:t += (1.2) 010 * 11 DrFr:t ε= (1.3) 0 2 202 D)1()r(LR:t ε−= where subscripts refer to the time period and : D = foreign deposits, F = foreign currency reserves; L = loan to domestic firms (also equal to imports at time 0) r = one-period total return to domestic deposits (1 + interest rate) r*= one-period total return to foreign deposits R = two-period total return to investments Eqs. (1.1)-(1.3) are defined in domestic currency. They can be easily written in foreign currency terms by deflating both sides with the current nominal exchange rate (ei ; i = 0, 1, 2, where e is the price of one euro in the currency of the emerging economy). Note that, since all and only foreign transactions are intermediated through the bank, the same equations also define the balance of payments of the economy across the three periods.5 Determination of interest rates on domestic deposits. In order to evaluate the resource constraint in the final period, substitute (1.1) and (1.2) in (1.3), to obtain: )1()r( r r 1R 2 2 * 1 1ε−=         ε − that is: Also, we could alternatively assume that the bank may finance these intermediate-deposit withdrawals borrowing (on the world financial market, at the going riskless rate, r2* ) based on the proceeds from final-period exports. This strategy would not improve on the one described in the text. 4 While we could assume that the central bank creates additional high-powered money to finance domestic credit creation, this complication would not add insight. 5 From the balance of payments perspective, D0 is the capital inflow, F0 is the capital outflow and L0 are imports, all at time 0. At time 1, r1*F0 is the capital and interest inflow and ε r1 D0 is the capital and interest outflow. Finally, at time 2, R L0 are exports and (1-ε) (r2)2 D0 is the capital and interest outflow. 15 At time 1may observe new information corresponding to two different set of events. The new information may refer to either: • profitability of the real investment in the emerging economy : R • state of the world in period 2: r2* If either of these variables is uncertain, we must modify the former maximization problem, which yields a generally different solution for r1 (specifically, equation (8) is no longer valid). Equation (2), though, continues to be valid in expectation. We must also evaluate Constraints C1-C.3 to to take into account that expected utility is now a function of both the investors' types and of the variability of either future profitability or future state of the world (or both). For tractability, we shall assume a binomial distribution (for either R or r2*), taking one source of uncertainty at a time, in Sections 5.1 and 5.2 respectively. In section 5.3 we state the main proposition on fundamental runs. 5.1 Uncertainty in final outcome To model the existence of uncertainty in final return to the domestic technology, we assume a simple binomial distribution. Define Ru > Rd (respectively "good" and "bad") as the possible values the final return may assume, with probabilities q and (1-q) respectively. Next, to reformulate the maximization problem of equation (4), we must encompass that at time 0, the investor computes expected utility out of four possible occurrences: • she is impatient and patient investors do not run [payoff to impatient: r1 ] • she is patient (prob. 1-ε ) and the good outcome occurs (prob. q ) [payoff: (r2u )2 ] 11 • she is patient and the bad outcome occurs, but no run follows [payoff: (r2d )2] • the bad outcome occurs and everyone decides to run12 To compute the expected utility, we need the probability of each outcome, which we know except for the probability of the run. However, assume the joint event of being a patient type 11 r2u and, below, r2d are one-period total returns earned by remaining invested in the contract for two periods, based on returns to the underlying technology of Ru or Rd . These returns are computed from equation (2). 12 Resources available at time 1, before withdrawals and repayments, are: A1 ≡ r1* F0 + q L0 , where q is the liquidation value of the investment in the intermediate period, and 0 < q ≤ 1. Hence the intermediate return per unit of deposit in case of anticipated liquidation of the bank is A1/D0 . This assumes that impatient depositors and "runners" are treated alike when the bank is liquidated. 16 who has observed the bad outcome. In this case, the agent decides she will not run if the following constraint (evaluated at time 0) is satisfied 13: • C.4.1 (r2d )2 ≥ r1r2* . In this section, we concentrate on the case when a run does not occur, and in sections 5.3 and 6 we examine in more detail what variables (or news) and policies would make the constraint C.4.1 more or less likely to be satisfied. It might be meaningful to investigate an efficient run, i.e. when an investor rationally anticipates a run if a bad outcome occurs, but nevertheless finds it attractive (in expected utility terms, computed at time 0) to invest in the emerging country. However, while an efficient run may be interesting in the case of a single bank, it is much less plausible extended to an aggregate banking system. Hence, we rule it out, assuming that an aggregate run (one affecting the entire banking system) is too costly to accept, so banks will not accept deposits if such an event were likely (that is, if C.4.1 is not satisfied). Accordingly, the assumption of constraint C.4.1 is entirely appropriate in our context, and so we compute the investor’s expected utility under the possible occurrence of three events only: (4.1)                     ε− ε− −+              ε− ε− ε−+ε * 1 1 d * 1 1 u 1r r 1 1 R U)q1( r r 1 1 R Uq)1()r(UMax Note that as a result of the uncertainty, contract rate r2 will de facto be indexed to the realized value of R. This might seem counterfactual, given the usual specification of deposit contracts. However it need not be if one interprets the value of r2(Rd) as either the result of a partial write-off of debt or, in a richer model, as isomorphic to a currency devaluation occurring if too little output had been produced to repay the foreign debt based on a constant (from time 0 to time 2) exchange rate. 5.2 Uncertainty in the world interest rate Several authors have indicated a role for external factors in precipitating banking crises in emerging markets. For instance, Eichengreen and Rose (1998) find that increasing interest rates in industrialized countries are strongly associated with the onset of banking crises in 13 The constraints states a necessary and sufficient condition for the non-occurrence of a run. Note that the cutoff point for an investor not to run is indeed r1 r2* since she evaluates the return without 17 developing countries. In this section, we extend our model to take account of volatility in the second period foreign interest rate, r2*. To do so, we retain equation (4) but we modify those constraints that depend on the value of r2*. Thus, for constraint C.1, we specify the expected utility of staying out as: (9) () () d* 2 u* 2 ** 2 * 1rU)p1(rUp)1()r(U)r,r(EU 1−+ε−+ε= , where we have assumed, similarly to the previous section, a binomial distribution for the foreign interest rate in the second period.14 Assuming again that constraints C.1-C.3 (the former modified to include equation (9)) are satisfied, will the patient investor confirm his decision to remain invested in the emerging country when r2* is observed? This question is critical only in case of a "high" realized value, i.e. r2*u. Alternatively, if patient depositors "stay in", they earn - with certainty - (r2)2. Therefore, they choose not to withdraw iff the following constraint is satisfied: • C.4.2: continue to stay in: (r2 )2 > r1r2*u In this case, the investor’s maximization problem is again described by equation (4). 5.3 Likelihood of a run upon observing new information We first define a fundamental run as follows: Definition: Fundamental Run If a bank has accepted deposits, a fundamental run will occur when { C.1, C.2, C.3 } are satisfied and either C.4.1 or C.4.2 are not. Let us re-examine constraint C.4.1: (r2d )2 ≥ r1r2*. For given second period foreign interest rates, the constraint will be less likely to bind, the higher the return to the depositors if a bad outcome to the technology, r2d , or the lower the first period bank contract rate. This observation provides the intuition for the following: internalizing the consequences of a run on the value of the bank in liquidation. 14 The bank could alternatively offer a second period interest rate indexed to the world rate, i.e. r2 = r2(r2*, R). However, the resource constraint, eq. (2), must still hold. Hence the bank must also index first period contract rates to the same rate, i.e. r1 = r1(r2*, R). This implies r1 could not be paid until r2* is observed, which we plausibly assume cannot be done in our context. This rules out indexing r2 . 18 Proposition 2.1 A fundamental run in case of uncertain profitability of the real investment would be more likely: • the more variable is R • the more risk averse are the investors. Proof: See Appendix, A.2. We apply similar reasoning to constraint C.4.2: (r2 )2 > r1r2*u . The constraint clearly becomes less binding, the “steeper” is the domestic term structure (for given foreign rates) and the lower are the foreign rates. This provides the intuition for the following: Proposition 2.2 A fundamental run in case of uncertain value of world interest rates would be more likely: • the more variable are world interest rates • the more risk averse are the investors. Proof: See Appendix, A.3. Table 2 provides numerical examples concerning Proposition 2.2. We examine three cases characterized by uncertainty in r2*, which may take values 1.2 or 1.0 with equal probability (equivalent to interest rates of 20% and 0 respectively). The three cases assume equal degrees of risk aversion and parameters differing only in ε, the proportion of "impatient" investors, respectively 0.1, 0.5, 0.9. We observe in the first two cases that capital will not flow into the economy: constraint C.2 binds in both cases. On the other hand, (ε = 0.9) the inflow occurs in the third case, but will be reversed (a run occurs) in the "bad" state of the world (i.e. world interest rates up to 20%). 19 Table 2. Uncertain world interest rates, with or without a run Data: R1*1.1 prob(r2*= r2*u) =0.5 r2*u1.2 R 1.5 r2*d1 γ -1.5 ε 0.1 Domestic rates: Expected Utility of: Constraints: R11.29843 (r1, r2)-0.38173 C.1 0.13391 R21.21241 (r1*, r2*u, r2*d)-0.51564 C.2 -0.08602 X E(r2*) 1.10000 C.3 0.04166 RRA 2.50 C.4.2 -0.08818 Data: R1*1.1 prob(r2*= r2*u) = 0.5 r2*u1.2 R 1.5 r2*d1 γ -1.5 ε 0.5 Domestic rates: Expected Utility of: Constraints: R11.20206 (r1, r2)-0.46290 C.1 0.08039 R21.16655 (r1*, r2*u, r2*d)-0.54329 C.2 -0.03551 X E(r2*) 1.10000 C.3 0.03857 RRA 2.50 C.4.2 -0.08164 Data: R1*1.1 prob(r2*= r2*u) = 0.5 r2*u1.2 R 1.5 r2*d1 γ -1.5 ε 0.9 Domestic rates: Expected Utility of: Constraints: R11.11900 (r1, r2)-0.55364 C.1 0.01731 R21.12552 (r1*, r2*u, r2*d)-0.57094 C.2 0.00652 E(r2*) 1.10000 C.3 0.03590 RRA 2.50 C.4.2 -0.07600 RUN Note: In the last column, an X denotes that a time-0 constraint is binding, hence foreign investors would not want to invest in the country; a RUN instead denotes that the bank would not want to accept foreign investments, because if it did there would be some non-zero probability of a run. 20 6. Policies to prevent currency runs In the previous sections we have identified that for certain parameter values, a "fundamental" run may occur (if the bank accepts the funds that foreign depositors would supply). As the possibility of a run can be anticipated at time 0, we argued in section 5.1 that the bank would more likely consider the significant bankruptcy costs associated with a run and reject deposits altogether if either constraint C.4.1 or C.4.2 is not satisfied. Nevertheless, it is clearly desirable to institute policies that widen the range of parameters values for which such constraints do not bind, and thus allow greater delivery of “safe” capital inflows. Is it possible to at least reduce the likelihood of runs, while avoiding radical therapy (à la Mahatir) - i.e. without discouraging capital inflows tout court? In general, we might consider several policies: • tax short-term inflows, or impose a non-remunerated reserve requirement15 • increase reserves ex ante, (which implies reducing loans to firms) • finance withdrawals by borrowing in the intermediate period; repay in the final period by taxing realized output (assuming the taxing authority has the authority to access foreign exchange obtained from exports). Note that in each of these three cases the resource constraint (eq.2) changes accordingly.16 The more interesting and widely debated policy suggestion is certainly the first and so we propose to examine it analytically within the framework of our model. 6.1 A tax on short-term inflows We now investigate the imposition of a tax on short-term inflows (sometimes denoted a "Chilean tax") that reduces the return paid to the foreign investors withdrawing in the intermediate period (either because they discover they are "impatient" or because they have observed unfavorable information at time 1-). In our model, this tax may be structured in one of two ways: 15 While the two policy instruments are operationally different, there is a one to one correspondence between them, expressed by the formula: t(m) = r1* [λ/(1-λ)](M/m) , where t is the tax rate, m is the actual length of the investment, M is the period for which the reserve deposit has to be maintained and λ is the coefficient of required reserve. See Edwards (1999) for references to the original sources of this formula and some examples. 21 • The tax reduces the amount paid to first period withdrawals while the bank holds no reserves against the tax, hence the tax revenues are invested for two periods in the firm. • Hold reserves equal to the amount of gross first period interest. Pay net interest to depositors while the tax goes to the government (consolidated with the central bank), which then earns the prevailing foreign rate r2* on its revenues. In the first case the tax is arbitraged away since contract terms are endogenous. Hence we consider only the second case. Proposition 3 establishes the effects of the tax on the term structure of deposit interest rates and verifies that the tax does not discourage investments (by making the constraints C1-C.3 binding). Proposition 3. A tax on short-term inflows has the following effects: • lowers r1 (net of tax) • for any other given parameter values, makes the constraint C.1 more binding • for γ < 0, lowers, and for 0 < γ <1, increases second period interest rates • may discourage foreign deposits that would have occurred had the tax been zero. Proof: See Appendix, A.4. The results in Proposition 3 are to some extent intuitive. However, they are really a foundation for a subtler thought-provoking statement that requires a more realistic set of assumptions in our model. We relax the assumption of homogenous foreign investors (apart from their liquidity type) and assume instead they have two different degrees of aversion to risk; characterized by high (H) or low (L) risk aversion. Imagine the following experiment, a situation in which both H and L investors wish to invest in an emerging economy. Introduce a tax on short-term inflows and gradually increase the tax rate. The tax will affect the constraints of both groups of investors. Whose constraints will start biting first, those of the H or of the L investors? And is the answer to this question related to the results stated in Propositions 2.1 and 2.2? The answer, as shown in the statement of Proposition 4, turns out to be rather favorable. Proposition 4. A tax on short-term inflows helps to select the less risk averse investors. 16 In particular in the latter case the constraint would become more slack as borrowing to finance withdrawals allows reducing, or in the limit to eliminating, initial reserves. This in turns yields an increased amount of output in period 2. 22 Proof. See Appendix, A.5. Discussion. Figure 1 shows the highest tax for which H accepts the contract (it remains acceptable to L) for given r1* = r2* = 1, R = 1.5 (the graph does not change appreciably with other values of r1* , r2* , or R ). With higher ε, a lower tax rate separates the H and L investors. This is because a higher ε places greater weight on first period, taxed, consumption in the utility of investors, so a tax increase decreases utility more strongly than when second period consumption is more heavily weighted (i.e. lower ε). Figure 1 In the figure, we measure the difference in risk aversion between the two investors as x = γL - γH , for γL → 0 . We find that with x < .2, the investors' risk-aversion coefficients are insufficiently different for the tax to separate the investors. The tax is also impotent below ε of approximately .05. x tax ε 23 In Table 3, we develop some numerical examples to calculate the lowest tax rate required to induce investor H or L respectively to drop out17. Table 3. Choice of tax rates to select out investors with different aversion to risk Parameter values:H drops out L drops out Rr 1*r 2*ε γHTax rate r1r1(1-t) Tax rate r1r1(1-t) 1.5 1.0 1.0 .3 -1.5 .65 1.77 .62 .83 2.22 .39 1.5 1.05 1.05 .3 -1.1 .68 2.11 .67 .80 2.54 .51 1.3 1.0 1.0 .7 -3.6 .22 1.21 .94 .34 1.35 .89 1.3 1.05 1.05 .7 -1.5 .27 1.37 .99 .33 1.45 .98 1.5 1.0 1.0 .7 -3.7 .19 1.15 .93 .36 1.32 .84 1.5 1.0 1.1 .7 -3.7 .16 1.13 .95 .35 1.34 .88 1.5 1.1 1.0 .7 -1.2 .29 1.40 .99 .35 1.48 .96 Notes. In each line the interest rates are optimal for investor L (who has γL → 0), conditional on a particular tax rate. The same rates are then “proposed” to investor H (who has risk aversion γH < γL , see column 5). We ensure that constraints C.1-C.3 are satisfied with this rate. We continue to raise the tax and recompute interest rates accordingly for the H investor until we reach her “drop out” point, characterized by the tax rate and corresponding beforeand after-tax total returns (columns 6 through 8). We continue to raise the tax rate, and recompute interest rates accordingly, until investor L also drops out18. See the corresponding tax rate and beforeand after-tax total returns in columns 9 to 11 respectively. Also note that as the L investor has RRA→1, the optimum (r2)2 →R always. To understand numerically the implications of Proposition 4, let us examine for instance the fourth line of Table 3. With world interest rates at 5%, a 2 period compound return in this economy of 30% (R=1.3), and a fairly high proportion of impatient investors (70%), we see that imposing a tax rate on first-period withdrawals equal to, say, 30%, is sufficient to discourage the more risk averse investors, while the less risk averse still wish to invest. The wedge between the two tax rates (the one discouraging investor H and the one discouraging 17 Recall that the analysis has been carried out using a tax rate applied to total returns. To find the tax rate (tN) on interest rates that yields the same after-tax return as a tax (t) on total returns, simply solve the following equation for tN: )t1(i1)t1(r N −+=− . 18 More precisely, to calculate the tax inducing investor L indifference, use the limit of equation (A.24’) in the Appendix A.5, evaluated as γL → 0, i.e.: () * 2 * 1 * 2 * 1 * 1 rln)1( )t1(r trrR ln r R ln ε−−         − − ε−         . 24 investor L) increases with the difference in attitudes towards risk. From the examples it is also apparent that, when the proportion of impatient investors is low (30%), then only very high tax rates would do the job (see the first 2 rows of the table). We can now state, without proving, a corollary of Proposition 4: Corollary. A tax on short-term capital inflows may reduce the range of fundamental parameter values for which a run would occur. This follows from Proposition 4 together with Proposition 2.1 and 2.2. Discussion The main objection to a Chilean tax is, of course, that it may throw the baby out with the bath, i.e. discourage capital inflows tout court. So the important task in devising a tax is to fine tune it so it discourages only the most run-prone (i.e., the more risk-averse) investors. As Table 3 shows, there is room for such tuning, especially if the differences in the attitude to risk of different investor groups are wide enough. We would also like to draw attention to one result in Proposition 3, namely that, if the relative risk aversion of the investors is “low” (strictly between 0 and 1), the imposition of a tax would increase long term (2-period) rates at the expense of short-term rates, thus reducing the degree of financial fragility even without considering a change in the composition of the investors. Finally, there are other ways in which a Chilean tax may be effective. It may, for example, discourage herding behavior, thus helping investors to coordinate on the good rather than the bad equilibrium, as suggested by Cordella (1998). Another way is to help select the less impatient investors, that is by discouraging those with higher liquidity demand.19 Intuition suggests that these are all complementary features of the tax, hence in a more general analysis their effects would likely reinforce each other. However we leave this to future research. 7. Conclusions Although capital inflows have vital importance in financing investment in emerging economies, they also play a critical role in the development of financial crises. In this paper we have focused on the role of short-term inflows intermediated by the domestic banking 19 A model with this feature has been developed by Smith (1984), in the context of a closed economy. 31 (A.16) [] *U 2 * 1 )1( * 1 rr r R R>         γ− γ Divide both sides by r1*: (A.17) *U 2 )1( 1 * 1 r r R>         γ− Part a). We show below that constraint C.4.2 is more likely to bind with an increase in variance. The variance of the binomial return distribution implicit in equation (8), is easily shown to be: 2 222 ))(1(*)(DU rr p p rVar −−= . By observation, (A.17) becomes more binding with a higher r2U that would be implicit in a mean preserving spread which increased the variance of r2*. QED Part b) We show that an increase in risk aversion, i.e. a decline in γ , decreases the left hand side of equation (A.17), i.e. makes it more difficult to satisfy constraint C.4.2. Take the derivative of (A.17) with respect to γ. (A.18) 0 )1( *r R ln *r R 2 1 )1( 1 1> γ −                 γ− QED A.4 Proof of Proposition 3 (Imposition of a Chilean-type tax) Part i. Derivation of the equilibrium contract in presence of taxes. Imposing a Chilean type tax alters the balance of payments and balance sheet constraints. First, note that flows in period 0 are unaffected, thus (1.1) is unchanged. In period 1, the gross foreign exchange earnings from investing reserves abroad in period 0 are divided between the (net of tax) payment to impatient investors and the tax payment to the government, expressed in the equation20 )t1(DrtDrFr 01010 * 1−ε+ε= Finally, in period 2, the project yields export goods. The proceeds of this sale reimburses (as in the base case) patient investors. In this case, however, tax receipts augment patient investors' returns. This implies that equation (1.3) changes: (1.3') 01 * 0 2 20 DtrrD)1)(r(RL 2 ε−ε−= 20 Note that this is identical to equation (1.2) in the text; i.e. the balance sheet and balance of payments restrictions remain identical. Note also that since the government invests the tax revenue abroad at period 1 it will earn foreign riskless rate r2* affecting period 2 accounts. 32 where the second term on the right hand side is the return from investing abroad the tax revenue from impatient investors. Combining equations (1.1), (1.2), and (1.3') and solving for r22 (2') () [] trrR r)1( r 1 R r* 2 * 1 * 1 1 2 2− ε− ε − ε− = Note this resource constraint reflects that a higher r2* increases the return on the government tax revenue and hence the return to patient investors. Maximization of expected utility now requires to solve the following: (4') () () []              − ε− ε − ε− ε−+−ε trrR r)1( r 1 R U)1()t1(rUMax * 2 * 1 * 1 1 1 In the case of CRRA preferences, the solution to this problem is obtained by satisfying the first order condition: (7') () () [] 0 r R trtrrR r)1( r 1 R )t1()t1(r * 1 * 1 1 * 2 * 1 * 1 1 1 1=      −         − ε− ε − ε− +−− −γ −γ Solving this equation for r1 yields: (8') γ− γ       − − ε−+ε− = 1 * 1 * 2 * 1 * 2 * 1 * 1 1 r)t1( trrR )1()trrR( Rr r Part ii. Net-of-tax r1 declines with a higher tax First, take the partial derivative of r1 (1 - t) w.r.t. t to show that net of tax r1 decreases: () () () () () () () () () () () () () γ− γ γ− γ γ− γ       − − ε−+ε− −                 − − ε−+ε−           − − γ− γ −       − − ε−+ε− = ∂ −∂ 1 * 1 * 2 * 1 * 2 * 1 * 1 2 1 * 1 * 2 * 1 * 2 * 1 * 2 * 1 * 2 * 1 1 * 1 * 2 * 1 * 2 * 1 * 1 1 rt1 trrR 1trrR Rr rt1 trrR 1trrR t1 rrR 1 rr rt1 trrR 1rr)t1(Rr t )t1(r Raising the second term to a common denominator, we find after simplifying: () () () () () () () () () () () () 2 1 * 1 * 2 * 1 * 2 * 1 1 * 1 * 2 * 1 * 2 * 1 1 * 1 * 2 * 1 * 2 * 1 * 1 1 rt1 trrR 1trrR rt1 trrR 1RR t1 rrR 1 rt1 trrR 1)t1(rrRr t )t1(r                 − − ε−+ε−                 − − ε−+ε−           − − γ− γ       − − ε−−+ε = ∂ −∂ γ − γ γ − γ γ − γ 33 Factoring the term in exponents and simplifying: () () () [] () () () () 2 1 * 1 * 2 * 1 * 2 * 1 * 2 * 1 * 2 * 1 * 2 * 1 1 * 1 * 2 * 1 * 1 1 rt1 trrR 1trrR RrrRrrR 1 rr rt1 trrR )1(Rr t )t1(r                 − − ε−+ε− −ε+         −− γ − γ −       − − ε− = ∂ −∂ γ− γ γ− γ Further simplifying: (A.19) () () () () () () () () 2 1 * 1 * 2 * 1 * 2 * 1 1 * 1 * 2 * 1 * 2 * 1 * 1 1 rt1 trrR 1trrR rt1 trrR 1 )1( rrRRr t )t1(r                 − − ε−+ε− ε+       − − γ − ε− −− = ∂ −∂ γ− γ γ− γ This expression is clearly negative for γ < 1. Part iii. Partial derivative of r22 and of (r2-r1(1-t)) with respect to a change in tax rate. Substitute eq. (8') into (2') for the equilibrium r22 in a world with taxes. Simplifying: (A.20) () () )1( )t1(r trrR R r 1 * 1 * 2 * 1 2 2 ε−+         − − ε = − γ γ And taking the partial derivative for use in part iv below: (A.21) () () () () () 2 1 * 1 * 2 * 1 2* 1 * 2 * 1 1 1 * 1 * 2 * 1 2 2 )1( )t1(r trrR )t1(r)1( rrRR )t1(r trrR t r           ε−+         − − ε−γ− −γ         − − ε = ∂ ∂ − γ γ − γ where we observe that, for the case where γ < 0, this derivative is also < 0, whereas it is positive for the case of where 0 < γ < 1. As regards the derivative of the after-tax term structure slope, we are unable to provide a signed derivative, however, we have computed the solution within the space of reasonable parameter values and have always find that an increase in tax will increase the slope of the term structure. Numerical simulations using Mathematica are available upon request. 34 Part iv. Investors may be discouraged if a tax is applied. An investor may be discouraged if for any parameter values, the imposition of a tax brings any constraint to become more binding. We show that if a tax is imposed for any given parameter values, C.1 becomes more binding. That is, 0 t )1C( < ∂ ∂, where: () () () γ γ γ γε−+ε γ − γ ε− +− γ ε =* 2 * 1 2 21 r)1( r r )1( )t1(r1C This will be true if (A.22) () () t r r)1( t )t1(r )t1(r t 1C 2 2 1 2 2 1 1 1∂ ∂ ε−+ ∂ −∂ −ε= ∂ ∂− γ − γ < 0. Since both (A.19) and (A.21) are negative, (A.22) is also negative. QED A.5 Proof of Proposition 4 (A Chilean tax selects less-risk averse investors) The proof is in 3 parts: i. For given interest rates, r1*, r2*, R, a given proportion of impatient investors, ε, and two investors with given levels of risk aversion, γH < γL < 0, we seek a level of tax, t, that satisfies constraint C.1 for investor L, but does NOT satisfy C.1 for investor H. ii. For these parameters, we show that in the absence of a tax, the H investor would accept the contract, i.e. when t = 0, constraint C.1 is satisfied for investor H. iii. We then show that no investor will be influenced by constraints C.2 or C.3, so we do not analyze these constraints. Part i. There exists a tax rate t simultaneously solving C.1 for investor L, but not for investor H. We consider a contract offering rates r1 and r2 that would be the optimal choice of the L investor as if she were the only investor. These rates are taken as a given by the H person in considering whether to accept the contract or not. That is, constraint C.1 is weighted by the risk-aversion coefficient of investor H. A tax rate, t, to keep H out must then satisfy (insert in C.1 equations (2') and (8'), equilibrium r1 and r2 that optimally satisfy the L investor): 35 H L L 1 * 1 * 2 * 1 * 2 * 1 * 1 H )t1(r )trrR( )1()trrR( Rr)t1( γ γ− γ                                     − − ε−+ε− − γ ε H L L 1 * 1 * 2 * 1 * 2 * 1 * 1 * 1 * 2 * 1 H )t1(r )trrR( )1()trrR( rR )1(r )trrR( 1 R1 γ γ− γ                                                           − − ε−+ε− ε− ε− − ε− γ ε− + H H * 2 r)1( H * 1 r γ         γ ε−+ε γ < Simplify the second term and create new variables A and B, where: * 1 * 2 * 1 r)t1( )trrR( A− − = and )1( B L L γ − γ = to obtain the following equation: {} {} [] H * 2 * 1 B B H B* 2 * 1 * 1 H HH H H Hr)1(r A)1( A)1( 1 R1 A)1()trrR( Rr)t1( γ ε−+ε <       ε−+ε ε−       ε− γ ε− +         ε−+ε− − γ εγγ γ γ γ Divide both sides by γH, factor the term from the left hand side to get: {} () () [] HH H H H * 2 * 1 B * 2 * 1 * 1 Br)1(r 1 A)1( 1 )trrR( r)t1( A)1( Rγγ γ γ γ ε−+ε>               ε− ε− ε−+         − − ε         ε−+ε Simplifying, we establish the condition for "H out": (A.23) [] [] H H H HH * 2 * 1 B Br)1( r R A)1( A)1(A γ γ − γ γ γ −ε−+ε         > ε−+ε ε−+ε We use similar steps to obtain the condition required for L to accept the contract ("L in"): (A.24) [] [] L * 2 * 1 B L BL L L LL r)1( r R A)1( A)1(A γ ε−+ε         ≥ ε−+ε γ ε−+ε γ γ− γ γγ− Notice that variable A is increasing in the tax rate. Notice also that if R = r1*r2* (the lowest real return by assumption A.1), * 1 r R A=, independent of the tax. That is, a tax can have no effect in this circumstance since both investors have C.1 constraints at the point of binding. 36 Now define, for later convenience, γH = (γL - x), where x is a positive constant. Rewriting (A.23) [][] [] x* 2 * 2 x * 1 * 1 x BB Bx B xrr)1( r R r R A)1(A)1( AA)1(AA L L L LL − γ γ− −γ − γγ− ε−+ε                 > ε−+εε−+ε ε−+ε Rearranging: (A.25) [] [] [] L L L LL B * 1 x* 2 * 2 x * 1 x B Bx B xA)1( r R rr)1( r R A)1( AA)1(AA γ γ− − γ − − γγ− ε−+ε         > ε−+ε         ε−+ε ε−+ε Now rearrange (A.24) so the right hand side is identical to that of equation (A.25): (A.24') [] [] L L L LL B * 1 * 2 BA)1( r R r)1( A)1(A γ γ− γ γγ− ε−+ε         ≤ ε−+ε ε−+ε Thus, to separate out investors having relatively risk-averse preferences, we require a tax rate t satisfying simultaneously (A.23) and (A.24'), i.e. to find a tax rate t0 solving the inequality: (A.26) [] [] [] x* 2 * 2 x * 1 x B Bx B x * 2 B rr)1( r R A)1( AA)1(AA r)1( A)1(A L LL L LL − γ − − γγ− γ γγ− ε−+ε         ε−+ε ε−+ε < ε−+ε ε−+ε In order to simplify, assume 0lim → L γ (which 0 →=> B): [] [] [] x* 2 x * 1 x x r)1( r R )1( )1(A )1( )1( − −ε−+ε         ε−+ε ε−+ε < ε−+ε ε−+ε or: (A.27) [] )1(Ar)1( r Rxx* 2 x * 1 ε−+ε<ε−+ε         − Now decompose A and solve for t0: (A.28) x * 1 r) 0 t1( ) 0 t * 2 r * 1 rR( x* 2 r x* 1 r x* 2 r x* 1 r)1()1( x* 2 r x R           − − ε< ε−−      ε−+ε Now define: [ ] x* 2 x* 1 x* 2 x* 1 xxx* 2 rr rrR)1(Rr Cε −ε−+ε =, 37 and solve (A.28) for t0: (A.29) 0 * 2 x/1 * 1 x/1 t rC r R C < − − Now, obviously C > 1, implying t0 < 1 for some parameter values. Thus (A.29) gives the condition for a tax that simultaneously satisfies C.1 for L but not for H. QED Note that since we use the C.1 constraint for the H investor as an equality, t0 gives the lowest tax that makes the contract unacceptable to the H investor. It does not inform about the maximum tax for which investor L accepts the contract. We examine this issue in the text. We now turn to the next part of the proof. Part ii. The H investor rejecting the above contract would accept the same contract in the absence of a tax. Consider again equation (A.23) and set t = 0. Note * 1 r R A= in this case. It implies the H investor has an incentive to participate if: (A.30) [] H * 2 * 1 B * 1 H B * 1 * 1H H H HH r)1( r R r R )1( r R )1( r R γ ε−+ε         >                 ε−+ε γ         ε−+         εγ γ− γ γγ− Since γH < 0 and γL → 0 => B → 0, rewrite (A.30) as: [] [] H H H H * 2 * 1 * 1r)1( r R )1( )1( r R γ γ− γ γ− ε−+ε         < ε−+ε ε−+         ε Simplifying: (A.31) H H * 2 * 1 r r Rγ γ <         This is true by assumption (A.1). QED Part iii. Constraints C.2 and C.3 never bind with the parameter values associated with the contract of part i. In the case a tax is imposed, constraint C.2 "do not go short" must be > 0. () () () () γγ γ γ− γ ε− −− γ ε − γ ε− +− γ ε =2 11 2 21 )t1(r )1( )t1(rr )1( )t1(r2C 38 It can be easily shown that the above will be satisfied so long as the spread is positive. The investor may rather invest in short-term deposits if the tax causes the spread to become negative. Simplify and insert equilibrium contract values, equations (A.20) and (8'), where the latter is multiplied by (1-t): () () 2 1 1 2 1 2 2 A)1(A R )1(A R ))t1(r(r                       ε−+ε −           ε−+ε =−− γ − γ − γ γ where       − − =* 1 * 2 * 1r)t1( trrR A. Using the assumptions from part i. above, that the contract offers an interest rate combination set optimally for an L investor with a small negative risk aversion coefficient that approaches 0, the spread above is: 2 2 1 2 2A R R))t1(r(r       −=−− Since A = R/r1* in the case of t = 0 and since A is increasing in the tax, this spread is always positive. QED In the case a tax is imposed, constraint C.3, "expect to stay in" must be > 0: () () () () γ γ γ γ− γ ε− −− γ ε − γ ε− +− γ ε =* 211 2 21 r)t1(r )1( )t1(rr )1( )t1(r3C Simplify and insert equilibrium contract values, equations (A.20) and (8'), where the latter is multiplied by (1-t): () () γ γ− γ γ γ −γ γ                       ε−+ε γ ε− −           ε−+ε γ ε− 1 * 2 1A)1(A Rr)1( )1(A R)1( where A is defined as above. Factor Rγ and prepare to combine terms: () () ()                           ε−+ε −                ε−+ε γ ε− γ γ− γ γ γ γ γ− γ −γ γ γ 1 * 2 11 A)1(A r A)1(A 1R)1( 39 Factor the A term and combine. For constraint C.3 to be satisfied the following must be true: () () () 0 A)1( Ar1 A R)1( 1 1 * 2 1 2>                           ε−+ε − γ ε− γ γ − γ − γ γ γ − γ γ γ For the important case when γ < 0, the numerator must be negative. Decomposing A, that means: () () () () 1 * 2 * 1 * 2 1 * 1trrRrr)t1( − γ γ γ − γ γ −<− Take the root of both sides and solve for r1* () Rr)r1(t1(r )1*( 2 * 2 * 1− γ γ <−− Dividing both sides by : () )r1(t1(r * 2 )1*( 2 γ − γ −− , we obtain: () )r1(t1( R rr * 2 )1(* 2 * 1 γ γ − −− < This is true by assumption A.2 QED.