Flight to Liquidity and Systemic Bank Runs

Flight to Liquidity and Systemic Bank Runs Roberto Robatto, University of Wisconsin-Madison November 15, 2016 Abstract This paper presents a general equilibrium, monetary model of bank runs to study monetary injections during financial crises. When the probability of runs is positive, depositors increase money demand and reduce deposits; at the economy-wide level, the velocity of money drops and deflation arises. Two quantitative examples show that the model accounts for a large fraction of 1) the drop in deposits in the Great Depression and 2) the $400 billion run on money market mutual funds in September 2008. In some circumstances, monetary injections have no effects on prices but reduce money velocity and deposits. Counterfactual policy analyses show that if the Federal Reserve had not intervened in September 2008, the run on money market mutual funds would have been much smaller. JEL Codes: E44, E51, G20 1 Introduction The bankruptcy of Lehman Brothers in September 2008 was followed by a flight to safe and liquid assets and runs on several financial institutions. For instance, Duygan-Bump et al. (2013) and Schmidt et al. (2016) document a $400 billion run on money market mutual funds. In response to these events, the Federal Reserve implemented massive monetary interventions. Flight to liquidity, runs, and monetary interventions characterized the Great Depression as well, although the response of the Federal Reserve was more muted at the time and the US economy experienced a large deflation (Friedman and Schwartz, 1963). Despite the interactions between bank runs, flight to liquidity, and monetary policy interventions, very few models analyze the interconnections among these phenomena. Most of the literature on banking crises assumes that banks operate in environments with only one real good, without fiat money. While this approach is useful for many purposes, in practice banks take and repay deposits using money, giving rise to non-negligible interactions with monetary policy choices. 1 E-mail: robatto@wisc.edu. I am grateful to Saki Bigio, Elena Carletti, Briana Chang, Eric Mengus, David Zeke, and many other participants in seminars and conferences for their comments and suggestions. UPDATES: https: //sites.google.com/site/robertorobatto/papers/robatto_flightliquidity.pdf 1 Few other papers deal with this observation. I review this literature in the next Section. 1

To fill this gap, I present a general equilibrium model of fundamental-based bank runs with money. When the fundamentals of the economy are strong, runs do not arise in equilibrium and the outcomes in the banking sector look very similar to the good equilibrium in Diamond and Dybvig (1983). If instead the fundamentals of the economy are weak, the equilibrium is characterized by runs on many banks (i.e., systemic runs). Runs are associated with a flight to liquidity (i.e., an increase in money demand and a drop in deposits), deflation, and a drop in money velocity. The objective of this paper is to use this model to study the effects of monetary injections on prices and allocations during systemic crises. To highlight the mechanics and transmission mechanisms of monetary injections, I make some stark assumptions to keep the model simple and tractable. In particular, output is exogenous, prices are fully flexible, and, in the baseline model, depositors preferences are locally linear. In this way, my results can be easily be compared with classical monetary models such as Lucas and Stokey (1987). The main results are related to the analysis of temporary monetary injections (that is, injections that are reverted when the crisis is over). During a crisis, temporary monetary injections produce unintended consequences: a reduction in money velocity and an amplification of the flight to liquidity (i.e., a drop in deposits). In particular, in the baseline model, the drop in velocity exactly offsets the direct effect of the monetary injections and, as a result, nominal prices are constant. I argue that these findings are important because the monetary interventions implemented during both the Great Depression and the Great Recession episodes are best characterized as temporary. The unintended consequences of monetary injections are related to the role of money in the microfoundation of the model. To understand this role, recall first the structure of typical threeperiod bank-run models (t =0, 1, 2) without money, such as Diamond and Dybvig (1983), Allen and Gale (1998), and Goldstein and Pauzner (2005). In these models, agents deposit all their wealth into banks at t =0. This is the case no matter whether depositors assign, at t =0, zero probability to runs at t =1(as in Diamond and Dybvig, 1983) or a positive probability (as in Allen and Gale, 1998, and Goldstein and Pauzner, 2005). Formally, depositors choices at t =0are corner solutions. In contrast, there is an explicit role for fiat money in my model; if the probability of runs is positive, depositors keep some money in their wallets rather than depositing it all in banks. That is, households money-holding decisions are interior rather than corner solutions. Whether depositors choices are interior or at a corner makes a critical difference to the analysis of temporary monetary injections. With interior solutions, depositors first-order conditions must hold with equality. In the baseline model, the linearity of depositors preferences imply that these first-order conditions are independent of consumption allocations and thus depend only on nominal prices. Therefore, in the event of a monetary injection, prices must remain the same in order to sustain the first-order conditions. But with more money in circulation, nominal prices remain the 2

same only if money velocity decreases. 2 Given the reduction in velocity, the flight to liquidity is amplified because there is a negative relationship between velocity and the flight to liquidity. This negative relationship is exemplified by the fact that velocity is low when fundamentals are weak and depositors fly to liquidity, whereas velocity is high when fundamentals are strong and depositors do not fly to liquidity. In contrast, with corner solutions, depositors first-order conditions hold with inequality and thus prices do not need to remain the same to sustain the allocation. In that case, the standard logic applies. A temporary injection increases nominal prices and does not affect either velocity or the flight to liquidity. I also analyze the robustness of the results to the structure of depositors preferences. I numerically solve a richer version of the model with standard, strictly concave preferences. In a quantitative example calibrated to study the run on money market mutual funds in September 2008, temporary monetary injections still reduce velocity and, depending on the size of the monetary injection, amplify the flight to liquidity. In addition, temporary monetary injections increase nominal prices but less than one-to-one, due to the endogenous reduction in velocity. Finally, I calibrate the model to present two quantitative examples: one for the Great Depression and the other for the 2008 crisis. First, I compare the model with the Great Depression and show that it accounts for about three-quarters of the drop in deposits between 1929 and 1933. Second, I demonstrate the relevance of the model to the Great Recession by showing that some monetary injections produce an equilibrium with runs and flight to liquidity but no deflation, consistent with stylized facts of the 2008 crisis. 3 The model accounts for almost half of the $400 billion run on money market mutual funds documented by Duygan-Bump et al. (2013) and Schmidt et al. (2016). In the most conservative calibration, counterfactual policy analysis shows that, if the Federal Reserve had not set up facilities to provide liquidity to mutual funds, the run would have been $141 billion smaller but deflation would have occurred. According to the model, the Federal Reserve avoided deflation in 2008 at the expense of an amplification of runs and of the flight to liquidity. 1.1 Additional comparisons with the literature Alvarez, Atkeson, and Edmond (2009) present a model in which a monetary injection produces an endogenous drop in money velocity, as in my paper. However, they focus on a different context (segmented-asset markets rather than bank runs) and thus the logic is somehow different. In their 2 Velocity is defined by the equation of exchange money velocity = prices output. Fixing output, a monetary injection would raise prices if velocity were constant. 3 Between September and December 2008, core inflation was approximately constant at around 2%. To simplify the analysis, I look at the scenario in which monetary injections in the model produce price stability, rather than 2% inflation. All results can be extended to a richer model in which anticipated inflation equals 2%. 3

model, agents have different individual velocities of money and the economy-wide velocity corresponds to the weighted average. A monetary injection is absorbed by agents with low velocity and as a result their weight increases and aggregate velocity drops. In contrast, in my model, all agents absorb the monetary injection and the drop of velocity is required to sustain their first-order conditions. There are a few other papers that analyze monetary injections in the context of bank runs driven by fundamentals, which differ from my paper in important ways. Allen and Gale (1998), Allen, Carletti, and Gale (2013), and Diamond and Rajan (2006) study how monetary policy should respond to aggregate shocks when deposit contracts are nominal and thus not contingent on the price level. As in most of the literature, agents deposit all their wealth into banks, while agents in my model do not deposit all their money in the event of a crisis. Moreover, nominal contracts do not play any role in my analysis. In Antinolfi, Huybens, and Keister (2001), agents again deposit all their initial wealth into banks, and the focus is on determining the optimal interest rate on central bank lending in response to aggregate shocks. Carapella (2012) and Martin (2006) show that monetary injections can eliminate multiple equilibria in models of banking panics. Their results are derived in a Diamond-Dybvig, partialequilibrium economy with money and a general-equilibrium economy in which panics are related to nominal contracts and debt-deflation, respectively. Cooper and Corbae (2002) also analyze how monetary injections can eliminate multiple equilibria in models of banking panics. Despite the different approach to modeling crises (they use multiple equilibria, I rely on fundamentals), depositors choices in their model are interior solutions, as in mine. However, there are two important differences. First, their model is much richer than mine and as a result they focus solely on steady-states in which banks are either well-functioning forever or malfunctioning forever; second, their policy analysis does not consider temporary monetary injections as I do, but only permanent ones. In contrast, my simpler model allows me to study a scenario in which crises eventually end, and therefore I can distinguish between temporary and permanent injections. That said, when I consider permanent monetary injections in my model, I obtain a result similar to Cooper and Corbae (2002); that is, permanent injections create an inflationary pressure by increasing future prices, thereby offsetting deflation and the flight to liquidity. Finally, in a related paper (Robatto, 2015), I build an infinite-horizon, monetary model of bank runs driven by panics, in the sense of multiple equilibria. In some circumstances, temporary monetary injections produce unintended consequences as well; 4 however, the richness of that model required to study multiple equilibria in an infinite-horizon economy hides the logic behind the 4 Thus, the unintended consequences of monetary injections are robust to the nature of the crisis, panic versus fundamentals. 4

Figure 1: Preferences of impatient households u (C) slope =1 slope = >1 C C effect of monetary injections. Moreover, the main focus of that paper is to study the monetary policy stance that eliminates multiple equilibria, similar to the main research question in Carapella (2012) and Cooper and Corbae (2002). 2 Model: the core environment This section presents the core environment without banks and Section 3 derives the equilibrium. Section 4 extends this core environment by introducing banks. The objective is to present a very simple framework that allows me to explain the intuition of the unintended consequences of monetary injections. Section 6 extends the model to relax some of the assumptions, showing that the main forces are still at work and can be quantitatively relevant. Time is discrete and there are three periods indexed by t 2{0, 1, 2}. The economy is populated by a double continuum of households indexed by h 2 H =[0, 1] [0, 1]; the double continuum is required when introducing banks in Section 4. The core environment combines preference shocks at t =1, in the spirit of Diamond and Dybvig (1983), with a cash-in-advance constraint; cash is required to finance consumption expenditure after agents are hit by preference shocks. As a result, a precautionary demand for money arises at t =0, so that households can finance consumption induced by preference shocks at t =1. In order to deal with money in a finite-horizon model, I introduce a technology to transform money into consumption goods at t =2. 5 5

2.1 Preferences Let C1 h and C2 h denote consumption of household h at t =1and t =2, respectively. Households utility depends on a preference shock that is realized at the beginning of t =1: 8 < u C1 h + C2 h (impatient household) with probability apple. utility = (1) : C2 h (patient household) with probability 1 apple Note that both patient and impatient households derive linear utility from consumption at t =2. The function u ( ) is piecewise-linear, as represented in Figure 1: 8 < C if C<C u (C) = >1, C>0. (2) : C + C C if C C The assumption >1captures impatience. If C<C, the marginal utility at t =1is >1and thus larger than the marginal utility at t =2, which equals one. If instead C C, both marginal utilities are one. This structure gives rise to an important driving force, namely, a desire to consume at least C if h is impatient. 6 The local linearity delivers neat results, in particular for policy analysis. Nonetheless, the main results are robust to a more standard, smooth utility function. In this case, though, some analyses can be performed only numerically. More discussion is provided in Section 6. The preference shock is i.i.d. across households and I assume that the law of large numbers holds, so that the fraction of impatient agents in the economy equals apple. Moreover, I assume that the law of large numbers also holds for each subset of H with a continuum of households. 7 The preference shock is private information of household h. 2.2 Assets, production and markets There are two assets with exogenous supply: money and capital. The supply of money is M (1 + µ t ). In this economy without banks, µ t =0for all t; in Section 4, I introduce a central bank that can inject money by choosing µ t > 0. Money is the unit of account; thus, without loss of generality, prices and contracts are expressed in terms of money. Capital is in fixed supply K at t =0. The fixed-supply assumption is made for convenience as 5 In a related paper (Robatto, 2015), I present an infinite-horizon model of banking that motivates this assumption. That is, money has a continuation value because it can be carried over to the next period. 6 Another way to understand the role of u ( ) is to note that this function is globally concave and thus produces risk aversion. 7 This is consistent with the results of Al-Najjar (2004) about the law of large numbers in large economies. 6

Figure 2: Timing of production and markets t =0 t =1 t =2 Walrasian market Q 0 : price of capital Market for consumption goods P 1 : price of consumption (cash-in-advance constraint) Production 1 unit capital! A 1 units of consumption Production 1 unit! capital 1 unit money Q 2/P 2 units of consumption! 1/P 2 units of consumption Preference shocks Shocks to capital it permits abstracting from endogenous investment decisions. Capital is hit by idiosyncratic, uninsurable shocks at t =1. 8 The effect of these shocks is to reallocate capital among agents, leaving the aggregate stock of capital unchanged at K. 9 For a fraction 2 (0, 1) of agents, the stock of capital held by these agents reduces by a factor of 1+ L, where 1 apple L apple 0; for the other 1 agents, capital increases by a factor of 1+ H, where H 0. 10 Since the shocks are idiosyncratic, they must satisfy the restriction 1+ L +(1 ) 1+ H =1 (3) In the rest of the analysis, I use L and to describe the stochastic process of the idiosyncratic shocks, while H is determined residually by Equation (3). These shocks do not play any major role in the bankless economy but are crucial to produce banks insolvencies and runs in the economy with banks. The timing of trading and production is represented in Figure 2. At t =0, there is a Walrasian market in which capital and money can be traded. The price of capital is denoted by Q 0 ; this price does not play any major role, but is required to express the value of capital and thus write budget constraints at t =0. At t =1(after preference shocks and capital shocks are realized), each unit of capital produces A 1 units of consumption goods. Consumption goods are sold at price P 1 and consumption ex- 8 To keep the model simple, I assume that markets are exogenously incomplete, rather than providing an endogenous motivation for the lack of insurance to idiosyncratic shocks to capital. 9 These shocks are equivalent to idiosyncratic shocks to the productivity of capital. Adding aggregate shocks to capital does not change the results qualitatively. 10 More precisely, the shock L hits a fraction of agents holding a share of the capital stock, and the shock H hits a fraction of agents holding a share 1 of capital. In equilibrium, all agents are alike at t =0and thus they hold the same amount of capital. Thus, a fraction of agents is hit by the negative shock and a fraction 1 by the positive shock. 7

penditures are subject to a cash-in-advance constraint; as in Lucas and Stokey (1987), households cannot consume goods produced by their own stock of capital. Capital is illiquid at t =1, i.e., it cannot be traded. 11 At t =2, each unit of money produces 1 /P 2 units of consumption goods and each unit of capital produces Q 2/P 2 units of consumption goods. Q 2 and P 2 are exogenous parameters, but are motivated by an infinite-horizon formulation in which fiat money and capital can be carried over and used in the next period. 12 For future reference, let 1+r2 K ( ) be the nominal return on capital at t =2for an agent that is hit by the idiosyncratic shock to capital. This return is defined by: 2.3 Endowments 1+r K 2 ( ) =(1+ ) Q 2 + A 1 P 1 Q 0. (4) Without loss of generality, I assume that all households have the same endowment of money and capital at t =0. Thus, each household h is endowed with money M and capital K. 2.4 Restriction on parameters I assume that the parameters Q 2 and P 2 that govern the value of capital and money at t =2are proportional to the quantity of money in circulation at t =2: Q 2 = 1 M (1 + µ 2 ) K, P 2 = M (1 + µ 2) A 1 K. (5) The restrictions in (5) are motivated by an infinite-horizon formulation; that is, Q 2 and P 2 would be, respectively, the price of capital and of consumption goods that would arise in t +1 in an infinite-horizon economy. 13 Moreover, these restrictions imply monetary neutrality at t =2. That is, the real value of capital Q 2/P 2 is independent of µ 2 and corresponds to the present-discounted value of output, Q 2 P 2 = 1 A 1, and the price level P 2 increases one-to-one with a change of the money supply. 11 Similar to Jacklin (1987), trading restrictions are required to provide a role for banks. If households could trade capital at t =1and use the proceeds of trade to consume, there would be no role for banks. 12 See the infinite-horizon, monetary model of banking in Robatto (2015). 13 In particular, in the infinite-horizon economy, these prices would arise in a steady-state in which banks are active and there are no runs. Note that the expressions for Q 2 and P 2 in (5) are similar to those derived for Q 0 and P 1 in the good equilibrium with banks; see Proposition 4.1. 8

I also impose a restriction on the parameter C that governs the utility of impatient households defined in (2): C = A 1K apple. (6) A 1 K/apple is the level of consumption at t =1that can be achieved if all impatient households consume the same amount (total production at t =1is A 1 K and there is a mass apple of impatient agents). Equation (6) implies that there is a feasible allocation in which consumption of impatient households is equalized at C and thus their marginal utility equals one; that is, no impatient household has marginal utility >1 in this allocation. 14 3 Equilibrium in the bankless economy The utility maximization problem of household h is given by: max apple M0 h,kh 0,Ch 1 ( u C h 1 + M h 0 P 1 C h 1 + Q 0 K h 0 E 1+r K 2 h P 2 {z } =C2 h if h is impatient +(1 apple) M0 h + Q 0 K0 h E 1+r2 K h P {z 2 } =C2 h if h is patient ) (7) where the expectation is taken with respect to the shocks to capital,. In (7), I use the fact that the optimal consumption of patient households at t =1is zero and thus C h 1 refers to the consumption at t = 1 if the household is impatient. The maximization in (7) is subject to the budget and cash-in-advance constraints: M h 0 + K h 0 Q 0 apple M + KQ 0 {z } value of endowments (8) P 1 C h 1 apple M h 0. (9) At t =0, the household has access to the Walrasian market where it can adjust its portfolio of money and capital, subject to the budget constraint (8); M h 0 and K h 0 are the amount of money and capital that the household has after trading. At t =1, consumption is subject to the cash-inadvance constraint (9). At t =2, consumption is financed with unspent money (M0 h P 1 C1 h if the household is impatient and M0 h if it is patient) and capital bought at t =0plus its return r2 K h, 14 For technical reasons, some results require the utility function u (C) to be differentiable at C = A 1 K/apple and its derivative to equal one. To guarantee these results, Equation (6) can be replaced with C = A 1 K/apple, with >0 but arbitrarily small. 9

where h 2 L, H is the shock to capital held by agent h; the return on capital includes the proceeds from selling output A 1 K0 h (produced by capital at t =1) at price P 1, and the output produced by capital at t =2. To solve problem (7), I guess that the cash-in-advance constraint (9) holds with equality, which is verified if the opportunity cost of holding money is positive; this opportunity cost is represented by the return on capital, E r2 K ( ). The first-order conditions imply: E 1+r K 2 h 1 P 2 = appleu 0 C h 1 1 P 1 +(1 apple) 1 P 2 (10) Households are indifferent between investing an extra dollar in capital or in money at t = 0. Investing in capital gives a return E 1+r K 2 h, discounted by the factor and evaluated in units of time-2 consumption (i.e., the return is divided by 1 /P 2 ). Investing in money allows households to increase consumption at t =1if the household is impatient (with probability apple) or at t =2if the household is patient (with probability 1 An equilibrium of this economy is a collection of prices Q 0 and P 1 and households choices M h 0, K h 0, C h 1 such that (i) M h 0, K h 0, and C h 1 solve the problem (7) given prices, (ii) the money and capital markets clear at t =0, M = R M h 0 dh and K = R K h 0 dh, and (iii) the goods market clears at t =1, R C h 1 dh = A 1 K. In equilibrium, all households are alike at t =0and thus market clearing implies that they hold the same amount of money and capital, M h 0 = M and K h 0 = K for all h. At t =1, only impatient households consume; since there is a mass apple of them and total output is A 1 K, consumption is C h 1 = A 1 K/apple. Next, I solve for the price level at t = 1. apple). I use the fact that only impatient households spend money and consume at t =1. Thus, money spent is applem and consumption expenditure is P 1 R C h 1 dh = P 1 A 1 K. Equating money spent with consumption expenditure, I can solve for the price level P 1 : P 1 = applem A 1 K. (11) To solve for Q 0, I first note that C h 1 = C (using Equation (6)) and thus u 0 C h 1 =1. Plugging u 0 C h 1 =1and Equation (11) into Equation (10), I can solve for the expected return on capital E 1+r K 2 h and, using Equation (4), for the price of capital Q 0. 15 Finally, I comment on welfare in the bankless economy. At t =1, the consumption of impatient households is equalized while the consumption of patient households is zero. This allocation is the same as the one that a social planner would choose. As a result, there is no welfare gain from 1 15 The expected return on capital is E i 1+r2 K h = 1 [1 + (1 apple) ] and the price of capital is Q 0 =. M K h apple+ (1 apple) 1+ (1 apple) 10

introducing banks. Nonetheless, banks can provide deposits, allowing for withdrawals at t =1 and paying a return at t =2and thus providing insurance against preference shocks. As a result, deposits reduce money demand and thus affect equilibrium prices, because banks can offer some insurance against households preference shocks. The baseline model is simple by design to provide an intuitive analysis of the interactions between banks and the money market, rather than welfare. Regardless, Section 6 extends the model in a way that produces a welfare loss in the bankless economy in comparison to the firstbest, opening up a welfare-increasing role for banks. In the extension, patient households have some utility from consumption at t =1. The results of the baseline model can be interpreted as the limiting case in which such utility is arbitrarily small. 4 Introducing banks I now extend the core environment of Section 2 by introducing a unit mass of banks indexed by b that act competitively and a central bank that can change the money supply. Similar to the previous sections, I use the superscript b to denote variables that refer to bank b. Depending on parameters, the equilibrium is either good (no runs at t =1) or bad (some banks are subject to runs at t =1). If the equilibrium is bad, runs are driven by fundamentals, as in Allen and Gale (1998), rather than panics as in Diamond and Dybvig (1983). The interaction between banks and households is standard. Households endowments are the same as in Section 2, whereas banks have no endowment. Each bank is associated with a unit continuum of households and takes prices as given. 16 Banks collect deposits from households at t =0and serve withdrawals at t =1, after households observe their own preference shock. Deposits not withdrawn are repaid at t =2. Recall that capital is subject to shocks at t =1and therefore capital held by banks is hit by these shocks as well. I denote b 2 L, H to be the shock to capital held by bank b. Similarly, h is the shock to capital held by household h. As in the bankless economy, the shock h to capital held by households does not play major role, because households are risk-neutral at t =2. In contrast, b is crucial because banks may become insolvent (and thus subject to runs) if they are hit by a shock b that is sufficiently negative. 16 Since there is a unit mass of banks and each bank is associated with a unit continuum of households, there is a well-defined link between the unit mass of banks and the double continuum of households introduced in Section 3. 11

4.1 Budgets and interaction between households and banks t =0: trading and deposits. Bank b can buy money M b 0 and capital K b 0, using deposits D b 0: K0Q b {z } 0 + M0 b apple D0 b {z} {z} capital money deposits (12) subject to the non-negativity constraints M b 0 0, K b 0 0, and D b 0 0. Household h makes its portfolio decisions by choosing money, deposits, and capital: M0 h {z} money + D h 0 {z} deposits + Q 0 K h 0 {z } capital apple M + KQ 0 {z } value of endowments (13) subject to the non-negativity constraints M0 h 0, D0 h 0, and K0 h 0. Each household can hold its deposits D0 h only at one bank. This assumption can be justified by the costs of maintaining banking relationships. Formally, the cost is zero if household h holds deposits at one bank, and infinite if household h holds deposits at two or more banks. 17 This assumption implies that households face the risk that their own bank may be hit by the negative shock L and possibly subject to a run. If households could deposit at all banks, they would diversify away this risk. In equilibrium, perfect competition in the banking sector implies that the household is indifferent between choosing any bank. For future reference, I denote b (h) to be the bank at which household h deposits at t =0. t =1: withdrawals and consumption. Households observe their preference shocks and then decide their withdrawals, W h 1. For each household, withdrawals cannot exceed deposits D h 0 chosen at t =0. Moreover, for each bank b, total withdrawals by its depositors cannot exceed the amount of money M b 0 chosen at t =0by the bank: Z W1 b = n o W1 h dh apple M0 b depositors (14) of bank b where the integral is taken with respect to households that hold deposits at bank b. The inequality in (14) arises from the fact that, at t =1, there is no market in which banks can sell capital in exchange for money. At each bank, withdrawals are repaid based on a sequential service constraint. 18 If there is a run, 17 The results are qualitatively unchanged if households can deposit at e.g. two or three banks, but it is crucial that households cannot hold deposits at a large number of banks. 18 The sequential service constraint is imposed as a physical constraint as in Wallace (1988), rather than as a restric- 12

the sequential service constraint gives rise to a limit on withdrawal determined by the position in line. Households at the beginning of the line can withdraw all their deposits (and thus W1 h apple D0), h but those at the end of the line cannot withdraw any money because the bank does not have enough cash to serve them. Thus: 8 < D W1 h 0 h if there is no run, or if h is at the beginning of the line in a run apple : 0 if h is at the end of the line in a run. After making withdrawals, households choose consumption expenditure P 1 C1 h, subject to a cashin-advance constraint; that is, P 1 C1 h cannot exceed the sum of money M0 h (chosen at t =0) and withdrawals W1 h : P 1 C1 h apple M0 h + W1 h. (15) t =2: return on deposits and consumption. At t =2, banks are liquidated and thus households receive a return 1+r b 2 b on deposits that have not been withdrawn at t =1. This return is possibly bank-specific because it is affected by the idiosyncratic shocks to capital b. Section 4.2 describes this return in greater detail, along with the possible profits made by banks after repaying depositors, denoted by 2. After deposits are repaid at t =2, households consume C h 2. Similar to the bankless economy, capital bought at t = 0, K h 0, plus its return r K 2 h and unspent money are used to finance consumption. In addition, consumption is financed by: (i) deposits not withdrawn D h 0 W h 1 plus the return r b 2 b paid by the bank; (ii) banks profits 2, if any; and (iii) lump-sum transfer T 2 from the central bank, if any (see Section 4.3). Therefore, household consumption at t =2is: C h 2 = Q 0K h 0 1+r K 2 h P 2 {z } capital + return " + 1 P 2 D0 h W1 h 1+r2 b b {z } deposits not withdrawn, + actual return 4.2 Restrictions on the demand-deposit contract + M0 h +W1 h P 1 C1 h {z } unspent money + 2 +T 2 #. (16) Since the main focus of the paper is understanding the effects of monetary injections, I impose three restrictions on the demand-deposit contract to simplify the analysis. These restrictions imply that the behavior of banks in the baseline model is fairly mechanical and thus I can abstract from the profit-maximization problem of banks. Section 6 extends the model and relaxes some of these restrictions. First, I assume that banks invest a fraction apple of their deposits in money at t =0, in order to tion on contracts. 13

serve withdrawals by the fraction apple of households that receive the impatient preference shock at t =1. I assume that banks make this choice even if there is a positive probability of a run. Using the banks budget constraint (12), the remainder fraction of deposits 1 apple is invested in capital. Second, I do not allow for any suspension of convertibility of deposits at t =1. Third, I impose a cap on the return r b 2 b paid by banks, as explained next. Without imposing any restriction on r b 2 b, all resources available at t =2would be used to repay deposits not withdrawn. In the case that arises in equilibrium in which all the money M b 0 held by banks is withdrawn at t =1, the return on deposits r b 2 b would equal the return on capital held by bank b, r K 2 b, defined in Equation (4). To solve for the equilibrium under the quasi-linear specification of preferences, I must impose a cap on r b 2 b. This cap is required to make sure that impatient households prefer to withdraw their deposits at t =1, rather than waiting until t =2. If I allow banks hit by H to pay a large return to depositors, impatient households might prefer to wait until t =2. This problem is avoided if the cap on r b 2 b is: 1+r b 2 b apple 1/. In the relevant equilibrium cases, a bank hit by H > 0 pays 1+r b 2 H =1/ ; any remaining profits, if any (denoted by 2 ), are distributed lump-sum to households. 19 A bank hit by L < 0 has fewer resources and the cap is not binding. To sum up: 8 < 1/ if b = H 1+r2 b b = (17) : 1+ L Q 2 +A 1 P 1 Q 0 = 1+r2 K ( L ) if b = L where the last equality follows from (4). Note that, if b = L < 0 and L is sufficiently low, then r b 2 L < 0. To see this, consider the limit case in which L = 1 and thus r b 2 L = 1. In equilibrium, r b 2 b < 0 triggers a run on bank b; withdrawing money and holding it until t =2gives zero return, while not withdrawing gives a negative return. Profits 2 are defined by: That is, for the fraction 1 2 =(1 ) (1 apple) D b 0 1+r K 2 H 1/ (18) of banks hit by H, profits are the difference between the return 1+r K 2 H, earned by investing a fraction 1 apple of deposits in capital, and the return 1/ paid to depositors. 19 Alternatively, some of these profits could be paid to impatient households that have withdrawn at t =1, as long as these payments do not violate the incentive compatibility constraint of depositors to truthfully reveal their own type when there are no runs. Due to linearity of preferences at t =2, these approaches are equivalent. 14

In Section 6, I consider a richer model with a standard, smooth utility function; this extension allows me to solve the model without imposing any cap on deposits. Moreover, I check numerically that investing a fraction apple of deposits is optimal even if the probability of a run is positive. 4.3 Central bank (Readers only interested in the model without policy intervention can skip this section.) Recall that the money supply is M (1 + µ t ), where µ t is chosen by the central bank. If there is no policy intervention, µ t =0for all t and money supply is constant at M. If there is a policy intervention, the central bank changes the money supply by choosing µ t. Interventions are announced at t =0, before the Walrasian market opens; the central bank fully commits to the policy announcement. If µ 0 > 0, the central bank is injecting µ 0 M units of money at t = 0, because the initial endowment of money is M and the money supply at t =0is M (1 + µ 0 ). The monetary injection is delivered using asset purchases, that is, purchases of capital K CB 0 on the market at price Q 0 : Q 0 K0 CB apple µ {z } 0 M. (19) asset purchases All results are unchanged if the central bank uses the newly printed money, µ 0 M, to offer loans to banks b, as long as such loans are fully collateralized using capital. 20 Buying capital directly is equivalent to offering loans that are used by banks b to buy capital, which is in turn used as collateral with the central bank. At t =1, I restrict attention to the case in which the money supply does not change because there is no market in which capital can be traded. 21 Thus, µ 1 = µ 0. At t =2, the central bank can again change the money supply by varying µ 2. Monetary injections at t =2are implemented using lump-sum transfers (or taxes, if negative) to households. 22 Moreover, any profits from the purchase of capital K CB 0 are distributed lump-sum to households 20 In particular, the central bank could offer nonrecourse, collateralized loans and charge a return on the loans. The return could either be state-contingent (i.e., contingent on the realization of the shock b of bank b) or fixed and equal to 1+r K 2 H (where 1+r K 2 H is the return earned by the bank if b = H ). In the latter case, if b = L, the central bank makes the return de facto state contingent by seizing the collateral and taking a loss. 21 I do not consider loans to banks at t = 1. Since runs are driven by shocks that make banks fundamentally insolvent, loans to banks at t =1would not eliminate runs unless the central bank provided loans to such banks and were willing to take losses. However, this policy is more akin to a bailout rather than a monetary injection and central banks are typically restricted from using it. 22 The parameter restrictions in (5) rule out the possibility that the central bank can increase consumption by printing money. 15

as well. 23 Thus, transfers T 2 to households are: T 2 = K CB 0 Q 2 + A 1 P 1 {z } gross return on capital +(µ 2 µ 0 ) M {z } change money supply at t=2 (20) The last term in Equation (20), (µ 2 µ 0 ) M, denotes the change of the money supply at t =2. For instance, if µ 0 > 0 and µ 2 =0, the monetary injection at t =0is temporary and thus the central bank taxes households at t =2to reduce the money supply to the initial level M (recall that banks and households are endowed with M units of money at t =0). If µ 0 =0and µ 2 > 0, the central bank is just intervening at t =2. If µ 2 = µ 0 > 0, the monetary injection implemented at t =0is permanent. 4.4 Market clearing conditions The market clearing conditions are as follows. Capital market, t =0: Money market, t =0: Deposits, t =0: Goods market, t =1: Z Z K0db b + K0 h dh + K0 CB = K. (21) Z Z M0db b + M0 h dh = M (1 + µ 0 ). (22) Z Z D0db b = D0dh. h (23) Z C1 h dh = A 1 K. (24) If there is no monetary policy intervention, the amount of assets bought by the central bank, K CB 0, in (21) and µ 0 in (22) are zero. 4.5 Equilibrium The notion of equilibrium is similar to the one used in Section 3. Given a monetary policy {µ 0,µ 2 }, an equilibrium is a collection of: prices Q 0, P 1, and r K 2 ; households choices M h 0, K h 0, D h 0, W h 1, C h 1, C h 2 ; banks choices M b 0, K b 0, and D b 0, return on deposits r2 b b, and profits 2 ; and central bank s asset purchases K0 CB and profits T 2 ; such that: households maximize utility; 23 Since the central bank is a large player in the market, I assume that the idiosyncratic shocks to K0 CB cancel out. Thus, the overall stock of capital K0 CB held by the central bank is unchanged at t =1and t =2. 16

banks serve withdrawals at t =1until they run out of money (that is, if withdrawals W1 h are constrained at zero for some households, Equation (14) must hold with equality); the actual returns on deposits r2 b b satisfy Equation (17); and profits satisfy (18); the market clearing conditions, (21)-(24), and the budget constraint of the central bank, (19), hold. I consider symmetric equilibria in which all banks have the same amount of deposits at t =0. 4.6 Small shocks to capital: good equilibrium As a benchmark, I study the economy in which the negative shock to capital, L, is small. In this economy, a good equilibrium arises. 24 Since the shock L is small, banks hit by L remain solvent and thus pay a positive return at t =2, r b 2 L 0. As a result, there are no runs. In equilibrium, banks offer deposits to households, which in turn withdraw at t =1only if they are hit by the impatient preference shock. The logic of this good equilibrium is therefore very similar to Diamond and Dybvig (1983). Proposition 4.1. (Good equilibrium) Assume that >1 L (1 apple) /apple and fix µ 0 =0and µ 2 =0. If L 1, a good equilibrium exists and is characterized by: prices: Q 0 = 1 t =0: deposits D h 0 = D b 0 = D, where M K Q, P 1 = M A 1 K P ; (25) D M/apple; (26) money holdings M0 h =0and M0 b = appled = M; capital K0 h and K0 b residually determined by the budget constraints; t =1: withdrawals and consumption 8 < D, C if h is impatient W1 h,c1 h = :(0, 0) if h is patient; t =2: returns E 0 1+r K 2 ( ) =1/ and r b 2 b 0 for all b. As in Diamond and Dybvig (1983), banks provide insurance against preference shocks, allowing impatient households to withdraw money and consume at t =1and patient households to 24 I use the terms bad and good equilibria to distinguish the cases with and without runs. Nonetheless, recall that there is no multiplicity of equilibria and thus runs are driven by fundamentals. 17

receive a return on deposits at t =2. Therefore, households hold no money (M0 h =0). Given the price level P, D is the amount of deposits required to finance household s consumption expenditure at t =1if the household is impatient. 25 That is, at t =0, households deposit all their endowment of money and a part of their endowment of capital into banks, in exchange for a promise to be able to withdraw at t =1or to be repaid at t =2. 26 The expected return on capital equals 1/ ; equivalently, the discounted return equals one. Given consumption C1 h = A 1 K/apple for impatient households, their marginal utility at t =1also equals one; see Equation (6). Thus, the marginal utility of impatient households at t =1and t =2 are equalized. Banks invest a fraction apple of deposits into money, in order to serve withdrawals by the fraction apple of impatient households at t =1. The remaining fraction of deposits, 1 apple, is invested in capital. At t =2, the return on deposits is paid using the return on capital. Banks hit by the positive shock H have a very large return on their investments; in particular, their gross return is greater than the expected return on capital 1/. However, due to the cap imposed in Section 4.2, the return paid at t =2to depositors is 1+r2 b H =1/. Due to a similar but antithetical logic, the gross return on investments for banks hit by L is less than 1/ ; nonetheless, the L shock is small and thus r2 b L 0. Similar to the bankless economy, the price of consumption goods P 1 is determined by equating consumption expenditures, R P 1 C1 h dh, to total money spent. Consumption expenditure can be rewritten as P 1 A 1 K using the market clearing for goods. Unlike in the bankless economy, the entire money supply M is spent. This follows from the fact that banks hold the entire money supply at t =0(M0 b = M) and that all money withdrawn at t =1is spent. As a result, P 1 = M/ A 1 K. 4.7 Bankless economy and good equilibrium: a comparison This section compares the price level and money velocity between the bankless equilibrium in Section 3 and the good equilibrium of Proposition 4.1. Banks offset the precautionary demand for money that arises in the bankless economy, reducing the demand for money and thus its equilibrium value. As a result, the price level P 1 is higher in the good equilibrium than in the bankless economy, 25 The assumption >1 L (1 apple) /apple guarantees that is large enough (note that 1 L (1 apple) /apple > 1 because L < 0) that households are willing to hold deposits. If is instead too low and does not satisfy the restriction, households do not want to hold deposits. That is because the cap on the return on deposits, 1+r b 2 ( ) apple 1/, implies that the expected return on deposits is lower than the expected return on capital. 26 Due to the cap on the return on deposits, the expected return on deposits is lower than E 0 1+r K 2 ( ). Thus, households invest directly in capital the wealth that they want to carry to time 2. Without the cap on deposits, the expected return on deposits would be equal to E 0 1+r K 2 ( ), and thus households would be indifferent among any choice of deposits greater than or equal to D. 18

as summarized below. 27 Corollary 4.2. The price level is lower in the equilibrium of the bankless economy, in comparison to the good equilibrium: (P 1 in bankless equilibrium) < P where P is the price level in the good equilibrium; see (25). This result follows from the monetary nature of the model. To explain further, let v denote money velocity, defined implicitly by the equation of exchange: M (1 + µ0 ) v = P 1 A 1 K. (27) Money velocity is endogenous and differs between the two equilibria. Less money is used for transactions in the bankless equilibrium than in the good equilibrium. As a result, money velocity is lower in the bankless equilibrium. 28 4.8 Large shocks to capital: bad equilibrium When the idiosyncratic shock to capital L is sufficiently negative, the equilibrium is characterized by runs on banks hit by L. I denote this scenario as a bad equilibrium. Recall that banks invest in money and capital at t =0, in amounts M b 0 and K b 0, respectively. If the stock of capital K b 0 of bank b is hit by L and if L is sufficiently negative, the bank does not have enough resources to pay a positive return on deposits at t =2. If this is the case, the return on deposits is negative: r b 2 L < 0. From the depositors points of view, running on this bank is profitable, because the return on money withdrawn is zero while the return on deposits not withdrawn is r b 2 L < 0. The fact that some banks will be subject to runs at t =1is known by households at t =0, as in Allen and Gale (1998). What households do not know is which bank will be subject to a run, because the idiosyncratic shocks are realized at t =1. 29 In particular, for each bank, the probability 27 A similar result arises in the monetary models of Brunnermeier and Sannikov (2011), Carapella (2012), and Cooper and Corbae (2002). 28 The price of capital Q 0 in the bankless equilibrium is lower than Q as well, where Q is defined in Proposition 4.1. If banks are not active, households preference shocks are not insured. Thus, the illiquidity of capital (i.e., its inability to provide insurance against preference shocks) reduces demand for it and thus its price. If instead banks are active, as in the good equilibrium, they provide sufficient insurance against preference shocks. In this case, the illiquidity of capital is irrelevant to households, and its nominal price Q 0 must be higher to clear the market. 29 In Allen and Gale (1998), there are no runs at t =1in some states of the world, whereas in the bad equilibrium of my model there are always runs on some banks at t =1. I can extend the model to allow two aggregate states at t =1: one state in which idiosyncratic shocks are realized and some banks are subject to runs, and another state in which idiosyncratic shocks are not realized and no bank is subject to runs. However, the main results would be unchanged. 19

of a run is related to the probability distribution over the idiosyncratic shocks to capital. Unlike in Allen and Gale (1998), households in my model fly to money and reduce their holdings of deposits in comparison to the good equilibrium. The flight to money allows households to (partially) selfinsure against preference shocks in the event of a run at t =1. The reduction in deposits implies that households deposit less money and a smaller fraction of their endowment of capital. In the limit case represented by the bankless economy, K h 0 = K and thus households do not deposit any of their endowment of capital. The flight to money and away from deposits implies that money holdings by households are at an intermediate level, in comparison to the economy without banks and the good equilibrium. Thus, the nominal prices Q 0 and P 1 are at intermediate levels as well, by a logic similar to that described in Section 4.7. The next Proposition summarizes these results; the exact parameter restrictions are provided in the Appendix. 30 Proposition 4.3. Fix µ 0 =0and µ 2 =0. Under some other parameter restrictions, a bad equilibrium exists and is characterized by: prices Q 0 <Q and P 1 <P ; t =0: households flight to liquidity, holding M h 0 > 0 and D h 0 <D ; t =1: banks hit by L are subject to runs (i.e., both patient and impatient households want to withdraw at t =1); households holding deposits at banks not subject to runs and those at the beginning of the line at banks subject to runs consume C h 1 > C, whereas households at the end of the line in a run consume C h 1 < C; t =2: returns on deposits are 1+r b 2 b =1/ if b = H and r b 2 apple 0 if b = L. The model in Allen and Gale (1998) has one bank (or, many identical banks subject to the same aggregate shocks) and an equal service constraint. I deviate from these assumptions by using a sequential service constraint and idiosyncratic shocks. Thus, contrary to Allen and Gale (1998), there is a welfare loss in the bad equilibrium generated by a misallocation of consumption within impatient households at t =1. Impatient households that are at the end of the line in a run cannot withdraw at t =1; as a result, they face a very tight cash cash-in-advance constraint. It follows that these households consumption is lower than households that can withdraw money at t =1, 30 While the good equilibrium exists if L > 1, the analysis when L < 1 is slightly more complicated. The equilibrium described here arises if L is lower than a threshold ˆ, where ˆ < 1. For intermediate values, 1 < L < ˆ, the logic of the equilibrium is similar to the one of purified mixed equilibria in game theory. If 1 < L < ˆ, and if I guess that all banks hit by L are subject to runs, r2 b L > 0 due to the endogenous response of asset prices. Thus, the guess is not verified because households do not run when r2 b L > 0. In this case, the equilibrium has r2 b L =0and only some of the banks hit by L are subject to runs. Since households are indifferent about running or not when r2 b L =0, the fraction of banks subject to runs is determined endogenously to make sure that r2 b L =0holds in equilibrium. 20