Liquidity, Asset Price and Banking (preliminary draft) Ying Syuan Li National Taiwan University Yiting Li National Taiwan University April 2009 Abstract We consider an economy where people have the needs for liquidity because of the unanticipated consumption opportunity, yet banks may not provide su cient liquidity due to imperfect technology in commitment, enforcement and record-keeping. We show a link between the liquidity generated by bank lending, the value of the asset used as collateral and aggregate economic activity. When the record-keeping is extremely limited so banks can only posses the collateral once defaults occur, higher in ation exerts adverse e ects on aggregate output because it reduces the value of the asset and the loan-to-value ratio. When the record-keeping is superior so that banks can exclude defaulters from the banking sector besides seizing the collated, higher in ation reduces the value of the asset, but raises the value of money, the loan-to-value ratio, and aggregate output. The reason is that in ation raises the cost of being excluded from the banking system and, thus, relaxes the borrowing constraint and increases liquidity to the aggregate economy. JEL Classi cation: E41; E50 Keywords: Liquidity; Credit; Asset price; Collateral; Banking Department of Economics, National Taiwan University, 21 Hsu-Chow Road, Taipei 10020, Taiwan. E-mail Yiting Li: yitinglintu.edu.tw. Ying Syuan Li: g904906alumni.nthu.edu.tw.
1 Introduction In a world where people have needs for liquidity due to the unanticipated shocks of consumption or investment opportunity, the provision of liquidity would be essential for resource allocation. Liquidity could be privately created through issuing nancial claims directly by borrowers and extending credit by the nancial intermediaries, many of which are backed by some forms of collateral. 1 While some people have liquidity needs, others may have idle money. Financial intermediary can transfer purchasing power from those who do not have consumption or investment opportunity to those who do. Banks or potential creditors, however, may not provide su cient liquidity due to imperfect technology in commitment, enforcement and record-keeping. Under this situation, the use of collateral is a common means to secure bank lending, and the ease of acquiring a loan and the terms of a loan contract usually depend on the value of collateral. Previous studies have tried to explain the movement in asset prices and how they are related to aggregate economic activity in an environment with limited credit. For example, Kiyotaki and Moore (2001, 2005) have shown assets that could generate liquidity from circulating before maturity are essential for the smooth running of the economy. Along this line we study an environment where the nancial intermediary takes deposits from people with idle money and provide credit to those with liquidity shocks. Banks face limited enforcement so that they cannot force repayment on the loans, but can ask borrowers to pledge some assets as collateral. The relationship between the loans and the value of collateral determines the amount of liquidity generated through backed lending, which may well depend on whether the record-keeping technology is e cient enough for undertaking other punishments besides seizing the collateral. 2 Thus, there is a link between the asset price, loan rate, liquidity, and aggregate output, and the relationship is a ected by the enforcement and record-keeping technology. We wish to study the above issues altogether in an environment where money and nancial intermediation are essential. We rst consider a technology of full enforcement so that banks can force repayment at 1 Assets that may back a loan include capital, real estate, net worth and nancial assets; see, e.g., Kiyataki and Moore (1997, 2001, 2005), Bernanke and Gertler (1989) and Iacoviello (2005). 2 Kocherlakota (1998) shows the coexistence of credit and money requires imperfect knowledge of individuals histories. Kocherlakota and Wallace (1998) consider that individual histories are made public only with a lag to show the coexistence of money and credit. 1
no cost. Therefore, default is not possible and thus agents face no borrowing constraint. We then consider that banks cannot force agents to repay the loans. Banks, however, have limited enforcement in the sense that borrowers have to pledge some assets as collateral to secure the loans. Once default occurs, banks are entitled to the collateral. Moreover, when the recordkeeping technology is very costly or the update on repayment records has an extremely long lag, the only punishment on defaults is to seize the collateral. If an agent defaults on his loan, he enjoys more leisure but loses his assets and, thus faces the current-period trade o of bene t and cost of default. Another case is that banks have superior record-keeping in the nancial records and they share information on agents repayment histories. When default occurs, banks not only posses the collateral but also exclude the defaulters from the banking sector forever. Thus, defaulters face the trade o of short-run bene t and long-run cost. For each case, we derive the conditions to ensure voluntary repayment and show that this may involve credit rationing when the borrowing constraints are binding. As assets are used as collateral, our main focus is whether and how much extra loans a unit of asset can generate under various enforcement and record-keeping technology and equilibria (whether borrowers are credit constrained). This provides a link between the value of asset and its role as collateral to facilitate borrowing and generate liquidity. We assume that the banking system has the technology to fully recognize the true quality of assets, but individuals do not have an access to this technology. As a result, assets are not used as a medium of exchange in the goods market. 3 We nd that, in the equilibrium where agents are not credit constrained, the market value of assets is determined by fundamentals, since assets are used only as a store of value. The liquidity premium of an asset, de ned as the di erence between its market value from the present value of dividends, is positive under limited enforcement. That is, assets that can serve as collateral to secure loans commands a liquidity premium because it provides liquidity to the economy through transferring funds from saver to the borrowers. The positive liquidity premium associated with the asset is due to its role as collateral, rather than from being used as a medium 3 One extension would be to explicitly model the investment in the technology to recognize the quality of assets and derive endogenously under which condition do assets circulate as a medium of exchange, such as in Lester et al. (2008). 2
of exchange. 4 The loan-to-value ratio (the relationship between the amount of loan to the value of collateral) is in uenced by the dividend-price ratio of the asset pledged as collateral and the loan rate which, in turn, depends on borrower s incentives to default. If the loan rate is higher, agents have higher incentives to default since the repayment cost is increased. Hence, banks should lend less (set a lower loan-to-value ratio) to control the bene t of defaulting. Moreover, the loan-to-value ratio is positively related to the market value and dividends of the real asset. One can interpret the loan-to-value ratio as the rate at which the assets can generate liquidity to the economy. Our results thus provide a microfoundation to the assumption in Kiyotaki and Moore (2001, 2005) that the fraction of liquidity that an asset can generate is exogenously given. When the record-keeping is extremely limited, higher in ation exerts adverse e ects on the economy with constrained credit through two channels: rst, as in a standard monetary model, the lower value of money reduces the incentives to produce; second, higher in ation also reduces the market value of the assets that is used as collateral and, therefore, agents face a more stringent borrowing constraint and fewer real loans to support their purchases. Interestingly, higher in ation also reduces the loan-to-value ratio, which is caused by the higher incentives to default due to less valuable collateral. When the record-keeping is superior, the loan-to-value ratio may be greater than one, since banks have other punishment devices besides seizing the collateral. However, in ation have some opposite e ects on prices and allocations from the case with extremely limited record-keeping. Higher in ation raises the loan rates and reduces the value of the asset, but surprisingly it raises output, the value of money and the loan-to-value ratio. The reason is that in ation raises the cost of being excluded from the banking system (since defaulters need to bring enough money to self-insured against the consumption shock) and, thus, relaxes the borrowing constraint and increases liquidity to the aggregate economy. 4 Lester et al. (2008) and Rocheteau (2008) focus on how recognizability of assets a ects their acceptability in the trade of goods. 3
2 Model We use Lagos and Wright (2005) and Berentsen et al. (2007) as the basic frameworks. Time is discrete and there is a [0; 1] continuum of in nitely-lived agents. Each period is divided into two subperiods, that di er in terms of economic activity. The discount factor across periods is 2 (0; 1): All consumption goods are nonstorable and perfectly divisible. In each subperiod, agents trade in the Walrasian markets. In the beginning of the rst subperiod, an agent gets a preference shock that he either consumes or produces. With probability n an agent can produce but cannot consume while with probability 1 n an agent can consume but cannot produce. 5 Consumers get utility u(q) from q consumption. Producers incur disutility c(q) from producing q units of output. Assume u(0) = c(0) = 0; u 0 > 0; c 0 > 0; u 0 (0) = 1, u 00 < 0; c 00 0: Trading history is private information because all goods trades are anonymous. Such anonymity makes money essential (see Kocherlakota 1998). Agents cannot commit to future actions so that there is no credit between private agents. In the second subperiod, all agents can produce and consume a consumption good (called general good ), getting utility U(x) from x consumption, with U 0 (x) > 0; U 0 (0) = 1; U 0 (1) = 0 and U 0 (x) 0: Agents can produce one unit of the good with one unit of labor which generates one unit of disutility. There are two kinds of in nitely-lived assets in the economy: at money, and a real asset that yields a dividend of units of general good each period. In the second subperiod, agents can trade money, real assets and general goods. The banking system does not have record-keeping on the trading histories in the goods markets of agents. Therefore, individuals cannot issue trade credit. Moreover, assume that individuals could not verify the real asset or the nancial claims backed by the assets (see e.g., Lester et al. 2008). As a result, only cash is used as the medium of exchange. There are competitive banks that accept nominal deposits and make nominal loans. Agents who are sellers in the rst subperiod can deposit their money holding in the bank at the nominal interest rate i d, and are entitled to withdraw fund at the second subperiod. Those who are buyers may borrow money from the bank at the nominal loan rate i; and needs to repay the loan at 5 This is a simple way to capture the uncertainty of trade such as in a decentralized market with anonymous bilateral matching, as assumed in Lagos and Wright (2005). 4
the second subperiod. A government is the sole issuer of at currency. The currency stock evolves deterministically at a gross rate by means of lump-sum transfers, or taxes if < 1; M t = M t 1 ; where > 0 and M t denote the per capita currency stock in period t: Agents receive lump-sum transfer of new money T t = ( 1)M t 1 in the second subperiod. Let t denote the value of money in terms of the general good. We denote the real transfer t = t T t : For notational ease variables corresponding to the next period are indexed by +1; and variables corresponding to the previous period are indexed by 1: We rst consider a technology of full enforcement so that banks can force repayment at no cost. Therefore, default is not possible and thus agents face no borrowing constraint. We then consider that banks cannot force agents to repay the loans. Banks, however, have limited enforcement in the sense that borrowers have to pledge some assets as collateral to secure the loans. Once defaults occurs, banks are entitled to the collateral. Moreover, the cost of collecting and maintaining the nancial records a ects whether banks can adopt other punishments on defaulters. We rst assume that banks face imperfect record-keeping so that once default occurs they can only take the borrower s assets pledged as collateral, and no other punishments are available. Another case is that banks have perfect record-keeping in the nancial records and they share the information on agents repayment histories. When default occurs, banks can take collateral and exclude the defaulters permanently from the banking system; i.e., defaulters cannot make deposits or get any credit forever. We derive the conditions to ensure voluntary repayment and show that this may involve binding borrowing constraints; i.e., agents face credit rationing. 3 Equilibrium Let denote the market value of real asset in terms of general good. equilibria where end-of-period real balances are time-invariant: We study stationary 1 M 1 = M 1A 1 = A: Thus, 1 = = M=M 1 = and 1 = = A=A 1 = 1 due to constant supply of real assets. 5
Let V (m; a) denote an agent s expected life time utility from entering he rst subperiod with money holding m and asset holding a. Let W (m; a; `; d) be the expected value of an agent from entering the second subperiod at time t with m units of money, a units of real assets, ` loan and d deposits, where loan and deposits are in terms of at money. Subperiod 2 In the second subperiod an agent produces h goods and consumes x; repays loans, redeem deposits and adjusts his holdings of at money and real assets. An agent solves the following problem: W (m; a; `; d) = max x;h;m +1 ;a +1 U(x) h + V +1 (m +1 ; a +1 ) s.t. x + m +1 + a +1 = h + (m + T ) + ( + )a + (1 + i d )d (1 + i)`: Substituting h from the budget constraint into the objective function, we have W (m; a; `; d) = (m + T ) + ( + )a + (1 + i d )d (1 + i)` + max x;m +1 ;a +1 fu(x) x m +1 a +1 + V +1 (m +1 ; a +1 )g: Note that due to the assumption of quai-linearity, W is linear in m; a; `; d; the initial portfolio entering the second subperiod. The rst order conditions are U 0 (x) = 1; (1) V m+1 (m +1 ; a +1 ); = if m +1 > 0: (2) V a+1 (m +1 ; a +1 ); = if a +1 > 0: (3) Equation (1) implies x = U 10 (1) for all agents. Note that m +1 and a +1 are independent of x and the initial holdings of m and a. Therefore, the distribution of holdings of money and real assets are degenerate at the beginning of the following period. The envelope conditions are W m = ; (4) W a = + ; (5) W` = (1 + i); (6) W d = (1 + i d ): (7) 6
In the second subperiod, the market clearing conditions for general good, money and real assets are, respectively, h + A = x ; (8) m = M 1 ; (9) a = A: (10) Subperiod 1 Let q b and q s denote the quantities consumed by a buyer and produced by as seller trading in the rst subperiod, respectively, and p be the nominal price of goods traded in the rst subperiod. An agent holding a portfolio of (m; a) entering the rst subperiod has expected lifetime utility V (m; a) = (1 n)[u(q b ) + W (m + ` + pq b ; a; `)] +n[ c(q s ) + W (m d + + pq s ; a; d)]: (11) Since there is a centralized market in the rst subperiod, all agents face the same market clearing price p. As a producer, taking p as given, an agent s problem is max q s;d c(q s ) + W (m d + pq s ; a; d) (12) s.t. d m: (13) Let d denote the multiplier on the deposit constraint, then the rst order conditions are c 0 (q s ) + pw m = 0; W m + W d d = 0: Given p, q s depends on W m, which in turn depends on. Using (4) and (7), the rst order conditions of sellers objective function can be rewritten as p = c0 (q s ) ; (14) d = i d : Equations (14) implies that production takes place until the marginal cost of production, c0 (q s), equals the marginal revenue, p. The production q s is independent of m and a due to the linearity 7
of the envelope condition. For any i d > 0, the deposit constraint is binding and so sellers deposit all money balances. As a consumer, taking p as given, an agent chooses the quantity of consumption q b and borrowing ` to maximize his utility while he faces budget and borrowing constraints: max q b ;` u(q b ) + W (m + ` + pq b ; a; `) s.t. pq b m + ` (15) ` ` (16) The rst order conditions are u 0 (q b ) pw m p = 0; W m + W` + ` = 0; where is the multiplier on the buyer s budget constraint, and ` is the multiplier on the borrowing constraint. as Using (4), (6), (14), the rst order conditions of buyer s objective function can be rewritten u 0 (q b ) = c 0 (q s )(1 + ) (17) i = ` (18) If = 0, then (17) reduces to u 0 (q b ) = c0 (q s). That implies trades are e cient. If > 0, the buyer spends all the money, and his budget constraint is binding. That is, q b = (m + `)=p < q b, and trades are ine cient. Combining (17) and (18) we get If ` = 0, the borrowing constraint does not bind and u 0 (q b ) c 0 (q s ) = 1 + i + ` : (19) u 0 (q b ) c 0 (q s ) = 1 + i: (20) Note that the nominal loan rate i > 0 acts as a tax on consumption. Equation (20) implies that the marginal bene t of borrowing an additional unit of money, u0 (q b ) c 0 (q s), equals the marginal cost of borrowing an additional unit of money, (1 + i). 8
If ` > 0, the borrowing constraint binds, ` = `, and from (19), u 0 (q b ) c 0 (q s ) > (1 + i): That is, the marginal bene t of borrowing an additional unit of money, u 0 (q b ) c 0 (q s), exceeds the marginal cost, (1 + i). Buyers are willing to get more loans, but banks may not be willing to lend. Thus, buyers consume q b = (m + `)=p. In the rst subperiod, good market clearing condition is nq s = (1 n)q b : (21) Competitive Banks In the rst subperiod, banks accept deposits from the would-be sellers and make loans to the would-be buyers. Since banks are perfectly competitive with free entry, they take all interest rates as given. There is no strategic interaction among banks or between banks and agents, and no bargaining over the terms of the loan contract. The representative bank solves the following problem per borrower: max ` (i i d )` s.t. ` `; u(q b ) + W (m; a; `; d) ; where is the reservation value of the borrower, which is the surplus from obtaining loans at another bank. If banks have full enforcement on repayment, the borrowing constraint is ` = 1: When banks have limited enforcement, we consider two cases that di er in the e ciency of the record-keeping. We will derive the condition to ensure voluntary repayment which determines `: The rst order condition to the bank s problem is i i d L + [u 0 (q b ) dq b d` + W`] = 0; where L and are the Lagrangian multipliers on the lending constraint and borrower s participation constraint, respectively. For i i d > 0, banks would like to make the largest loan possible to borrowers and, therefore, would choose a loan size such that > 0. The zero pro ts condition for competitive banks is i = i d : (22) 9
From (14) and (15), dq b d` = c 0 (q s). We rewrite the rst order condition of bank s maximization problem as u 0 (q b ) c 0 (q s ) = 1 + i + L : If banks can force repayment without any cost, the lending constraint is not binding, and L = 0: The loan supplied by the bank implies u0 (q b ) c 0 (q = 1 + i. If s) L > 0, the lending constraint is binding and u0 (q b ) c 0 (q s) > (1 + i): With limited enforcement, banks may have to conduct credit rationing in order to prevent the default problem. In a symmetric equilibrium, loan market clearing condition is (1 n)` = nd: (23) 3.1 The optimal portfolio choices To nd an agent s optimal portfolio, we rst derive the expected marginal value of each asset in the rst subperiod market. In the Appendix we derived the marginal values of holding money and assets: V m (m; a) = (1 n) u0 (q b ) p + n(1 + i d ): The bene t of bringing an additional unit of money includes receiving the expected gains u0 (q b ) p from spending the money on goods as a buyer, and the interest payment from making deposits as a seller. Using (14), the marginal values of money can be described as V m (m; a) = [(1 n) u0 (q b ) c 0 (q s ) + n(1 + i d)]: (24) Using (2) lagged one period to eliminate V m (m; a) from (24), an agent s optimal money holdings satisfy 1 [(1 n) u0 (q b ) c 0 (q s ) + n(1 + i d)]; = if m > 0 The marginal value of holding real asset is V a (m; a) = ( + ) + (1 n)[ u0 (q b ) c 0 (q s ) (1 + i)] ` a : (25) The bene t of carrying an additional unit of real asset include two terms: rst, the market value and dividend that the holder of the asset is entitled to when entering the second subperiod. Second, since assets can be pledged as the collateral to secure loans, holding one more unit of 10
asset enables the buyer to receive extra loans which generates the net gain from buying goods minus the interest payments. The main point is whether and how much extra loans an additional unit of asset can generate under various enforcement and record-keeping technology and equilibria (whether borrowers are credit constrained). This provides a link between the value of asset and its role as collateral to facilitate borrowing. To see this, recall that from (20), when an agent is not credit constrained, the marginal bene t of receiving an additional dollar of loans, u0 (q b ) c 0 (q s) ; equals the cost, (1 + i): Therefore, V a (m; a) = + ; and the bene t of holding an asset includes only its market value plus dividend. When an agent is credit constrained, his marginal bene t of taking an additional unit of loans exceeds the marginal cost, u0 (q b ) c 0 (q s) > (1 + i). The marginal value of holding real asset is (25), which is greater than + ; and the di erence stems from whether agents are credit constrained. satisfy Using(3) lagged one period to eliminate V a (m; a) from (25), an agent s optimal asset holdings 1 f + + (1 n)[ u0 (q b ) c 0 (q s ) (1 + i)] ` g; = if a > 0: (26) a Thus, in an equilibrium where agents hold money and real asset the following two conditions must be satis ed: 1 = (1 n)[ u0 (q b ) c 0 (q s ) = + (1 n)[ u0 (q b ) c 0 (q s ) 1] + ni d ; (27) (1 + i)] ` a : (28) 3.2 Equilibrium with full enforcement We study the equilibrium with full enforcement as a benchmark. When banks can force agents to repay the loans with no cost, ` = 1, and so agents are unconstrained. Substituting (20) and zero-pro t condition i = i d into (27), we have i = : (29) 11
Use the goods market clearing condition (21) and (29) to get = u0 (q b ) c 0 (q s ) 1: (30) De nition 1 When banks can force agents to repay the loans, a monetary equilibrium with credit is a list of agent s choices (m; a; `; d), terms of trade (q s ; q b ; x; h), a sequence of prices ; ; p; loan rate i, and deposit rate i d that satisfy (13), (15), the optimal portfolio choices (1), (14), (20), (27), and (28), market clearing for good market in the two subperiods, money, loan, and asset (8), (21), (9), (23), and (10), and bank s zero pro t condition (22). Note that in the equilibrium with full enforcement agents are not credit constrained and so 0 < ` = nc 0 (q s )q b < ` = 1: Proposition 1 With full enforcement, the equilibrium value of real asset is the present value of dividends; that is, = u where u = 1 : (31) Proof. Substituting = +1, (14), and (20) into (28) and the result follows. If banks can force agents to repay the loans, the borrowing constraint does not bind. Even though real assets can be used as the collateral, in equilibrium it does not a ect the credit that agents receive. Therefore, the market value of the real asset is determined only by the fundamentals, and is not a ected by whether it is used as collateral. however, when banks cannot force borrowers to repay the loans. This is not the case, 4 Equilibrium with limited enforcement and record-keeping We now assume that banks have limited ability to force repayment of loans so that borrowers have incentives to default. Banks have limited enforcement in the sense that borrowers have to pledge some assets as collateral to secure the loans. Once defaults occurs, banks are entitled to the collateral. We consider two types of punishment on defaulters, that depends the e ciency in record-keeping. First, the limitation in record-keeping is so high that the only punishment available to banks is to possess defaulter s assets that have been pledged as collateral. We then consider the case where banks have a superior record-keeping technology and they share 12
the information on agents repayment histories. When default occurs, banks are able to take the defaulters assets and exclude the defaulters permanently from the banking system; i.e., defaulters cannot make deposits or get any credit forever. We will characterize equilibria for the two cases, and discuss how the asset price, interest rate and borrowing constraints are a ected by the limitations in enforcement and record-keeping. 4.1 Extremely limited record-keeping When the record-keeping technology is very costly or the update on repayment records has an extremely long lag, the only punishment on default is to seize the defaulters assets. If an agent defaults on his loan, he enjoys more leisure in the second subperiod, but loses his asset and the dividend. The agent thus face the current-period trade o of bene t and cost of default. For a buyer entering the second subperiod who repays his loan, the expected discounted utility in a stationary equilibrium is W (m; a) = U(x ) h b + V +1 (m +1 ; a +1 ): (32) Now consider a buyer who defaults on his loan. Since he does not make repayment, the bene t of defaulting is that he can work less than otherwise. The cost is that his asset was seized by the banks. Due to the extremely ine cient record-keeping, banks punishment is limited to the current period. Thus, an agent who defaults on his loans will start the next period just as a non-defaulter. That is, a defaulter would choose the same portfolio as non-defaulters and his expected discounted utility from next period is also V +1 (m +1 ; a +1 ). 6 Thus, a deviating buyer s expected discounted utility entering the second subperiod is cw (m; a) = U(bx) b hb + V +1 (m +1 ; a +1 ); where the hat indicates a deviator s optimal choice. For the existence of equilibrium with credit, borrowers must voluntarily repay their loans, which requires W (m; a) W c (m; a). Thus banks need to choose the loan size such that agents have no incentives to default; that is, banks chooses the real borrowing constraint ` such that W (m; a) = W c (m; a): (33) 6 Of course we need to make sure that the loss in asset is so large that the defaulter s labor is not enough to choose the same portfolio as the non-defaulters. 13
Since the utility function in the second subperiod is quasi-linear, bx = x. Thus, (33) implies b hb h b = 0: Given the borrowing constraint, it may be the case where agent s demand for loans is lower than the credit limit imposed by banks so that the borrowing constraint is not binding. Hence, ` < ` in an unconstrained equilibrium. If at least some agents face a binding constraint, then we are in a constrained equilibrium, and ` = `: Lemma 1 The real borrowing constraint ` satis es (1 + i)` = ( + )A: (34) The left-hand side of (34) is the leisure gained by not repaying loans (or, saving in the utility cost of producing for repayment), and the right-hand side is the cost of losing the collateral, which includes the market value of the asset and dividends. Under the extremely limited recordkeeping, the only punishment available is the loss of collateral. Thus, the credit limit is obtained by equating the current-period cost and bene t for defaulting the loans. From (34) we can derive the relationship between the amount of real loan to the real value of collateral, the loan-to-value ratio, as follows. Proposition 2 The loan-to-value ratio 1 under the extremely limited record-keeping is where r p = is the dividend-price ratio. 1 = 1 + r p 1 + i ; (35) The loan-to-value ratio is in uenced by the dividend-price ratio of the assets pledged as collateral and the interest rate which, in turn, depends on borrower s incentives to default. If the loan rate is higher, agents have higher incentives to default since the repayment cost is increased. Hence, banks should lend less (set a lower loan-to-value ratio) to control the bene t of defaulting. Moreover, the loan-to-value ratio is positively related to the market value and dividends of the real asset. One can interpret the loan-to-value ratio as the rate at which the assets can generate liquidity to the economy. Our result provides a microfoundation to Kiyotaki and Moore (2001). 14
De nition 2 When the record-keeping is extremely limited, a monetary equilibrium with unconstrained credit is a list of agent s choices (m; a; `; d), terms of trade (q s ; q b ; x; h), a sequence of prices t ; t ; p t ; a interest rate i, and a deposit rate i d that solve (13) and (15), satisfying the Optimal portfolio choices (1), (14), (20), (27), and (31), market clearing for good market in the two subperiods, money, loan, and asset (8), (21), (9), (23), (10), (34), and bank s zero pro t condition (22) where 0 < ` = nc 0 (q s )q b < `. If agents are credit constrained, ` = L > 0 and (19) holds. Equation (34) implies ` a = + (1 + i) : (36) De nition 3 A monetary equilibrium with constrained credit is (q b ; i; ) satisfying (27), (28), (36) and (34), where 0 < ` = nc 0 (q s )q b = `. In a monetary equilibrium with constrained credit, to induce voluntary repayment, banks charge i. Agents borrow the real loans ` = `. Substituting = 1, (14), and (36) into (28), we have = 1 where and 1 = B 1 B B = 1 + (1 n)[ u0 (q b ) c 0 (q s ) 1 1 + i Denote B as the e ective discount factor by taking into account the credit market imperfection. We nd 1 > u since B > 1: Note that B > 1 because u0 (q b ) c 0 (q s) 1]: > 1 + i; which in turn, due to the limited enforcement. De ne the liquidity premium as the di erence between the market value of an asset from its present value of dividends. Then the liquidity premium of the real asset in a constrained equilibrium ( 1 u ) is positive. The reason is that with limited enforcement the real asset is used as collateral that enables buyers to get liquidity from the bank to purchase goods. The positive liquidity premium associated with the real asset is due to the credit market imperfection, rather than from being directly used as a medium of exchange. Using u(q) = log q and c(q) = q 2 =2 we nd a threshold on A; A = (1 )n ; such that (i) if A > A, a monetary equilibrium with unconstrained credit exists; (ii) if A A, a monetary equilibrium with constrained credit exists. Since the real asset is used as collateral 15
to secure the loans, if the supply of real assets is not ample enough, agents are more likely to be credit-constrained. When the dividend is larger, the unconstrained equilibrium can be sustained with lower stock of assets. The reason is that higher dividend increases the market value of assets, together with the high dividend, which make banks more willing to lend more. The threshold A increases in n because higher n implies more agents with borrowing need, and so higher stock of assets are needed to allow for unconstrained borrowing. 7 Proposition 3 Monetary policy has the same e ect in a constrained and unconstrained equilibrium: i > 0; q b < 0; < 0; < 0; p > 0; < 0: The change in the dividends of asset a ects only asset price in an unconstrained equilibrium, but has the e ects on prices and allocations in a constrained equilibrium: i > 0; q b > 0; > 0; > 0; p < 0; > 0. Higher in ation exerts adverse e ects on the economy with constrained credit through two channels: rst, as in a standard monetary model, the lower value of money reduces the incentives to produce; second, higher in ation also reduces the market value of the real assets that is used as collateral and, therefore, agents face a more stringent borrowing constraint and less real loans to support their purchases. Interestingly, higher in ation also reduces the loan-to-value ratio, which is caused by the higher incentives to default due to less valuable collateral. In an unconstrained equilibrium, the market value of assets is determined by fundamentals, since assets are used only as a store of value. Thus, changes in the dividends do not a ect the equilibrium prices and allocations. In the constrained equilibrium, on a contrary, higher dividends lead to a higher asset price as well as loan-to-value ratio. Consequently, there is higher liquidity to support purchases and leads to higher output. Our results show a link between the value of assets and the aggregate production and consumption. We also demonstrate that the underlying mechanism is through relaxing the borrowing constraint from the role of assets as collateral. 4.2 Superior record-keeping We now consider a superior record-keeping technology that allows banks to exercise more severe punishment on defaulters: they not only posses the collateral but also exclude the defaulters 7 With the utility function u(q) = p q we nd that the threshold also depends on the in ation rate, though we are not able to get the closed-from solution. 16
from the banking sector forever. That is, the defaulters cannot borrow or deposit their savings in the banks. The technology, however, cannot exclude the defaulters from the asset market so that they can still trade real assets and money. Banks limited enforcement only allows them to take defaulter s assets that have been pledged as collateral; they have no ability to possess defaulter s asset holdings in the future. In this case, defaulters thus face the trade o of short-run bene t and long-run cost. A non-defaulter s expected discounted utility in a stationary equilibrium is (32) and (??). If an agent defaults his loan in the second subperiod, he enjoys more leisure but loses his assets, and is excluded from the banking system for the rest of life. The latter punishment adds a longrun cost to the defaulters. For a deviating buyer, his expected discounted utility in a stationary equilibrium is fw (m; a) = U(ex) e hb + V e +1 ( em +1 ; ea +1 ); where the tilde indicates a deviator s optimal choice. For the existence of equilibrium with credit, borrowers must voluntarily repay their loans, which requires W (m; a) W f (m; a). Thus, banks need to choose the loan size such that agents have no incentives to default; that is, banks chooses the real borrowing constraint ` satisfying W (m; a) = f W (m; a): Again, since the utility function in the second subperiod is quasi-linear, ex = x : Lemma 2 The real borrowing constraint ` satis es ` = r p (1 + i) A + (1 )(1 + i) f(1 n) (q b; eq b ) + c0 (q s )[eq b (1 n)q b ]g: (37) where (q b ; eq b ) = u(q b ) u(eq b ) c 0 (q s )(q b eq b ) 0: Proposition 4 The loan-to-value ratio 2 under the superior record-keeping is 2 = r p (1 + i) + (1 )(1 + i) A f(1 n) (q b; eq b ) + c0 (q s )[eq b (1 n)q b ]g; (38) The loan-to-value ratio under the superior record-keeping consists of two terms: the rst term is due to the loss of collateral, and the second term captures the long-term loss of not 17
being able to make deposits and loans. Note that the rst term does not equal to 1 ; when defaulter s loss is con ned to the collateral. The reason is that deviators choose not to hold real assets. Since real assets provide extra bene ts as collateral to the non-deviators, yet no such gains to deviators, (26) implies that if non-deviators choose to hold real assets, then it never pays to deviators to buy real assets. Therefore, the rst term in (38) is di erent from 1 : The loan-to-value ratio under the superior record-keeping, 2 ; is also positively related to the market value and dividends of the real asset, and the expected gains from trading in the rst subperiod market. From the numerical examples we nd that in the constrained equilibrium changes in dividends have the same e ects on the prices and allocations as in the economy with extremely limited record-keeping. The real asset s liquidity premium is positive, again due to the binding credit constraint. In this case, the loan-to-value ratio may be greater than one, since now banks have other punishment devices besides seizing the collateral. However, since defaulters would be excluded from the banking system forever, in ation have some opposite e ects on prices and allocations. Higher in ation raises the loan rates and reduces the value of the real asset as in the previous case, but surprisingly it raises output, the value of money and the loan-to-value ratio in an constrained equilibrium. The reason is that in ation raises the cost of being excluded from the banking system (since defaulters need to bring enough money to self-insured against the consumption shock) and, thus, relaxes the borrowing constraint and increases liquidity to the aggregate economy. We also compare the prices and allocations across constrained equilibria under di erent record-keeping technology. In the economy with extremely limited record-keeping, the output, loan rate, real loan, and loan-to-value ratio are all lower, but liquidity premium of asset is higher. When the record-keeping is so limited that collateral is the only commitment device for repayment, assets could command higher value than otherwise. One may be puzzled by the fact that the loan rate is lower when banks have less punishment on defaulters. The reason is that higher loan rate implies higher interest payment that makes defaults more likely. Banks have to control the loan rate and the loan size to ensure repayment. In the model we consider here, banks choose a lower loan rate and fewer real loans to induce repayment. Compared to the constrained equilibrium with limited enforcement, when banks can enforce repayment, the value of money, loan rate and output are all higher, and the asset value is lower. The implication is 18
that, when enforcement and record-keeping technology is less e cient, the loan rate would be lower, the assets that could be pledged as collateral command higher value, but the real loans and output are all lower. 5 Conclusion We have shown a link between the liquidity generated by bank lending, the value of assets that are used as collateral and aggregate economic activity, and how the relationship is a ected by the enforcement and record-keeping technology. When the record-keeping is extremely limited, higher in ation exerts adverse e ects on the economy with constrained credit particularly because it reduces the market value of the assets that is used as collateral and the loan-to-value ratio. Therefore, agents face a more stringent borrowing constraint and fewer real loans to support their purchases. When the record-keeping is superior, however, higher in ation raises the loan rates and reduces the value of the asset, but surprisingly it raises output, the value of money and the loan-to-value ratio. The reason is that in ation raises the cost of being excluded from the banking system and, thus, relaxes the borrowing constraint and increases liquidity to the aggregate economy. 19
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