A Theory of Portfolio Choice and Partial Default Kieran James Walsh

y

University of Virginia Darden School of Business July 2016

Abstract I develop a portfolio choice model that allows for partial default and accommodates trade in a rich set of assets. I characterize the solution to an in…nite horizon, consumption/portfolio problem with Markov shocks and many assets. The characterization facilitates a simple solution algorithm and allows me to establish properties of the model. For example, agents default more in bad economic times. I de…ne Recursive Default Equilibrium, which extends the equilibrium concept of Dubey, Geanakoplos, and Shubik (2005) to include an in…nite number of time periods and many assets. I prove existence for a tremble-re…ned equilibrium that rules out o¤-equilibrium pessimism. Keywords: partial default, portfolio choice, dynamic programming, existence of equilibrium, small open economy, international capital ‡ows JEL Codes: C61, C62, E44, F34, F41, G11

I would like to thank John Geanakoplos, Tony Smith, and Aleh Tsyvinski for their guidance, widsom, and support, which greatly contributed to this work. This paper has greatly bene…ted from detailed comments from and discussions and collaboration with Alexis Akira Toda. I also thank Costas Arkolakis, Andrew Atkeson, Brian Baisa, Lint Barrage, Javier Bianchi, William Brainard, David Childers, Michael Curran, Maximiliano Dvorkin, Gabriele Foà, Andrew Kloosterman, Udara Peiris, David Rappoport, Eric Young, and seminar participants at Yale, Darden, UCLA, U-Maryland, UWisconsin, the Federal Reserve Board of Governors, BU, Geneva Finance Research Institute, BIS, 2014 ICMAIF, 2014 SED, 10th CEPR MGI Workshop, 2015 INFINITI, NRU-HSE-ICEF (Moscow), the 12th Yale Cowles GE conference, and the St. Louis Fed for insightful comments and useful discussions. y Email: [email protected]

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1

Introduction

Portfolio composition played a large role in the global …nancial crisis that developed over 2007-2008, both across borders1 and within the U.S.2 In large part due to the crisis, there is a growing consensus that gross ‡ows and portfolio composition are at least as important as aggregates (like GDP) or net measures (like the current account) in conducting monetary and macroprudential policy.3 Therefore, to be useful in performing counterfactual policy exercises or in forecasting, macroeconomic models should accommodate trade in a rich set of …nancial securities. However, the majority of quantitative macroeconomic and international …nance models include at most two types of assets, an equity claim on GDP and a one-period bond. Furthermore, many (if not most) sovereign default models contain only a single bond with …xed maturity and a constant and ‡at risk-free yield curve. In this paper, I develop a new framework for the study of gross portfolio ‡ows, default, and haircuts in a small open economy facing domestic and foreign shocks. Speci…cally, I study the properties of an in…nite horizon consumption/savings model that allows for many assets, which may be long duration, default on liabilities, and Markov shocks. Agents in a small open economy face uncertain …nancial returns, which are determined by large external investors. In each period, the domestic agents choose how much to consume, how to allocate savings in a portfolio of assets and debt instruments, and how much to default on the external investors. Risk spreads on defaultable bonds are determined in equilibrium, balancing the risk premium required by international investors with the domestic appetite for debt and default. Why do domestic borrowers ever choose to repay the external lenders? As in Dubey, Geanakoplos, and Shubik (2005) and its successive papers,4 I assume there is a utility penalty from default, which is proportional to the amount defaulted. I extend their analysis by incorporating their equilibrium concept and model of debt and default in an in…nite horizon, small open economy setting with long horizon assets and liabilities. My main contribution is in using recursive portfolio choice methods to characterize the solution to my model of default and portfolio choice. Speci…cally, I extend the technique and computation of Toda (2014), who builds on Samuelson (1969) and others, to an environment with default. I show that for an individual agent there 1

See, for example, Obstfeld (2012) and Shin (2012). See, for example, Mian and Su… (2011). 3 See, for example, Forbes and Warnock (2012a), Auclert (2016), and Kaplan, Moll, and Violante (2016). 4 See, for example, Peiris and Tsomocos (2015), Tsomocos (2003, 2008), Goodhart, Sunirand, and Tsomocos (2006), and Goodhart, Peiris, and Tsomocos (2015). 2

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are two state variables, wealth and the aggregate state of the world (the Markov shock). Proposition 1, which requires homothetic utility and proportionality of default penalties, says that at the agent optimum there is e¤ectively a separation of the portfolio choice decision from the consumption/savings/default decision. I show that the value function splits into two components, wealth “plugged in”the period utility function and a state-dependent constant, which is implicitly de…ned by a recursion that monotonically converges from unity. Up to these unknown constants and the optimal portfolio, consumption and default are given in closed form and are proportional to wealth, which essentially falls out as a state variable due to proportionality in wealth for the variables of interest. The optimal portfolio depends on the aggregate state but not on wealth or consumption. I explain how this characterization enables a rapid, global computation of the solution to the optimization problem. My second contribution is in using Proposition 1 to derive two general properties of the agent solution. Proposition 2 shows that as market prospects decline, agents consume less and default more. Therefore, in contrast with many international default models,5 default in my model always occurs in “bad times,” regardless of the persistence of the underlying shocks. Corollary 1 shows that delivery rates (one minus the haircut) implied by agent optimization depend only the current and previous realizations of the aggregate state and not on agent wealth. Because delivery rates are the only aspect of the domestic markets concerning the external pricers, this corollary implies that the Markov process is the only aggregate state variable. Following Proposition 1 and Corollary 1, the third contribution of the paper is in de…ning and studying the new concept of Recursive Default Equilibrium (RDE), which extends the equilibrium concept of Dubey, Geanakoplos, and Shubik (2005) to my Markov, in…nite horizon, portfolio choice setting. An RDE is essentially a delivery rate …xed point. Given delivery rates, the external investors pin down bond prices. Given these prices, domestic agent optimization yields new delivery rates. An RDE is a set of delivery rates that (i) depend only on the current and last realizations of the exogenous Markov process and (ii) imply themselves via agent optimization (essentially, rational expectations). Following the Selten (1975) inspired strategy of Dubey, Geanakoplos, and Shubik (2005), Proposition 2 shows that a trembling “"-boosted” RDE, which excludes self-ful…lling pessimistic equilibria, always exists. Taking the tremble to zero, there are re…ned, non-pessimistic RDE (Corollary 3). In Section 2 I outline the related literature and my contribution. Section 3 describes the agent problem and characterizes its solution and economic properties. 5

See, for example, the literature surrounding Eaton and Gersovitz (1981) and Arellano (2008).

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Section 4 de…nes equilibrium and shows existence. Section 5 provides an example.

2

Related Literature

My analysis contributes to three literatures. The …rst is the set of papers using recursive methods to characterize the solutions to consumption/savings problems. See, for example, Toda (2014), Schmidt and Toda (2015), Lichtendahl, Chao, and Bodily (2012), Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007), Alvarez and Stokey (1998), and Samuelson (1969). My paper is the …rst to characterize the consumption/savings/portfolio solution while simultaneously allowing for partial default, many assets, and Markov shocks. Second, my work contributes to the Dubey, Geanakoplos, and Shubik (2005) literature. Goodhart, Sunirand, and Tsomocos (2006) study equilibrium existence, fragility, and macroprudential and monetary policy in an incomplete markets model with banks, default, and money (see also Tsomocos (2003)). Following this paper (and Tsomocos (2008)), Peiris and Tsomocos (2015) build a multi-country model with incomplete markets, default, and money and prove existence for their concept of International Monetary Equilibrium with Default. My paper furthers this literature by incorporating the Dubey, Geanakoplos, and Shubik (2005) model into an in…nite horizon small open economy model (the previous works are two period general equilibrium models)6 and by showing how to compute equilibrium with default and many long horizon assets. Finally, my work is related to the international …nance literatures on default and capital ‡ows. A growing theme in international …nance is that gross ‡ows and their composition are important, perhaps more so than current accounts, in understanding cross-border …nancial risks, crises, and default. See, for example, Forbes and Warnock (2012a), Forbes and Warnock (2012b), Obstfeld (2012), Broner, Didier, Erce, and Schmukler (2013), Alfaro, Kalemli-Ozcan, and Volosovych (2014), and Ahmeda, Curcuru, Warnock, and Zlatea (2015). Consequently, theorists are now striving to extend single asset quantitative models of ‡ows and default to include many assets and thus maturity and debt/equity decisions. See, for example, Bianchi, Hatchondo, and Martinez (2013), Bai (2013), Evans and Hnatkovska (2012), Devereux and Sutherland (2011), Tille and van Wincoop (2010), Pavlova and Rigobon (2012), Stepanchuk and 6

One exception is Goodhart, Peiris, and Tsomocos (2015), who calibrate and solve a linear approximation to a long-horizon model of Germany and Greece, who faces of a proportional utility cost of default. The authors argue that debt restructing in the shadow of crisis bene…ts both Greece and Germany.

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Tsyrennikov (2015), Rabitsch, Stephanchuk, and Tsyrennikov (2015), and Arellano and Ramanarayanan (2012).7 In contrast with these papers, my framework allows for rapid computation of an exact, global solution to a model with partial default, endogenous spreads, and many assets: the multiplicative nature of my model and the assumptions of the Dubey, Geanakoplos, and Shubik (2005) default model mean that adding assets in my framework does not expand the state space. In Walsh (2015) I use my model to explain four sets of facts about emerging market international …nance that are puzzles in or beyond the scope of the existing literature. Calibrating the small open economy to Latin American and the external agents’stochastic discount factor to US stock and bond market data, my model simultaneously generates procyclical gross capital ‡ows, plausible haircuts and default probabilities, and high levels of gross external debt.

3

Model and De…nition of Equilibrium

Consider an economy with an in…nite number of time periods, t = 0; 1; 2; : : : and an exogenous underlying shock process st . At time t, st is randomly equal to one of S possible values: st 2 S = f1; : : : ; Sg. st is a Markov process with transition matrix , where ss0 denotes element (s; s0 ) of . I assume that ss0 > 0 for all (s; s0 ). In the recursive formulation below, I drop references to t and let s and s0 denote, respectively, the shock realization today and tomorrow. Two sets of agents populate the economy. First, there is a mass-one continuum of domestic agents (think of them as representing the citizens of a small open economy). The agents have identical utility functions, face the same budget constraints, and are atomistic and anonymous. This implies that an agent’s portfolio and default decisions do not impact his risk spread. However, in equilibrium, the collective actions of the small open economy agents will be consistent with the aggregate laws of motion investors take as given. In particular, individuals’debt and default policies generate the aggregate delivery rates on anonymous bond pools, which in turn determine spreads.8 7

The consumer credit literature has also recently emphasized the importance of extending default models to include multiple assets and thus a gross/net debt distinction. See, for example, Mitman (2016), Livshits (2015), and Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007). 8 Kim and Zhang (2012), following Jeske (2006) and Wright (2006) and others, also analyze a small open economy model in which capital ‡ows are determined by atomistic households. In their model, however, default is determined by the central government, and they abstract from partial default and assets beyond a one quarter bond. In the consumer …nance setting, Chatterjee, Corbae, Nakajima, and Ríos-Rull (2007) study a quantitative model of debt and default with atomistic but

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The second group of agents consists of rich, external investors. I assume that these foreign agents are extremely wealthy relative to the total value of the domestic market, which is thus a “small open economy.” Rather than explicitly modeling the utility maximization of these foreign agents, I simply represent them with an S S pricing kernel or stochastic discount factor (SDF) matrix M = fmss0 gs;s0 2S , M 0. This pricing kernel yields the economy’s prices and spreads, given default rates and the underlying shock and dividend processes. This is a reduced-form for a model in which the external investors maximize utility from consumption but where …nal consumption is independent of domestic market choices and assets. In either case, the point is that domestic market excess demand for assets does not exert pressure on prices. Rather, prices are uniquely determined such that the external investors are indi¤erent to the small open economy portfolio. In particular, they are willing to take positions opposite to those of the active agents. At di¤erent prices, the external investors would e¤ectively perceive arbitrage opportunities and asset markets would never clear. This does not mean that my analysis is partial equilibrium: given bond default rates, the pricing kernel yields risk spreads. These risk spreads in turn imply portfolio and default policies. However, in the aggregate, these policies may not coincide with the original bond default rates. Therefore, solving the model entails …nding a default rate …xed-point.

3.1

Asset Structure

The agents of the small open domestic economy borrow from the international investors via anonymous bond pools, one for one-period bonds (asset 0) and many for long-term bonds (assets 1 through J1 ). J1 also denotes the set f0; 1; : : : ; J1 g. The long-term bonds are decaying perpetuities with distinct decay rates j 2 (0; 1]. To borrow via a pool, an agent simply sells a share of the pool at the market rate qj and promises to make payments in the future equal to either 1 for the one-period bond or the stream 1; j ; j2 ; j3 ; : : : for perpetuity j. Thus, the short-term bond, asset 0, is a = 0 long-term bond. Because there are many, anonymous domestic agents, each borrower takes the market price as given. However, the agents may deliver less than promised and thus default. From the perspective of the lenders, buying shares of the pools is risky, as they may deliver less than 100% in some states. For simplicnon-anonymous agents. In their work, agents face a schedule of interest rates corresponding to di¤erent loan sizes. However, in part due to having only a single asset, they cannot perfectly match debt statistics and do not target data on interest rates. The main di¤erences in my analysis are that I allow for many assets and partial default, which in my setting is akin to missing payments rather than fully going bankrupt.

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ity, I assume that default is spread equally across outstanding bonds. Speci…cally, the pools have the same stochastic one-period ahead delivery rate dss0 2 [0; 1], which everyone takes as given. Let d = fdss0 gs;s0 2S 2 [0; 1]S S be the delivery rate matrix. Note that to simplify notation, I for now assume that delivery rates depend just on the current and subsequent realization of the Markov process st . Below, I prove that this is indeed the case in equilibrium. Given delivery rates, there are unique pool or bond prices that make the lenders willing to meet the demand for credit (in particular, the pricing kernel will yield unique prices, given delivery rates). Conversely, given prices, the current portfolio choices of agents will a¤ect future delivery rates. Each agent is small relative to the pools and therefore does not internalize the impact of his actions on the prices he faces. However, in my de…nition of equilibrium, I impose a rationality or consistency requirement on lenders: they must correctly forecast the pool delivery rates, stateby-state. There are also J J1 default-free but potentially stochastic assets (assets J1 + 1 through J). J also denotes the set f0; 1; : : : ; Jg. I assume J + 1 S, so there are not redundant assets. These assets may be either in …xed supply and priced by the international investors via M or constant returns to scale “AK”technologies available to the domestic agents. Because the pricing kernel is exogenous, there is no need to be explicit about their dividend processes. However, I assume that the t to t + 1 returns on these assets depend only on s and s0 , an assumption which implicitly restricts the underlying processes. If, for example, dividends are growing over time, the growth rate must just be a function of s and s0 . Let Rj (s; s0 ) 2 (0; 1) ; j 2 fJ1 + 1; : : : ; Jg ; denote the gross returns on the J

3.2

J1 default-free assets.

Asset Pricing

While the returns of the default-free assets are e¤ectively exogenous, risky bond prices (and thus spreads) are determined in equilibrium by the international investor SDF and the portfolio and default decisions of the domestic agents. I say that bond prices (qjd (s), j 2 J1 ), which depend on delivery rates d, are consistent with the international

7

SDF if they make international investors indi¤erent to lending: qjd (s) =

S X

ss0 mss0

d j qj

(s0 ) + 1 dss0 ;

(1)

s0 =1

which in matrix form implies ! q dj = [I

j

(

M

d)]

1

(

M

d) 1

where is the Hadamard product (element-by-element multiplication), and 1 is a vector of ones. By incorporating dss0 multiplicatively in equation 1, I have made two assumption regarding default. First, default comes in the form of a proportional haircut on a bond’s principal and coupon. In other words, if the current delivery rate is d, only a fraction d of the current coupon is paid, and a fraction d of the remaining value is written-o¤. Second, the haircut is the same across maturities. It is straightforward to extend my analysis to exogenously allow for di¤erent deliveries on principal vs. interest or to allow for heterogeneous haircuts across maturities, but doing so complicates notation. To ensure prices are …nite and continuous in d, I impose the following assumption: Assumption q: The gross one-period risk-free interest rate is always strictly greater than maxj2J1 j , that is, max

j2J1 ;s2S

3.3

j

X

ss0 mss0

< 1:

s0 2S

Portfolio Choice and Consumption

Let c denote the period consumption of a domestic agent. An agent’s period utility from consumption takes the constant relative risk aversion (CRRA) form, u (c) =

c1 1

;

and agents discount future utility ‡ows at rate , 0 < < 1. Throughout, I assume that > 1.9 Let D 0 denote the amount defaulted by an agent, and let = 9

Proposition 2 aside, my analysis would proceed similarly with 1. However, in the proofs the < 1 and = 1 cases must be dealt with separately from the > 1 case, so I maintain > 1 for simplicity of exposition. Proposition 2, however, requires > 1, as I explain below.

8

( 0 ; 1 ; :::; J )0 2 be the portfolio weights on the J + 1 assets. I assume the following about the portfolio set : Assumption T:

is a compact and convex set such that (i) 8 > < > :

9 > 10 = 1 = J+1 ; ; 0 2R > ; 0 8j 2 J1 j

2 RJ+1

(ii) for all j 2 J1 and any 2 , the portfolio e with ej = 0 and ei = for i 2 Jn fjg is feasible, and (iii) there exists 2 such that 0.

P

i=

i6=j i

imposes arbitrary short10 = 1 says that the portfolio weights sum to one, sale constraints, j 0 8j 2 J1 says the agents may not have net long positions in the defaultable assets, and part (iii) means there are feasible portfolios without shorting/borrowing. Part (ii) says that given any portfolio it is always possible to deleverage on an arbitrary defaultable bond j 2 J1 by proportionally decreasing the rest of the portfolio to maintain 10 = 1. An immediate implication is that there are feasible portfolios with j = 0 for all j 2 J1 , that is, portfolios that don’t use the defaultable assets. Part (ii) is a weak assumption in that it accommodates convex J X collateral constraints of the form kj;i i gj j for all j 2 J and positive ki;j and i=J1 +1

gj . To see why, choose 2 such that j < 0 for some j 2 J1 , and form e with ej = 0 P and ei = i = i6=j i for i 2 Jn fjg. 10 e = 1 by construction. Since j < 0 and 10 = 1, P it must be that i6=j i > 1. Thus, e since 0 , and ej 0 8j 2 J1 . Finally, the j collateral constraint is strictly satis…ed, and the others continue to hold by linearity. Note, however, that while I use part (ii) to prove existence of equilibrium (via Corollary 2), the partial equilibrium solution (Proposition 1) just requires a compact and convex , j 0 8j 2 J1 , and feasibility of not borrowing/shorting. I impose the additional structure at the outset only for ease of exposition.10 Why do I impose j 0 8j 2 J1 ? As I will show in the proof of Proposition 1, this assumption facilitates the analysis by ensuring that the budget set is convex. This assumption is without loss of generality in many instances. Suppose J1 = 0 so that the only defaultable asset is the one period bond. Provided a non-defaultable, non-shortable one period bond is available for purchase, the assumption does not a¤ect optimal consumption/default/utility (or the net bond position) in equilibrium: 10

could also depend on s, but I exclude this for parsimony.

9

if the agents are not borrowing, there can be no equilibrium spread, and the assets are the same. If the agents are borrowing at a risk premium in equilibrium, the buying constraint is not binding. Thus, having the two assets with the long/short constraints is equivalent to having a single defaultable bond. Next, I de…ne the pre-default returns promised by the agents, given aggregate delivery rates and the corresponding prices. Speci…cally, let Rjd (s; s0 ), j 2 J1 , denote the bond pool return promised by a domestic agent: Rjd

0

(s; s ) =

d j qj

(s0 ) + 1

max qjd (s) ; 1=R

;

(2)

where R is an arbitrary number such that J X

1 > R > max

2 s;s0 2S j=1+J1

j Rj

(s; s0 ) :

0 and j 0 8j 2 J1 , R is above the highest possible return an agent Since qjd could ever get in any state for any d. I include the maximum in the de…nition of Rjd (s; s0 ) to keep returns bounded when deliveries are very low. As I will show below, this assumption does not impact portfolio or default decisions and is thus without loss of generality. Intuitively, once qjd (s) 1=R the interest rate on asset j exceeds the highest possible return, which means it never makes sense to borrow using the asset. Let J1 J X X d 0 d 0 0 R ( ; s; s ) = j Rj (s; s ) + j Rj (s; s ) j=0

j=1+J1

be the gross return, net of promises, on portfolio . Let Rds be the S J matrix of returns from state s to s0 . With this notation, Rd ( ; s; s0 ) is the s0 element of the vector Rds . Also, de…ne R`d

0

( ; s; s ) =

J1 X

Rjd (s; s0 ) min ( j ; 0)

j=0

to be the defaultable liability part of the gross return, that is, the promise function. Due to Assumption T, this function is linear in on , and R`d ( ; s; s0 ) 0 for any 2 . By Assumption q and the de…nition of Rjd , Rd ( ; s; s0 ) and R`d ( ; s; s0 ) are continuous in d and (see the appendix for the proof):

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Lemma R: Under Assumption q, for any s; s0 2 S, the functions Rd ( ; s; s0 ) and R`d ( ; s; s0 ), [0; 1]S S to R, are continuous in ( ; d). For the rest of Section 3, I take d (and thus Rjd (s; s0 )) as exogenous. Therefore, for exposition, I suppress d superscripts until I return to equilibrium in Section 4. Also, for the rest of Section 3, I make the following assumption about returns (given delivery rates): Assumption R1: For all s 2 S, the columns of Rs are linearly independent. Assumption R1 (like Assumption R2 in Section 4) ensures that the optimal portfolio is unique. Note that versions my main results, Propositions 1-3, hold without these rank conditions. However, dropping these assumptions can introduce set-valued optimal portfolios, which complicate notation without adding insight in my context. Furthermore, in many cases of interest, these assumptions are without loss of generality. For example, since J + 1 S and M 0, Assumption R1 is irrelevant when non-defaultable dividends are linearly independent and the only defaultable bond is one period. In this case, the non-defaultable returns are linearly independent by no arbitrage, and collinearity between R0d (s; s0 ) and the risk-free one period bond is irrelevant since 0 0 by Assumption T and because d 1.

3.4

Default

The cost of default is a utility penalty, which is proportional to the level of default. In particular, the utility cost of defaulting an amount D 0 at t is ! e

D;

where is > 0 is a constant, and ! e (de…ned below) is the pre-default liquid net wealth of the defaulter. In short, there is a proportional, linear cost of default, but the marginal cost of default declines as the wealth of the borrower grows. This cost speci…cation is a reduced-form for the myriad of losses that may accompany breaking a deal, including embarrassment, moral injury, legal fees, reputation decline, or material penalties (output loss, trade loss, or jail, for example). The ! e term in the cost of default is included to facilitate analytic solutions below, but it also has a natural interpretation. What the ! e term implies is that the marginal cost of default is declining as the agent grows in wealth. Just as jails and punishments have become less draconian as society has progressed, I assume that punishment in the model “…ts 11

the crime:”the cost of default is proportional to the marginal utility of consumption, which declines as wealth grows. Without the ! e term, agents would quickly “grow out” of default. In particular, as I will show below, this speci…cation ensures that the default-wealth ratio is stationary. It might seem that this speci…cation would generate lots of default in good economic times. I will show below that this is not the case.11 A second interpretation of the ! e term concerns the presence of a minimum consumption level, which is a result of my model of default. Mechanically, the default cost introduces a minimum level of consumption, as a fraction of liquid net wealth ! e : default occurs if and only if u0 (c) ! e . If u0 (c) > ! e , the agent will want to default further and consume more at a net utility gain. If u0 (c) < ! e , the agent should default less and cut consumption until either D = 0 or the wedge closes. Be1= ! e . All else equal, cause u0 (c) = c , the minimum consumption level is c = because default is unpleasant, agents want to ful…ll their promises. However, if keeping promises entails low consumption, the agents will renege until c is possible or there is no further debt on which to default. Without the ! e term, the minimum consump1= tion level would be . Thus, the inclusion of the ! e implies that c grows with wealth. This is like consumption habit depending on previous consumption/wealth.12

3.5

Domestic Agent Optimization Problem and Solution

Given return and promise functions R ( ; s; s0 ) and R` ( ; s; s0 ), the recursive optimization problem of an agent is v (!; s) = max c;Ds0 ;

c1 1

+ Es v (e ! 0 + D s0 ; s0 )

subject to (i) : ! e 0 = R ( ; s; s0 ) (!

(ii) :

(iii) : 0

2

Ds0

! e 0 Ds0

c)

R` ( ; s; s0 ) (!

11

c) :

For simplicity, I have removed much of the heterogeneity present in the original model of Dubey, Geanakoplos, and Shubik (2005). For example, they allow the default cost to vary across agents. While it is straightforward in my setting to allow for idiosyncratic multiplicative wealth shocks, I have not explored how or if I can accommodate other forms of heterogeneity. Doing so would be interesting because it would allow me to explore the role of adverse selection in international capital markets. See Dubey and Geanakoplos (2002) and Dubey, Geanakoplos, and Shubik (2005) for discussions of adverse selection in this class of models. 12 See Campbell and Cochrane (1999) for an example of history dependent utility stemming from consumption habit.

12

where v (!; s) is the value function, ! is post-default liquid net worth or wealth, and ! e 0 is pre-default liquid net wealth.13 Let cr and Dr;s0 denote the consumption and default fractions of post-default wealth, cr = c=! Dr;s0 = Ds0 = (!

c) ;

and let ` denote the fraction of pre-default wealth comprised of defaultable liabilities: ` ( ; s; s0 ) =

R` ( ; s; s0 ) (! ! e0

c)

=

R` ( ; s; s0 ) ; R ( ; s; s0 )

where the second equality follows from constraint (i) in the optimization problem. These de…nitions are important, because, as we will see, the solution to the problem is proportional to wealth. The …nal piece of notation needed is the de…nition of Us (z), which measures the utility from an optimized portfolio and default rate, given a particular level of wealth and arbitrary marginal utilities z 2 RS++ : Us (z) = max Es ;Dr;s0

"

(R ( ; s; s0 ) + Dr;s0 )1 zs0 1

(R ( ; s; s0 ))

Dr;s0

#

subject to 2 0

R` ( ; s; s0 ) :

Dr;s0

As I show in the proof of Proposition 1, which is in the appendix, Us (z) is the continuation value of the recursive problem, after pulling out (! c)1 . Extending the analyses of Toda (2014), Alvarez and Stokey (1998), and Samuelson (1969), Proposition 1 characterizes the domestic agent solution. Proposition 1 Assume Us (1) (1 ) < 1 for all s 2 S, and maintain Assumptions q, T, and R1. Then there are S constants z 2 RS++ such that the domestic agent solution satis…es: 1

1. v (!; s) = zs !1 13

:

As usual, Es [ ] denotes E [ js ].

13

2. The optimal portfolio does not depend on !: (s) = arg max Es 2

"

zs0 (R ( ; s; s0 ) + Dr;s0 (s; ; z ))1 = (1 (R ( ; s; s0 )) Dr;s0 (s; ; z )

where Dr;s0 (s; ; z) is optimal default per unit of savings ! and Dr;s0 (s; ; z) = min max R ( ; s; s0 )

zs0

1=

)

#

c, given

; and z,

1 ; 0 ; R` ( ; s; s0 ) :

3. Consumption is proportional to ! and depends just on Us (z ): c (!; s) = !cr (s) = !

1 ) Us (z ))1=

1 + ( (1

4. Default is proportional to ! e 0 and depends just on z and ` ( (s) ; s; s0 ): Ds0 (!; s) = ! e 0 (!; s; s0 ) min max

zs0

1=

!

!

1; 0 ; ` ( (s) ; s; s0 ) ;

where ! e 0 (!; s) is optimal pre-default wealth, following (!; s): ! e 0 (!; s; s0 ) = !R ( (s) ; s; s0 ) (1

cr (s)) :

5. z is a …xed-point of the recursion

zs = 1 + ( (1

) Us (z))1=

:

What the proposition says is that e¤ectively there is a separation of portfolio choice from the consumption/saving/default decision. Furthermore, relative to individual wealth !, agents make the same decisions. In particular, they choose identical portfolio weights. Relative to !, decisions just depend on , the cost of default, and Us (z ) and zs , which measure economic conditions going forward. (1 ) Us (z ) and zs could be high (meaning times are bad) when equity returns are expected to stay low or when borrowing rates are high. The key assumption in deriving the proposition is the ! e 0 term in the cost of default, which allows me to both include partial default and solve the problem using portfolio choice techniques. A consequence of this proposition is that given returns it is computationally easy 14

to globally calculate a solution to the agent problem. I know the shape of the value function, and, given the z’s, …nding the optimal portfolio at each node s (solving for the Us ’s) is computationally straightforward. The recursion that determines the z’s converges quickly in practice (monotonically from 1), and, unlike with standard value function iteration, does not include a guess of the shape of value function. Given the z’s and the portfolio solution, the other variables are given in closed form. The proof of Proposition 1 is essentially “guess and check.” The conjecture of 1 v (!; s) = zs !1 yields the policy functions and recursively de…nes z via an operator S S T z :R ! R . Following Toda (2014), I show that the sequence of functions 1, T 1, T 2 1, T 3 1, . . . monotonically converges to a …xed point z =T z , which constitutes a solution to the problem. The main di¢ culties in extending Toda (2014) to my setting is default. Ultimately, however, the regularity condition that ensures existence, Us (1) (1 ) < 1, is the same as the one in Toda (2014). This is because default necessarily makes the recursion operator for the z’s more tightly bounded.

3.6

Properties of the Domestic Agent Solution

In this section, I establish three properties of the agent solution that are implications of Proposition 1. The following proposition (proved in the appendix) characterizes the relationship between the state of the economy and default. Proposition 2 (Default in Bad Times) For any level of pre-default wealth ! e and default limit ! e `, zs1 zs2 implies Ds1

c (e ! + Ds1 ; s1 )

Ds2

c (e ! + Ds2 ; s2 ) :

To the extent that z measures economic conditions, this proposition tells us that as market prospects deteriorate, the agents consume less, save more, and default more. When zs is high, meaning utilities Us (z ) and v (!; s) = zs ! 1 = (1 ) are low, the marginal utility of consumption is high. When the marginal utility of consumption is high, it is worth defaulting and paying the marginal penalty, which is relatively low. In other words, agents default when they are desperate, in marginal utility terms. Unlike in the Eaton-Gersovitz models, potential defaulters are not implicitly playing a game with lenders who are threatening coordinated exclusion from capital markets. In Eaton-Gersovitz models one might default because times are good and expected to stay so for a while: in such instances, it may be unnecessary to borrow in the 15

near future, meaning market exclusion is a very weak penalty. My model predicts that agents will default only when investment opportunities look bleak. This is true for any degree of shock persistence and holds in spite of the fact that the marginal cost of default is declining in wealth. Note that the ! e term e¤ectively neutralizes low wealth as a cause of default. When agents are poor, both the marginal utility of consumption and the marginal cost of default rise. We see from Proposition 1 that my speci…cation ensures that these e¤ects exactly cancel. Without the ! e term, the solution to the model would not be so simple, but the relationship between default and bad times would be even stronger. Note that > 1 is important for this interpretation of Proposition 2. If < 1, then the value function v (!; s) = zs ! 1 = (1 ) is increasing in zs , and portfolio utility Us (z ) is positively related to zs . In this case, while default is still increasing in zs , high zs represents good times. See Schmidt and Toda (2015) for general results on the relationship between market prospects and consumption/savings in related models. Next, I show that the bond pool delivery rates implied by agent optimization depend only on s and s0 . Consequently, equilibrium delivery rates will depend only on s and s0 . For post-default wealth !, let F d be the operator that maps any delivery rate d into a new delivery rate implied by agent optimization: 1

Dsd0 (!; s) R`d ( d (!; s) ; s; s0 ) (!

cd (!; s))

;

which is one minus the default rate (total default divided by defaultable liabilities). Note that I am here explicitly indicating that all endogenous variables depend on expected deliveries d. Corollary 1 and its proof (in the appendix) will help us characterize and prove existence of equilibrium below: Corollary 1 Delivery rates F d implied by the agent solution depend just on the exogenous shock process, (s; s0 ), not on wealth. Speci…cally, for any d 2 [0; 1]S S , agent optimization implies (under the assumptions of Proposition 1): 0

F d = @1 2 RS

min max

zs0d =

1=

`d ( S

d

:

1; 0 ; `d (s) ; s; s0 )

d

(s) ; s; s0

1 A

s;s0 2S

Corollary 2, which I prove in the appendix, establishes that including the maximum in the denominator of Rjd (s; s0 ) = j qjd (s0 ) + 1 = max qjd (s) ; 1=R does not 16

impact optimal agent allocations or implied delivery rates. That is, the return upper bound, which is useful in proving existence, is without loss of generality. ed (s; s0 ) = Corollary 2 For any d 2 [0; 1]S S , Rjd (s; s0 ) and R j yield identical agent solutions and implied delivery rates.

4

d j qj

(s0 ) + 1 =qjd (s)

De…nition and Existence of Recursive Default Equilibrium

Given the Markov structure, it is clear that s is a state variable for the economy. Given delivery rates that depend only on (s; s0 ), Proposition 1 tells us that agent optimization is homogeneous in wealth and that optimal policy functions are e¤ectively independent of wealth. Finally, Corollary 1 tells us that the delivery rates implied by agent optimization depend just on (s; s0 ). Therefore, it is natural to de…ne an equilibrium concept in which s is the only aggregate state variable and in which beliefs about future default rates are self-ful…lling and thus rational. Greatly simplifying computation, to forecast prices agents do not not need to forecast future aggregate wealth. This equilibrium, which I call “Recursive Default Equilibrium,” extends the concept of Dubey, Geanakoplos, and Shubik (2005) to a small open economy in…nite horizon setting with many risky assets, including long-term bonds. De…nition of Recursive Default Equilibrium (RDE) An RDE consists of delivery rates d, defaultable bond prices qjd (s) ( j 2 J1 ), and policy functions v (!; s), (!; s), c (!; s), and Ds0 (!; s), ! e 0 (!; s; s0 ) such that the following hold:

1. Optimality: Given prices, the policy functions solve the domestic agent optimization problem. 2. Market Clearing: Given delivery rates, prices are consistent with the international SDF. 3. Consistency: For all s; s0 2 S, if R`d d (!; s) ; s; s0 6= 0 (that is, if defaultable debt exists) then delivery rates are generated by domestic agent optimization and do not depend on !: dss0 = 1 dss0 is arbitrary when

R`d

Ds0 (!; s) ( (!; s) ; s; s0 ) (!

R`d ( (!; s) ; s; s0 ) = 0. 17

c (!; s))

:

I now turn to the matter of existence. As in Dubey, Geanakoplos, and Shubik (2005), it is not di¢ cult to see that trivial RDE often exist. Suppose d 0. d Then qj (s) 0, which implies the agents choose no defaultable borrowing, that is, R`d ( (!; `; s) ; s; s0 ) = 0. This is an RDE because the consistency condition holds trivially. We are left with what is an e¤ectively partial equilibrium problem, a solution to which exists by Proposition 1. In other words, the “pessimistic”RDE always exists in my model. Therefore, the more interesting question is, does an RDE with dss0 < 1 and positive defaultable debt ( R`d ( (!; s) ; s; s0 ) > 0) for some s; s0 exist? As described in Dubey, Geanakoplos, and Shubik (2005), the trivial equilibrium is the result of pessimism by the external investors regarding behavior “o¤ the equilibrium path,”that is, when R`d = 0. In the spirit of Selten (1975), Dubey, Geanakoplos, and Shubik (2005) o¤er a re…nement that precludes the pessimistic equilibrium and prove a re…ned equilibrium exists. In what follows, I adapt their technique to the RDE setting. The basic idea is to introduce an " tremble in the form of an auxiliary player (which I call the stalwart) who also sells into the defaultable bond pool (on the order of ") but never defaults. That is, the stalwart bounds delivery rates away from zero. To maintain the wealth proportionality of the problem (and the result of Corollary 1), I assume these trembles occur in proportion to future wealth ! e 0 (!; s; s0 ). This means that while the tremble is state-contingent, the tremble rate is not.14 The resulting “"-boosted” RDE cannot have a pessimistic equilibrium: if d 0 then R`d ( (!; s) ; s; s0 ) drops to zero, leaving only the stalwart in the bond pool, meaning the realized delivery rate is 1. Proposition 3 shows that an "-boosted RDE exists for any " > 0. Taking " ! 0, we construct a re…ned RDE that cannot be pessimistic because it is the limit of "-boosted RDEs. First I formally de…ne the "-boosted RDE: De…nition of "-boosted RDE Given " > 0, an "-boosted RDE consists of d(") delivery rates d ("), defaultable bond prices qj (s) ( j 2 J1 ), and policy functions v (!; s), (!; s), c (!; s), Ds0 (!; s), and ! e 0 (!; s; s0 ) such that the following hold:

1. Optimality: Given prices, the policy functions solve the domestic agent optimization problem.

14

Thus, if the tremble is "e = ! e 0 ", the delivery rate is 1

which depends only on (s; s0 ).

"e

D0 =1 (! c) R` ( )

18

D0 =e !0 ; " + `0 ( )

2. Market Clearing: Given delivery rates, prices are consistent with the international SDF. 3. Consistency: For all s; s0 2 S, delivery rates are generated by domestic agent optimization and do not depend on !: dss0 (") = 1 =1

dss0 = 1 when

d(")

R`

Ds0 (!; s) =e ! 0 (!; s; s0 ) " + ` ( (s) ; s; s0 ) min max (zs0 = )1=

1; 0 ; ` ( (s) ; s; s0 )

" + ` ( (s) ; s; s0 ) ( (!; s) ; s; s0 ) = 0.

Even though the presence of the stalwart rules out a pessimistic equilibrium, I can prove an "-boosted RDE exists, after slightly strengthening the regularity condition from Proposition 1. De…ne Ues (z) to be the same as Us (z) in Section 3, but with the additional constraints j = 0 for all j 2 J1 and Dr;s0 = 0. In other words, Ues (z) is optimized utility without access to the defaultable assets. Proposition 3 imposes an upper bound on (1 ) Ues (z) instead of (1 ) Us (z). Also, to simplify exposition, I impose the following assumption: Assumption R2: For all s 2 S and any d 2 [0; 1]S linearly independent.

S

, the columns of Rds are

Assumption R2 is not needed for existence and only serves to ensure the optimal portfolio is not set valued. Without this assumption, the portfolio problem, while concave, is not strictly concave, meaning (s) may be a set. Allowing this would complicate the notation and require use of Kakutani’s …xed point theorem (instead of Brouwer’s) but would leave the analysis otherwise unchanged. Furthermore, as explained in Section 3, Assumption R2 is weak in that it is always satis…ed in some cases of interest, for example, when non-defaultable dividends are linearly independent and the only defaultable bond is one period. Proposition 3 (proved in the appendix) establishes existence. The proof strategy is to de…ne a function G (z; d) that simultaneously updates, via agent optimization, marginal utility and delivery guesses (z; d) 2 RS++ [0; 1]S S . A …xed point (z ; d ) = G (z ; d ) then constitutes an "-boosted RDE. Proposition 3 Assume Ue (1) (1 ) < 1 for all s 2 S and maintain Assumptions q, T, and R2. Then for any " > 0 an "-boosted RDE exists. 19

A consequence of Proposition 3 is that non-pessimistic RDE exist. To see why, …rst note that because marginal utilities and delivery rates are bounded by [0; 1K] [0; 1] for any ", the limit as " ! 0 exists and is an RDE. By taking a convergent subsequence, we know that the RDE is “close” to an "-boosted RDE and thus a “re…ned” RDE in the sense that it cannot be pessimistic. An RDE supported by pessimism cannot survive the survive the tremble, which discontinuously pushes the delivery rate to 1. I formalize the limit in Corollary 3 and then conclude with an example with on equilibrium debt and default. Corollary 3 Maintain the assumptions of Proposition 3. The re…ned equilibrium constructed by taking the limit of "-boosted RDE as " ! 0 is an RDE.

5

Example

Suppose there are three states, bad, medium, and good (S = fB; M; Gg), and two assets, a stock and a one-period defaultable bond. For s; s0 2 S, let Ra (s; s0 ) and Rrf (s) be, respectively, the stock return and risk free rate. Assume the international investor is risk-neutral (but with stochastic time preference) so that for delivery rates d = fdss0 g 2 [0; 1]9 the promised risky interest rate Rb (s) is given by Rrf (s) ; Rbd (s) = X ss0 dss0 s0 2S

1= Let = 2, = :5, = :31, and assume 0 1, where b is the defaultb able bond portfolio weight (so 1 b is the stock weight). Then with the following return/shock structure,

0 0:70 B 0 Ra (s; s ) = @0:90 0:80 0 :40 B = @:25 :25

1 0 1 0:99 0:99 0:99 1:10 1:50 B C C 1:10 1:25A ; Rrf (s; s0 ) = @1:00 1:00 1:00A ; 1:07 1:07 1:07 1:00 1:40 1 :30 :30 C :50 :25A ; :25 :50

20

there is a non-pessimistic RDE consisting of 0

1 1 B d = @:96 1 1 1 0 :2949 B 0 0 c =e ! = @:3100 :2949

1 1 C 1A ; 1

0

1 0 1 0 11:50 B C B C b = @ 1A ; z = @10:23A ; 0 10:37 1 :3126 :3106 C :3126 :3106A ; spread (M ) = :01: :3126 :3106

That is, when the delivery rate from medium to bad is :96 (4% haircut) and otherwise 1, the resulting agent decisions yield the same delivery rates. The investment opportunity is excellent in M , which leads the agent to fully leverage in this state. Otherwise, the agent just holds stock. In the worst state of the world, B, consumption would fall below the minimum, so the agent wants to default. However, ` (G; B) = ` (B; B) = 0 because b (G) = b (B) = 0, so the default constraint binds in state B, except following state M . In state M , there is default risk, and bonds trade with a 1% risk premium, that is, Ra (M ) Rrf (M ) = :01.

6

References 1. ARELLANO, C. (2008): “Default Risk and Income Fluctuations in Emerging Economies,”American Economic Review, 98, 3, 690-712. 2. ARELLANO, C. AND A. RAMANARAYANAN (2012): “Default and the Maturity Structure in Sovereign Bonds,” Journal of Political Economy, 120, 2, 187-232. 3. ALFARO, L., S. KALEMLI-OZCAN, AND V. VOLOSOVYCH (2014): “Sovereigns, Upstream Capital Flows, and Global Imbalances,” Journal of the European Economics Association, 12, 5, 1240-1284. 4. ALVAREZ, F., AND N.L. STOKEY: “Dynamic Programming with Homogeneous Functions,”Journal of Economic Theory, 82, 167-189. 5. AUCLERT, A. (2016): “Monetary Policy and the Redistribution Channel,” Working Paper. 6. AUSUBEL, L.M., AND R.J. DENECKERE (1993): “A Generalized Theorem of the Maximum,”Economic Theory, 3, 1, 99-107. 21

7. BAI, Y. (2013): “Discussion on ‘Gross Capital Flows: Dynamics and Crises’by Broner, Didier, Erce, and Schmukler,” Journal of Monetary Economics, 60, 1, 134-137. 8. BIANCHI, J., J. HATCHONDO, AND L. MARTINEZ (2013): “International Reserves and Rollover Risk,”Working Paper. 9. BRONER, F., T. DIDIER, A. ERCE, AND S. SCHMUKLER (2013): “Gross capital ‡ows: Dynamics and crises,” Journal of Monetary Economics, 60, 1, 113-133. 10. CAMPBELL, J.Y., AND J.H. COCHRANE (1999): “By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior,”Journal of Political Economy, 107, 2, 205-251. 11. CHATTERJEE, S., D, CORBAE, M. NAKAJIMA, AND J.-V. RÍOS-RULL (2007): “A Quantitative Theory of Unsecured Consumer Credit with Risk of Default,”Econometrica, 75, 6, 1525-1589. 12. DEVEREUX, M., AND A. SUTHERLAND (2011): “Country Portfolios in Open Economy Macro-Models,”Journal of the European Economic Association, 9, 2, 337-369. 13. DUBEY, P., AND J. GEANAKOPLOS (2002): “Signalling and Default: RothschildStiglitz Reconsidered,”The Quarterly Journal of Economics, 117, 1529-1570. 14. DUBEY, P., J. GEANAKOPLOS, AND M. SHUBIK (2005): “Default and Punishment in General Equilibrium,”Econometrica, 73, 1, 1-37. 15. EATON J., AND M. GERSOVITZ (1981): “Debt with Potential Repudiation: Theoretical and Empirical Analysis,” Review of Economic Studies, 48, 2, 289309. 16. EVANS, M.D.D., AND V. HNATKOVSKA (2012): “A Method for Solving General Equilibrium Models with Incomplete Markets and Many Financial Assets,” Journal of Economic Dynamics & Control, 36, 1909-1930. 17. FORBES, K., AND F. WARNOCK (2012a): “Capital Flow Waves: Surges, Stops, Flight, and Retrenchment,” Journal of International Economics, 88, 2, 235-251.

22

18. FORBES, K., AND F. WARNOCK (2012b): “Debt- and Equity-Led Capital Flow Episodes,”NBER Working Paper. 19. GOODHART, C., P. SUNIRAND, AND D. TSOMOCOS (2006): “A Model to Analyse Financial Fragility,”Economic Theory, 27, 1, 107-142. 20. GOODHART, C., M.U. PEIRIS, AND D. TSOMOCOS (2015): “Debt, Recovery Rates and the Greek Dilemma,”Working Paper. 21. JESKE, K. (2006): “Private International Debt with Risk of Repudiation,” Journal of Political Economy, 114, 3, 576-593. 22. KAPLAN, G., B. MOLL, AND G.L. VIOLANTE (2016): “Monetary Policy According to Hank,”NBER Working Paper 21897. 23. KIM, Y.J., AND J. ZHANG (2012): “Decentralized Borrowing and Centralized Default,”Journal of International Economics, 88, 1, 121-133. 24. MIAN, A., AND A. SUFI (2011): “House Prices, Home Equity-Based Borrowing, and the US Household Leverage Crisis,”American Economic Review, 101, 5, 2132-2156. 25. LICHTENDAHL, K.C., R.O. CHAO, AND S.E. BODILY: “Habit Formation from Correlation Aversion,”Operations Research, 60, 3, 625-637. 26. LIVSHITS, I. (2015): “Recent Developments in Consumer Credit and Default Literature,”Journal of Economic Surveys, 29, 4, 594-613. 27. MITMAN, K. (2016): “Macroeconomic E¤ects of Bankruptcy and Foreclosure Policies,”American Economic Review, 106, 8. 28. OBSTFELD, M. (2012): “Does the Current Account Still Matter?” American Economic Review Papers & Proceedings, 102, 3, 1-23. 29. PAVLOVA, A., AND R. RIGOBON (2012): “Equilibrium Portfolios and External Adjustment under Incomplete Markets,”Working Paper. 30. PEIRIS, M.U., AND D. TSOMOCOS (2015): “International Monetary Equilibrium with Default,”Journal of Mathematical Economics, 56, 45-57. 31. RABITSCH, K., S. STEPANCHUK, AND V. TSYRENNIKOV (2015): “International Portfolios: A Comparison of Solution Methods,” Journal of International Economics, 97, 404-422. 23

32. SAMUELSON, P. (1969): “Lifetime Portfolio Selection By Dynamic Stochastic Programming,”Review of Economics and Statistics, 51, 3, 239-246. 33. SELTEN, R. (1975): “Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,” International Journal of Game Theory, 4, 25-55. 34. SHIN, H.S. (2012): “Global Banking Glut and Loan Risk Premium,”IMF Economic Review, 60, 155-192. 35. STEPANCHUK, S., AND V. TSYRENNIKOV (2015): “Portfolio and Welfare Consequences of Debt Market Dominance,” Journal of Monetary Economics, 74, 89-101. 36. TILLE, C., AND E. VAN WINCOOP (2010): “International Capital Flows,” Journal of International Economics, 80, 157-175. 37. TODA, A.A. (2014): “Incomplete Market Dynamics and Cross-Sectional Distributions,”Journal of Economic Theory, 154, 310-348. 38. TSOMOCOS, D. (2003): “Equilibrium Analysis, Banking and Financial Instability,”Journal of Mathematical Economics, 39, 619-655. 39. TSOMOCOS, D. (2008): “Generic Determinacy and Money Non-Neutrality of International Monetary Equilibrium,”Journal of Mathematical Economics, 44, 866-887. 40. WALSH, K.J. (2015): “Portfolio Choice and Partial Default in Emerging Markets: A Quantitative Analysis,”Working Paper. 41. WRIGHT, M.L.J. (2006): “Private Capital Flows, Capital Controls and Default Risk,”Journal of International Economics, 69, 1, 120-149.

7 7.1

Appendix Proof of Lemma R

For any d 2 [0; 1]S

S

, s 2 S, and j 2 J1 , recall that the pricing equation is qjd

(s) =

S X

ss0 mss0

s0 =1

24

d j qj

(s0 ) + 1 dss0 :

De…ne qs = Es [mss0 ], and let qdj be the vector of prices (across states). Let q = maxs2S qs . The pricing equation de…nes the operator : RS ! RS q=

j

(

M

d) q + (

M

d) 1:

Since 0 0 then is monotone. De…ne K = q= (1 j q). ss0 ; mss0 ; dss0 , if q 0 < K < 1 since 0 < ss0 ; mss0 and j q < 1 (by Assumption q). If q 1K, then ( q)s =

S X

ss0 mss0

( j qs0 + 1) dss0

s0 =1

=

S X

ss0 mss0

( j K + 1)

s0 =1

j Kqs

+ qs

j Kq

+q

= K; where the …rst inequality uses dss0 1, and the second uses the de…nition of q. Thus, is a self map on [0; 1K]. Choose q1 ; q2 2 [0; 1K]. By the triangle inequality and 0 ss0 ; mss0 ; dss0 , k q1

S X

q2 k1 = max s2S

max s2S

max s2S

=

ss0 mss0 j

(q1;s0

q2;s0 ) dss0

s0 =1 S X

s0 =1 S X

ss0 mss0 j

jq1;s0

ss0 mss0 j

max jq1;s0 0

s0 =1 S X

max s2S

s0 =1

s 2S

ss0 mss0 j

!

kq1

q2;s0 j q2;s0 j q2 k1 :

P Therefore, since maxs2S Ss0 =1 ss0 mss0 j < 1 by Assumption q, is a contraction mapping. By the contraction mapping theorem, the pricing equation has a unique and …nite …xed point qdj on [0; 1K], for any d 2 [0; 1]S S . Also, because is linear and thus continuous in both d and q, the …xed point must be continuous in d. To see this, …rst recall that by the contraction mapping property, k q1 q2 k1 kq1 q2 k1 for some 2 (0; 1). Fix d, and choose " > 0. By continuity of in d, we can …nd 0 d > 0 such that if kd d0 k1 < then qdj qj 1 < " (1 ). Therefore, by

25

the triangle inequality and contraction mapping property, if kd 0

qdj

qdj

0 d0 qj

qdj

= 1

0 d qj

qdj

=

)+

0 d qj

+

qdj

0 d0 qj

0 d qj

+

0 d qj 1

qdj " (1

1

0 qdj

0 d0 qj

d0 k1 <

then

1 1

1

=) 0

qdj

qdj

1

" (1 1

)

= ":

Thus, the …xed point qdj is continuous in d, since d and " were arbitrary. The lemma now quickly follows. Since qjd (s) 2 [0; K] is continuous in d, it is immediate that 0 d j qj (s ) + 1 d 0 Rj (s; s ) = max qjd (s) ; 1=R is non-negative, continuous in d, and bounded above. Rd ( ; s; s0 ) and R`d ( ; s; s0 ) are linear in gross returns and thus continuous in d as well. Rd ( ; s; s0 ) and R`d ( ; s; s0 ) are continuous in by linearity, the continuity of min ( j ; 0), and the boundedness of returns.

7.2

Proof of Proposition 1

The proof proceeds in three steps: (1) Guess form of value function and simplify the problem; (2) Characterize the policy functions; and (3) Con…rm form of initial guess, and show existence/construction of z. The agent’s recursive optimization problem is v (!; s) = max c;Ds0 ;

c1 1

+ Es v (e ! 0 + D s0 ; s0 )

subject to (i) : ! e 0 = R ( ; s; s0 ) (!

(ii) :

(iii) : 0

2

Ds0

! e 0 Ds0

c)

R` ( ; s; s0 ) (!

c) :

Step 1: Guess form of value function and simplify the problem Suppose that there exist S strictly positive constants z = fzs gs2S 2 RS++ such 26

that v (!; s) = zs ! 1 = (1

). Plugging in the guess, de…ning the policy “rates” cr = c=! Dr;s0 = Ds0 = (!

c) ;

and dividing both sides by ! 1 , the optimization problem becomes zs 1

= max cr

cr1 1

+ (1

cr )1

Us (z) ;

(3)

where Us (z) = max Es ;Dr;s0

"

(R ( ; s; s0 ) + Dr;s0 )1 zs0 1

(R ( ; s; s0 ))

Dr;s0

#

(4)

subject to (ii) : (iii) : 0

2 Dr;s0

R` ( ; s; s0 ) :

Note that a …nite optimized value Us (z) exists by the extreme value theorem because the objective function is continuous in (by Lemma R) and Dr;s0 , and the constraint set is compact (due to Assumptions T and Lemma R).15 In particular, 1 < Us (z) < 0 since > 1. Note that Us (z) is nonzero by the compactness of the budget set and …niteness of returns (Lemma R and the de…nition of Rjd (s; s0 )). Step 2: Characterize the policy functions First, consider the optimization problem of equation 3, which gives optimal consumption as a fraction of wealth. The problem is strictly concave because the second derivative is 1 cr 1 (1 ) (1 cr ) Us (z) ; which is negative because (1

) and Us (z) are both negative. Therefore, the …rst

15

There are discontinuities at R ( ; s; s0 ) 0, but this does not a¤ect the analysis: since by assumption it is always feasible to not borrow and returns are strictly positive, there is a lower bound U > 1 on the value from maximizing over and Dr;s0 . De…ne C to be ( ; Dr;s0 ) such that (ii) and (iii) hold and utility exceeds U. C is bounded because is bounded. C is closed because (ii) and (iii) are closed and since the inverse image of [U; 1) is closed (by continuity). The objective function is continuous on C, which is compact, and maximizing over C yields the same solution as just using constraints (ii) and (iii).

27

order condition gives the solution: cr (s) =

1 ) Us (z))1=

1 + ( (1

c (!; s) = !cr (s) :

Next, consider the problem of equation 4, which gives optimal portfolio choice and default. A solution exists by the argument in step 1. It is also unique. This is because the constraint set is convex by Assumption T and the linearity of Dr;s0 R` ( ; s; s0 ) (Assumption T) and because the problem is strictly concave. To see strict concavity, de…ne (R + D)1 R D; f (R; D) = z 1 which has gradient

"

z (R + D) + 5f = z (R + D)

1

R R

D

#0

:

There are two cases to consider, z and z > . When z , @f =@D < 0 if D > 0, so D = 0 binds, and the objective is f (R; 0) = zR1 = (1 ), which is strictly concave. When z > , @f (R; 0) =@D > 0, so there is default, and z (R + D) R . The inequality is strict when Dr R` binds. The Hessian is 52 f = where

" 1

= z (R + D) det 52 f =

2

=

+

2

( + 1) R 2

R

( + 1) R 2

D

+

1

R

#

;

> 0. Thus

2

=

2

( + 1) R 1 + R

R

2

1 2

+

D

2 2

R

D+2

z (R + D)

2

R 1

2 1

2 2

R

2

(( + 1) D + 2R)

2

R

:

Since ( + 1) D + 2R > R + D, we have that det 52 f >

2

R

2

z (R + D)

R

0;

where the second inequality follows from z (R + D) R . Since the (1; 1) elek ment of 52 f is negative, we have that ( 1) det (52 f )k > 0 for all k 2 f1; 2g, where 28

(52 f )k is the k th principal minor of 52 f . It immediately follows that 52 f is negative de…nite, meaning f is strictly concave in the D > 0 case too. Because R ( ; s; s0 ) is linear in and because the returns across states are linearly independent (Assumption R1),16 the strict concavity of f (R; D) implies that the objective function of equation 4 is strictly concave in ( ; Dr;s0 ). Since the budget set is convex, the problem of equation 4 thus has a unique solution. Given and ignoring the constraints 0 Dr;s0 R` ( ; s; s0 ), the …rst order condition with respect to Dr;s0 yields zs0 1= 1 : Dr;s0 = R ( ; s; s0 ) Including the constraints, the solution (given ) is zs0

Dr;s0 (s; ; z) = min max R ( ; s; s0 )

1=

1 ; 0 ; R` ( ; s; s0 ) :

Therefore, the optimal portfolio is given by (s) = arg max Es 2

"

zs0 (R ( ; s; s0 ) + Dr;s0 (s; ; z))1 = (1 (R ( ; s; s0 )) Dr;s0 (s; ; z)

)

#

Finally, de…ning ! e 0 (!; s) = R ( (s) ; s; s0 ) (!

` ( ; s; s0 ) = and using Dr;s0 = Ds0 = (! Ds0 (!; s) = (!

c (!; s))

R` ( ; s; s0 ) R ( ; s; s0 )

c), we also have c (!; s)) Dr;s0 (s; ; z) zs0

=! e 0 (!; s) min max

1=

1; 0 ; ` ( (s) ; s; s0 ) :

Step 3: Con…rm form of initial guess, and show existence/construction of z Plugging the policy functions into equation 3 and multiplying both sides by (1 ), we get zs = 1 + ( (1 ) Us (z))1= ; 16

This ensures that for all s, if

1

6=

2

then R ( 1 ; s; s0 ) 6= R ( 2 ; s; s0 ) for some s0 .

29

which de…nes the operator T z :RS ! RS where (T z)s = 1 + ( (1

) Us (z))1=

Therefore, if T has a …xed point z =T z on RS++ , I have con…rmed the initial guess. To construct a …xed point, I employ the method of Toda (2014), which is to show that T is continuous and that the sequence 1, T 1, T 2 1, T 3 1, . . . is monotonically increasing and bounded on [1; K1], for some K > 1 (steps (3.I) through (3.IV) below). (3.I) If z1 z2 then T z1 T z2 : After solving 4 we get Us (z1 ) = Es

"

z1;s0 (R ( (s; z1 ) ; s; s0 ) + Dr;s0 (s; (s; z1 ) ; z1 ))1 = (1 (R ( (s; z1 ) ; s; s0 )) Dr;s0 (s; (s; z1 ) ; z1 )

)

#

;

where (s; z1 ) is the arg max. Since (s; z2 ) is not the arg max, it provides less utility. Therefore, Us (z1 ) " Es

z1;s0 (R ( (s; z2 ) ; s; s0 ) + Dr;s0 (s; (s; z2 ) ; z2 ))1 = (1 (R ( (s; z2 ) ; s; s0 )) Dr;s0 (s; (s; z2 ) ; z2 )

)

#

Us (z2 ) ; where the second inequality follows from > 1 and linearity in z (when holding and Dr;s0 constant). It immediately follows that (1 ) Us (z1 ) (1 ) Us (z2 ). Since T is increasing in (1 ) Us (z), we have that T z1 T z2 . (3.II) T z 1 for z 0: This is immediate from (1 ) Us (z) 0 (see step 1) and the de…nition of T . (3.III) T z K1 for some K > 1 and any z 2 [1; K1]: De…ne y = max f (1 s2S

K = 1= 1

y 1=

) Us (1)g :

By the premise of the proposition we have y < 1, which implies K > 1. K is …nite because y > 0, which follows from (1 ) and Us (1) having the same sign. Note that Us (1) is nonzero by the argument in step 1. Now, choose z 2 [1; K1]. It is the case

30

that (1

) Us (z)

) Us (K1)

(1 K (1

) Us (1)

Ky: The …rst inequality is the result of monotonicity (3.I). The third inequality is from the de…nition of y. To see the second inequality, observe that "

K (R ( (s; K1) ; s; s0 ) + Dr;s0 (s; (s; K1) ; K1))1 = (1 Us (K1) = Es (R ( (s; K1) ; s; s0 )) Dr;s0 (s; (s; K1) ; K1) " # K (R ( (s; 1) ; s; s0 ) + Dr;s0 (s; (s; 1) ; 1))1 = (1 ) Es (R ( (s; 1) ; s; s0 )) Dr;s0 (s; (s; 1) ; 1) # " (R ( (s; 1) ; s; s0 ) + Dr;s0 (s; (s; 1) ; 1))1 = (1 ) = KEs (R ( (s; 1) ; s; s0 )) Dr;s0 (s; (s; 1) ; 1) K

)

#

KUs (1) ; where the …rst inequality uses the arg max argument from part (3.I) and the second uses K > 1. It follows that (1 ) Us (K1) K (1 ) Us (1), as claimed. Thus, by (1 ) Us (z) Ky and the de…nition of y we have (T z)s = 1 + ( (1

) Us (z))1=

1 + (Ky)1= = K; as claimed. (3.IV) T z is continuous in z on [1; K1]: As argued in step 1, the constraint set of problem 4 is compact (by Assumption T and Lemma R), the objective function is continuous in and Dr;s0 . Since the objective function is also continuous in z, Us (z) is continuous in z by the maximum theorem.17 Therefore, because T is continuous in Us (z), T is also continuous in z. 17

There are discontinuities at R ( ; s; s0 ) 0. Using the argument in footnote 15, this does not prevent the invocation of the maximum theorem because we can restrict the budget set to keep utility above a …nite lower bound.

31

By steps (3.I) through (3.III), it is the case that T1

T 21

T 31

:::

and T n 1 K1 for all n. By the monotone convergence theorem, there is a limit z = limn!1 T n 1. Hence, since [1; K1] is closed and T is continuous (step (3.IV)), z = T z is a …xed point on [1; K1].

7.3

Proof of Proposition 2

Fix pre-default wealth ! e and defaultable liabilities wealth fraction `. By Proposition 1, ! ! 1= zs 1; 0 ; ` : Ds = ! e min max It is immediate that if zs1

zs2 then Ds1

Ds2 as claimed. Also by Proposition 1,

c (e ! + Ds ; s) = (e ! + Ds ) cr (s) = (e ! + Ds ) = (e ! + Ds ) zs

1=

1 1 + ( (1

;

) Us (z ))1=

since ) Us (z ))1=

zs = 1 + ( (1

Combining this with the expression for Ds , c (e ! + Ds1 ; s1 ) diate from (e ! + Ds ) cr zs

1=

=

! e+! e min max

=! e min max

1=

32

; zs

: c (e ! + Ds2 ; s2 ) is imme!

zs

1=

1=

; (` + 1) zs

!!

1; 0 ; `

1=

:

zs

1=

7.4

Proof of Corollary 1

For any post-default wealth !, d 2 [0; 1]S ahead delivery rate is

d

(!; s) =

Ds0 (!; s) = (!

, and state s 2 S, the implied one period

Dsd0 (!; s) R`d ( d (!; s) ; s; s0 ) (!

(F d)(!;s;s0 ) = 1 By Proposition 1,

S

d

cd (!; s))

:

(s) does not depend on !, and

c (!; s)) Dr;s0 (s; ; z)

=! e 0 (!; s) min max

! e 0 (!; s) = R ( (s) ; s; s0 ) (!

zs0

1=

1; 0 ; ` ( (s) ; s; s0 )

c (!; s)) :

Plugging these expressions in F d, it follows that (F d)(!;s;s0 ) Rd

d

(s) ; s; s0

cd (!; s) min max (zs0 = )1=

!

=1

=1

R`d ( min max (zs0 = )1= `d (

d

(s) ; s; s0 ) (!

1; 0 ; `d d

d

1; 0 ; `d

d

(s) ; s; s0

cd (!; s))

(s) ; s; s0 ;

(s) ; s; s0 )

which depends on (s; s0 ) but not !.

7.5

Proof of Corollary 2

ed and Rd yield the same solution. Suppose If qjd (s) 1=R for all j and s, it is clear R j j then that for some j 2 J1 and s 2 S, we have qjd (s) < 1=R. In this case, for all s0 2 S ed (s; s0 ) = R j

(s0 ) + 1 > qjd (s)

d j qj

d j qj

(s0 ) + 1

1=R

= Rjd (s; s0 )

(5)

1 = R; 1=R

ejd (s; s0 ) > Rjd (s; s0 ) R. Now, let (s) be the optimal portfolio (with Rjd that is, R returns), and suppose towards contradiction that j (s) < 0. Form e with ej = 0 and 33

ei =

i

P (s) = i6=j i (s) for i 2 Jn fjg. e 2

R

d

e; s; s0 =

X

i

i2Jnfjg

1

=P

ei Rd (s; s0 ) +

i6=j i

Therefore, Rd e; s; s0 =

=

j

(s)

0 @

X

by Assumption T. Now, observe that J X

ei Ri (s; s0 )

i=1+J1

i

(s) Rid (s; s0 ) +

J X

i

i=1+J1

i2Jnfjg

1

(s) Ri (s; s0 )A :

Rd ( (s) ; s; s0 )

(s) Rjd (s; s0 ) +

d 0 j (s) Rj (s; s ) +

where I used

P

i6=j i

P

!0

1

i6=j i

0

@ j (s)

1 @

(s) X

1=

Rd e; s; s0

j

j

(s)

(s) Rid (s; s0 ) +

P

J X

i=1+J1

i2Jnfjg J X

i

i2Jnfjg

Therefore, by the de…nition of R,

i

ei Rd (s; s0 ) +

1

(s)

X

i=1+J1

1

ei Ri (s; s0 )A ; 1

i6=j i

(s)

and ei =

(s) < 0, and Rjd (s; s0 )

i

i

(s) Ri (s; s0 )A

P (s) = i6=j i (s).

R, it follows that

Rd ( (s) ; s; s0 ) >

j

(s) Rjd (s; s0 ) +

j

=

j

(s) Rjd (s; s0 )

R

(s) R

0: In other words, switching from (s) to e is feasible and provides a higher return in every state of the world. Thus, (s) cannot be optimal, which is a contradiction. This is true even when there is default. The improved return reduces the cost of default, and if the constraint Dr;s0 = R`d ( (s) ; s; s0 ) is binding, e still dominates since the reduction in Dr;s0 is o¤set by the increase in Rd e; s; s0 , of which R`d e; s; s0 is an additive component. Thus, if qjd (s) < 1=R, then it must be that j (s) = 0 with Rjd returns. So, ejd returns only impacts states of the dropping the maximum term and switching to R ed (s; s0 ) > Rd (s; s0 ) when q d (s) < 1=R, j (s) = 0 is world where j (s) = 0. Since R j j j d d e optimal whether or not Rj or Rj is used. The implication is that optimal portfolios and the optimized values of R`d ( ; s; s0 ) and Rd ( ; s; s0 ) are the same, whether or not I include the maximum term. As the remaining endogenous variables only depend 34

1

on these quantities, it is without loss of generality to de…ne promises as Rjd (s; s0 ) = d 0 d j qj (s ) + 1 = max qj (s) ; 1=R .

7.6

Proof of Proposition 3

For ease of exposition, I drop reference to " > 0, which can be arbitrarily small. Also, for now assume that returns are given by equation 2, which includes the maximum function in the denominator. As explained below, by Corollary 2 any equilibrium constructed with this de…nition will also be an equilibrium when the maximum is ed (s; s0 ) = j q d (s0 ) + 1 =q d (s) is used. The proof strategy is to de…ne dropped and R j j j a function G (z; d) that simultaneously updates, via agent optimization, marginal utility and delivery guesses (z; d) 2 RS++ [0; 1]S S . A …xed point (z ; d ) = G (z ; d ) then constitutes an "-boosted RDE. As we saw in the proof of Proposition 1, given any (z; d) 2 RS++ [0; 1]S S , the optimal portfolio d (s; z) and market conditions Usd (z) do not depend on ! and are given by d

(s; z) = arg max Es 2

Usd (z) = max Es 2

"

"

1

d zs0 Rd ( ; s; s0 ) + Dr;s = (1 0 (s; ; z) d Rd ( ; s; s0 ) Dr;s 0 (s; ; z) 1

d zs0 Rd ( ; s; s0 ) + Dr;s = (1 0 (s; ; z) d 0 d R ( ; s; s ) Dr;s0 (s; ; z)

) )

#

#

;

where d Dr;s max Rd ( ; s; s0 ) 0 (s; ; z) = min

Dsd0 (!; s; z) = ! 1

zs0

1=

1 ; 0 ; R`d ( ; s; s0 )

d cdr (s; z) Dr;s 0 (s; ; z)

cdr (s; z) = zs 1= : Furthermore, I showed that if the z’s are a …xed point of (T (z; d))s = 1 +

(1

) Usd (z)

1=

;

then v (!; s) = zs ! 1 = (1 ) and these policy functions solve the domestic agent solution. From the proof of Corollary 1, we see how to update delivery rates, given

35

any (z; d) 2 RS++

[0; 1]S

S

:

min max (zs0 = )1=

(F (z; d))(s;s0 ) = 1

" + `d (

where `d ( ; s; s0 ) =

1; 0 ; `d d

d

(s; z) ; s; s0 ;

(s; z) ; s; s0 )

R`d ( ; s; s0 ) : Rd ( ; s; s0 ) [0; 1]S

Now, de…ne the updating function G : RS++

S

! RS

RS

S

by

G (z; d) = (T (z; d) ; F (z; d)) : To show that there exists a …xed point (z ; d ) = (T (z ; d ) ; F (z ; d )), I prove that G is a continuous self map on the convex and compact set [1; K]S [0; 1]S S , where y = max s2S

n

K = 1= 1

(1 y 1=

o ) Ues (1) :

I address T in step 1 and F in step 2: Step 1: T is continuous in (z; d) 2 [1; K]S [0; 1]S S and bounded by [1; 1K] (1.i) Boundedness: In the proof of Proposition 1, I showed that 0 < (1 ) Usd (z) < 1 for any d and z 0. Thus (T (z; d))s > 1 is immediate. I also showed in part (3.III) of Step 3 of Proposition 1 that for any d, K 1, and z 2 [1; K]S , we have (1

) Usd (z)

K (1

) Usd (1) :

Since Ues (1) re‡ects a more constrained problem than Usd (1), it must be that Ues (1) Usd (1). Therefore, by > 1 (1

) Usd (z)

K (1 Ky;

) Ues (1)

where the second inequality is from the de…nition of y. Thus, (T (z; d))s = 1 +

(1

1 + (Ky)1= = K; 36

) Usd (z)

1=

by the de…nition of K. (1.ii) Continuity: By Lemma R, Rd ( ; s; s0 ) and R`d ( ; s; s0 ) are continuous in d ( ; d) 2 [0; 1]S S , which implies Dr;s . The con0 (s; ; z) is continuous in d and straint set of Usd (z) is compact by Assumption T and Lemma R. Therefore, since z (R + D)1 = (1 ) (R) D is continuous in D, R, and z, the maximum theorem gives us that Usd (z) is continuous in z and d. It immediately follows that T is continuous in (z; d). Note that by the convexity of the constraint set (Assumption T and Lemma R) and the strict concavity of the objective function (Assumption R2 and the argument in part 2 of the proof of Proposition 1), the maximum theorem also gives that d (s; z) is continuous in (z; d).18 Step 2: F is continuous in (z; d) 2 [1; K]S [0; 1]S S and bounded by [0; 1] (2.i) Boundedness: This follows immediately from the de…nition of (F (z; d))(s;s0 ) and " > 0. (2.ii) Continuity: Since min max (X= )1= 1; 0 ; Y = (" + Y ) is continuous in (X; Y ) for Y 0, it just remains to show that d

`

d

R`d (s; z) ; s; s = Rd ( 0

(s; z) ; s; s0 d (s; z) ; s; s0 ) d

is weakly greater than 0 and continuous in (z; d) 2 [1; K]S [0; 1]S S . We saw in part (1.ii) that d (s; z) is continuous in (z; d). Furthermore, because utility becomes in…nitely low when Rd ( ; s; s0 ) approaches 0 and some U > 1 is always feasible, Rd d (s; z) ; s; s0 > 0 for all s; s0 and any (z; d). R`d d (s; z) ; s; s0 0 by ded 0 d 0 …nition. Since R ( ; s; s ) and R` ( ; s; s ) are continuous in d and by Lemma R, `d d (s; z) ; s; s0 is non-negative and continuous in (z; d). Therefore, F is continuous in (z; d). Since G is a continuous self map on the convex and compact set [1; K]S [0; 1]S S , a …xed point (z ; d ) = (T (z ; d ) ; F (z ; d )) exists by Brouwer’s …xed point theorem. At this …xed point, z solves the agent problem given d , and the resulting solution implies the delivery rates d . Therefore, (z ; d ) constitutes an "-boosted RDE with bounded promises. Moreover, Corollary 2 implies that (z ; d ) is an "-boosted RDE ed (s; s0 ) = j q d (s0 ) + 1 =q d (s), without the maximum in the de…ning returns by R j j j d denominator. This is because if max qj (s) ; 1=R = 1=R then j = 0. If I increase ejd (s; s0 ), j = 0 is still optimal. The realized delivery the promise from Rjd (s; s0 ) to R 18

As in footnotes 15 and 17, a subtlety arises due to discontinuities when Rd ( ; s; s0 ) 0. However, because the objective function is u.s.c., with the only discontinuity at 0, and since does not depend on d, I can employ the generalized maximum theorem of Ausubel and Deneckere (1993). See, in particular, their discussion at the end of section 3 on page 103.

37

on asset j is 0 either way.

7.7

Proof of Corollary 3

By Proposition 3, for any " > 0 an "-boosted RDE exists. In equilibrium, for any ", (s; s0 ), and !, we have (letting superscripts denote equilibrium at tremble ") d

ss0

Ds"0 (!; s) =e ! "0 (!; s; s0 ) " + `" ( " (s) ; s; s0 )

(") = 1

min max (zs0 (") = )1=

=1

" + `" (

1; 0 ; `" ( "

"

(s) ; s; s0 ) ;

(s) ; s; s0 )

which is bounded by [0; 1] since " > 0 and ` 0 (by the arguments in Step 2 of the proof of Proposition 3). Furthermore, we have (z ("))s = 1 +

) Usd(") (z ("))

(1

1=

2 [1; K] ;

where K does not depend on " by the arguments in Step 2 of the Proposition 3’s proof. Therefore, by the Bolzano-Weierstrass theorem, we can …nd a sequence f"n g1 n=1 such 1 that "n > 0, limn!1 "n = 0, and fd ("n ) ; z ("n )gn=1 has a limit. De…ne dy ; zy = limn!1 (d ("n ) ; z (")). dy ; zy must constitute an RDE without "-boosting. To see this, …rst note that if y

dy

s; zy ; s; s0 = 0 for all (s; s0 ) then dy ; zy is trivially an RDE. Otherwise,

y

dy

s; zy ; s; s0

`d

`d

> 0 for some (s; s0 ). Choose " > 0. By convergence we can

…nd N1 such that dyss0

dss0 ("n ) < "=2 if n

N1 . Also, by convergence and by the

continuity of min max (zs0 = )1= 1; 0 ; ` = ( + `) in (d; z; ) (Steps 1 and 2 of Proposition 3’s proof), we can …nd N2 such that if n N2 then 1

min(max((zs0 ("n )= )1= 1;0);`"n ( "n +`"n ( "n (s);s;s0 ) min max

1

1=

(zsy0 = ) y `d (

1;0 ;`d dy

y

)

"n (s);s;s0 )

dy

(s;zy );s;s0

(s;zy );s;s0 )

!

< "=2:

Therefore, since

dss0 ("n ) = 1

min max (zs0 ("n ) = )1= "n + `"n ( 38

1; 0 ; `"n ( "n

(s) ; s; s0 )

"n

(s) ; s; s0 ) ;

by the triangle inequality

dyss0

0

B B1 @

min max

1=

zsy0 =

1; 0 ; `d

dy

dy

`

y

dy

s; zy ; s; s0

(s; zy ) ; s; s0

1

C C < ": A

But " was arbitrary, so min max dyss0

=1

zsy0 =

1=

`dy

y

1; 0 ; `d dy

meaning dy ; zy is an RDE.

39

(s; zy ) ; s; s0

dy

s; zy ; s; s0 ;