Abstract We study an overlapping generations model of an economy with banks and asset markets. In each period, two types of aggregate shocks hit the economy. A liquidity shock that aﬀects the demand for real balances of lenders, and a real shock that aﬀects the outcome of the investment projects held by borrowers. Banks arise to insure depositors against liquidity risk, which is by nature intertemporal. Markets, because banks cannot perfectly diversify, provide insurance against the risk of real shocks that can aﬀect borrowers. Real shocks, unlike liquidity shocks, determine the total amount of resources available at a certain date. We study how the demand for liquid assets (i.e. real balances) of banks interacts with asset prices and interest rates. We show that the economy in which banks, markets, and a central bank are present reaches the same equilibrium of an economy with complete asset markets and a safe asset. ∗

We would like to thank for helpful conversations, without implicating for any shortcoming, Costas Azariadis, Abhijit Banerjee, Paolo Siconolfi, Steve Spear, and Ping Wang. We gratefully acknowledge financial support from the Weidenbaum Center on the Economy, Government, and Public Policy at Washington University. † Department of Economics, Washington University, St. Louis MO 63130-4899, U.S.A., email, [email protected]. ‡ Department of Economics, Unversidad de San Andrés, Victoria (1644) Buenos Aires, Argentina, email: [email protected].

1

1

Introduction

Financial systems diﬀer substantially across countries. The diﬀerence is often summarized by the relative weight of banks and markets in overall financial activity. For example, the U.S. financial system is considered a market oriented one, while the German financial system is characterized by a strong relevance of (universal) banking. Interestingly, banks and markets are present, albeit to a diﬀerent extent, in all industrialized economies, and both contribute to economic activity; moreover, the diﬀerent mix of markets and other intermediaries has not prevented countries from reaching high levels of industrial development: The United Sates, France, and Germany, for example, are all very rich countries. It is reasonable to postulate that banks and markets play diﬀerent yet in part complementary roles in the economy, even though both share the common general goal of transferring resources from lenders to borrowers. In this paper, we build a model to characterize these roles. We postulate that banks specialize in insuring the economy against liquidity risk. When lenders face uncertainty over the relative distribution of income and consumption over time, they use intermediation for two reasons: smooth consumption given its intertemporal price, and pool risk. Part of aggregate savings constitute a reserve of liquid assets, real balances, available to lenders on demand, but lenders as a group can economize over the total amount of investment in (non-productive) real balances. Banks perform these roles for depositors. Markets, on the other hand, allow investors to share risk over the uncertainty generated by investment projects. In general, investment projects bear uncertainty and their realization, at a certain future date, can be more or less successful. In as much as projects’ realizations exhibit some negative correlation, there is a social benefit in trading risk with securities. We study a pure-exchange, two-period-lived overlapping generations model in which the population at each date is partitioned into two groups: lenders and borrowers (or entrepreneurs). Agents are born at either of two identical locations (islands). At the end of each period, a fraction of lenders born in one island is relocated to the other island. Spatial separation and limited communication, as in Townsend (1978), prevent trade across islands, and relocated agents must carry (fiat) currency. This friction generates a stochastic demand for real balances. Banks in this setting arise to insure depositors against liquidity shocks, and face stochastic withdrawals at the 2

end of each period. For this reason, they hold precautionary reserves of real balances, which can be dominated in rate of return. Banks also make loans to borrowers. Borrowers are of two types in each island. Each type is endowed with a stochastic investment project, and a random fraction of borrowers in each type results in an unsuccessful project in each period. We assume that the investment projects are negatively correlated across types. In addition, we assume that a bank cannot lend simultaneously to both types of borrowers. Following Allen and Gale (2001) we assume the existence of a (complete) set of Arrow securities on the set of states of nature generated by real shocks, in which banks are allowed to trade. Banks face real risk because they cannot perfectly diversify across borrowers. Hence, they have an incentive to trade risk with other banks in the Arrow security market. We study this environment in three diﬀerent settings.1 Initially, we do not allow the existence of the Arrow securities market, and show that real shocks and liquidity shocks interact by aﬀecting banks’ precautionary demand for real balances. We then introduce Arrow securities and let banks trade assets to share risk. We study how real and liquidity shocks interact in this setting. We show that banks demand less precautionary reserves in this case, interest rates are lower, but that the in turn the presence of liquidity shocks aﬀect asset prices. We study specifically how asset prices are aﬀected in equilibrium. Finally, we introduce a central bank that makes one-period liquidity loans to banks and show that in this case the equilibrium outcome is the same that obtains in an economy with complete markets and a safe asset. Therefore, in this setting banks, markets, and the central bank, all contribute distinctively to the determination of equilibrium.

1.1

A brief discussion of the literature

The literature on banks and markets, and in general financial systems, is very extensive. Here we discuss just some contributions that are particularly relevant to our work. Several empirical studies have analyzed the presence in the economy of banks and market-based intermediation. Levine (1997) surveys a large sample of the literature on finance and growth: the evidence that emerges from various studies is that financial systems participate in the growth process, and that both banks and market-based intermediation 1

We do not consider bank runs. For a study of these equilibria and references on the subject see Peck and Shell (2003).

3

individually contribute to the growth of output. La Porta et al. (1998) take a diﬀerent approach and investigate how the protection that diﬀerent legal systems, common-law and French-civil-law systems, shapes the evolution of the financial system. Through empirical analysis they show that the legal protection of small investors, marked in countries with a common-law based legal infrastructure, is an important determinant of the development of strong asset markets. On the other hand, civil-law based systems show a strong domination of banks in their financial sector. La Porta et al. (1998) do not fail to note, however, that some countries where asset markets did not develop nonetheless experienced robust and persistent economic development, and therefore their analysis should not be intended to determine whether one system is superior to the other. Rajan and Zingales (2002) expand and deepen the scope of the former analysis from a theoretical point of view, and introduce other elements besides the evolution of the legal system. They study what characteristic of the economic, legal, and technological environment explain the relative advantages of banks and markets. Weak legal protection of minority shareholders, small markets and firms, incremental technological progress, little transparency, tend to favor banks, that is relationship-based financial intermediation; vice versa, strong protection of minority shareholders, large markets and firms, wave-like technological progress, and transparency, tend to favor market-based financial systems. Note that Rajan and Zingales, like La Porta et al. (1998), do not intend to pick a winner, but describe the characteristics of the environment that make one type of financial system better fit to intermediate funds from savers to investors. Other authors concentrate on the institutions themselves rather than the environment. For example, Allen and Gale (2000) study, among other issues, how markets and banks collect information in diﬀerent ways and hence provide diﬀerent services. Also in the approach that we take in our analysis banks and markets provide diﬀerent services: banks specialize in insuring the economy against liquidity shocks, and markets in insuring the economy against cross-sectional risk. Banks and markets perform diﬀerent, albeit complementary, roles in the transfer of resources from savers to investors. A common theme in the study of financial systems is the emphasis on liquidity provision. Allen and Gale (1997), for instance, have in common with our analysis the separation between liquidity shocks and “real shocks”, and the overlapping generation structure of the economy. They study the diﬀerent reaction of the economy in response to real shocks, and show that 4

the volatility of consumption is diﬀerent is bank-based and market-based financial systems. In their model banks produce a Pareto Optimal allocation while markets do not. A similar conclusion, albeit for diﬀerent reasons, is reached by Holmstrom and Tirole (1997) who study the optimal provision of liquidity in the economy. They study a moral hazard model in which banks provide liquidity, and are superior to markets in this role. In their framework, unlike the others mentioned here, firms suﬀer liquidity shocks, and in the presence of aggregate uncertainty a commitment problem needs the coordinating role of banks for the eﬃcient allocation of liquidity (which markets are not able to provide). In one case, when there is only aggregate uncertainty, Holmstrom and Tirole (1997) show that government debt is needed as a tool for optimal liquidity provision. The reader will recognize the similarity with the role that the central bank plays in our model. Allen and Gale (2003) present a framework conceptually closely related to ours. They study a model with liquidity and real shocks, and relate the existence of diﬀerent intermediary contracts to their ability to complete markets. This idea is close to the one in the present paper. One way to reformulate our results is to emphasize that we start from an economy in which markets are incomplete, both relative to liquidity uncertainty and real shocks. In this formulation banks and the central bank complete the market with respect to liquidity shocks and asset markets complete markets with respect to “real” uncertainty. Unlike other papers mentioned, we study a monetary model where liquidity shocks are denominated in fiat money. This means that the preference shock that hits certain consumers does not take the form of an urgency to consume, but the urgency to hold a liquid asset, fiat money. We want to study the role of banks and markets at the aggregate level, and this formulation seems more appropriate for this purpose. Finally, of course the reasons we point out in our analysis are clearly not the sole possible distinction that can be made about the diﬀerent roles of markets and banks. Diamond and Rajan (2001), for example, point to diﬀerent commitments that are involved in market contracts and bank based contracts (deposits). The paper proceeds as follows. In the next section we introduce the general framework of our model. In the following three sections we study the model under three specifications of financial structure: first, we study an economy with only banks, then we introduce asset markets and finally we study the environment with banks, asset markets and a central bank operating discount window loans to provide an elastic currency to the economy. 5

In the conclusions, we discuss some limitations of our setting and indicate directions of future research.

2

The model

We study an economy populated by a sequence of two-period lived, overlapping generations, and an initial old generation. There is a unique consumption good in the economy. There are two separate but identical islands, A and B, and two regions in each island, called region 1 and region 2.2 At every period t = 0, 1, 2, ..., a new generation in each island and region is born. Each generation consists of two groups, each of a continuum of agents of unit mass. One group consists of risk neutral entrepreneurs, who invest when young and value consumption only when old. The second group consists of a continuum of risk averse lenders who value consumption when young and old. At time t = 0 there is an initial old generation of lenders each endowed with M units of fiat money. We follow Antinolfi, Huybens and Keister (2001), and Champ, Smith, and Williamson (1996), and assume that good cannot be transported between islands, and limited communication also prevents the transfer of assets. Only money, which is universally recognizable, can be transferred between islands.

2.1

Entrepreneurs

Entrepreneurs are risk neutral and value consumption only when old. Each has access to a (risky) technology, and has no endowment. An investment of k units of the consumption good at time t yields f (k) units at t + 1 with positive probability, or else 0. We assume that f (k) is strictly increasing, strictly concave, C 2 and satisfies the standard Inada condition. In addition, we assume that the productivity shocks that aﬀects investment projects are perfectly inversely correlated between the two regions: either entrepreneurs in region 1 get f (k) and entrepreneurs in region 2 get 0, or the converse is true. We let η be the probability that only in region 1 the technology is successful and so 1 − η is the probability that only in region 2 the technology is successful. Because entrepreneurs have no endowment, they need to borrow to invest. In case of successful projects, entrepreneurs in region j pay back the amount 2

An alternative interpretation is to think about these as sectors.

6

they borrowed one period earlier with interest. We denote Rtj the gross interest rate that the entrepreneur in region j = 1, 2 pays in the favorable state. An entrepreneur in region 1 solves the following expected income maximization problem: £ ¡ ¢ ¤ max η f kt1 − Rt1 kt1 . 1 kt

The first-order condition for this problem is ¡ ¢ f 0 kt1 = Rt1

Similarly, an entrepreneur in region 2 solves £ ¡ 2¢ ¤ 2 2 − − max (1 η) f k k R , t t t 2 kt

which gives the same first order condition: f 0 (kt2 ) = Rt2 . We can express the demand for funds by an entrepreneur of region j as ¡ ¢ ¡ ¢ ktj∗ = ψ Rtj ≡ f 0−1 Rtj .

2.2

Lenders

All lenders receive an endowment vector (ω 1 , ω 2 ) = (x, 0) , with x > 0. At the end of each period a fraction π t of young agents in each island is relocated to a diﬀerent island. The fraction π t represents the size of the aggregate liquidity shock in each island, and determines the existence of banks in a similar fashion as in Diamond and Dybvig (1983). In addition, because money is the only asset that can be transported between islands, the random liquidity shock determines a transaction role for money as in Townsend (1987). We assume that π t is drawn from the distribution function F (π t ) , which is twice continuously diﬀerentiable and has density f (π t ) . Lenders have preferences given by U (c1 , c2 ) = ln c1 + β ln c2 . Because they face the possibility of a liquidity shock, they deposit their endowment in a local bank. Banks promise a rate of return to depositors contingent on the state of nature prevailing in the region where the bank is located, whether the depositor is moving to the other island or not, and finally the fraction of total population relocated. We denote rtm (s, π) to be the return 7

on deposits to a mover when the aggregate state is (s, π t ), where s = s1 is the state when region 1 projects are successful, and s2 is the state when region 2 projects are successful. Likewise, we indicate with rt (s, π) the rate of return on deposits promised to a lender who does not leave the island. Consumers take these returns as given and choose optimally the amount to deposit, d. Their problem can be written as: ( " 2 #) Z 1 X max ln (x − d) + β η (si ) [π ln (rtm (si , π) d) + (1 − π) ln (rt (si , π) d)] f (π) dπ d

i=1

0

The solution to this problem is d∗ =

2.3

β x 1+β

Banks

Banks take deposits, decide their portfolio of loans and reserves, announce rates of return on deposits, and trade in asset markets. We assume perfect competition in the banking sector: banks act as Nash competitors and maximize the expected utility of depositors. We assume that banks in each region can lend to entrepreneurs of the same region. This assumption does not allow banks to perfectly diversify credit risk. We maintain the assumption for simplicity, and it is easy to think of a cost function for the intermediation process that would limit endogenously the number of entrepreneurs whom a bank finds profitable to lend to. We study the problem of a bank in three economies. In the first scenario, the bank faces only the problem of determining its demand for monetary reserves. In the second scenario we open a market for contingent claims where banks in diﬀerent regions (but not diﬀerent islands) are allowed to trade in the contingent claim market. Finally, we also add a central bank that provides an elastic currency to the economy through discount window loans.

8

3

The case of no asset market or central bank

In this section, banks take deposits, decide their portfolio of loans and reserves, and announce rates of return on deposits. Their problem is to choose the fraction of deposits to invest in real balances to maximize the expected utility of depositors. Let γ t be the fraction of deposits invested in real balances, and let αjt (s, π) be the fraction of cash balances that a bank of region j uses to pay relocated depositors. The first constraint that the bank faces is that it can only use real balances to satisfy the demand for withdrawals of the π t relocated depositors. Formally, π t rtmj (s, π) 5 γ jt αjt (s, π)

pt . pt+1

(1)

Banks use the remaining resources, possible remaining real balances and return on loans, to repay deposits and provide the promised return to the 1 − π t depositors who are not relocated to the other island. The constraint is given by ¢ pt ¡ + (1 − γ t ) Rtj (s) , (1 − π t ) rtj (s, π) 5 γ jt 1 − αjt (s, π) pt+1

(2)

where Rtj (s) = Rtj if s = sj , and 0 otherwise, with j = 1, 2. The problem of the bank is to choose rtmj (s, π) and rtj (s, π) to maximize the utility of depositors, taking the deposits, d, as given. The problem is: Z 1 X ¤ £ mj j − − max log (x d) + η (s) (s, π) d + (1 π) ln r (s, π) d f (π) dπ π ln r t t m,j j r

,r

0

s

subject to (1) and (2) in addition to the non-negativity constraints 0 ≤ α ≤ 1 and 0 ≤ γ ≤ 1. Substituting the constraints (1) and (2), which will hold with equality in equilibrium, and deleting irrelevant constants, the problem can be equivalently written as Z 1 X £ max η (s) π ln γ jt αjt (s, π) + j j γ ,α (s,π)

s

0

¶¸ µ ¢ pt ¡ ¡ ¢ j j j j + 1 − γ t Rt (s) f (π) dπ (1 − π) ln γ t 1 − αt (s, π) pt+1 9

subject to 0 ≤ αj ≤ 1 and 0 ≤ γ j ≤ 1, and where η (s) = η when s = s1 and η (s) = 1 − η when s = s2 . Note that α, the fraction of real balances used to repay relocated depositors, is chosen after the shocks are observed. Therefore, the optimal value of α is contingent of the choice of γ, the total amount of real balances available. On the other hand the optimal amount of real reserves is chosen before the observation of the realization of the shocks, and cannot be contingent on their value. The solution to the problem of the bank is given by

αjt (s, π) =

where

i h p π γ jt p t +(1−γ jt )Rtj (s) t+1

p γ jt p t t+1

1;

;

¡ ¢ π∗j γ jt , s ≤ π < 1

¢ ¡ π < π ∗j γ jt , s

j p

π∗j

t γ t pt+1 ¡ j ¢ i. γt , s = h ¡ ¢ pt γ jt pt+1 + 1 − γ jt Rtj (s)

In practice, π ∗j indicates the critical value of the liquidity shock such that a bank in region j exhausts the whole amount of real balances held as reserves. For shocks higher than π ∗j banks in region j face a liquidity shortage, and movers and non-movers receive diﬀerent returns on their deposits. The optimal choice of real balances is given by Z 1 j γ t = 1 − η (sj ) F (π) dπ. π ∗ (γ jt ,sj ) Because of the aggregate nature of the liquidity shock, the bank provides partial liquidity insurance to its depositors. The rate of return on money is lower than the expected return on loans, and at the margin banks balance the insurance benefit of holding currency reserves and their opportunity cost due to the higher expected returns on loans to entrepreneurs. Notice that cash reserves are used by the bank to provide insurance to both movers and nonmovers, who do not suﬀer the liquidity preference shock. In other words, the presence of credit risk increases the demand for real balances. The presence of credit risk makes money an attractive asset because money has value in every future state of the world. Lack of credit risk would mean that η (sj ) = 1, 10

and γ jt = 1 −

R1

π∗ (γ jt ,sj )

F (π) dπ, as in Antinolfi, Huybens and Keister (2001)

and Champ, Smith and Williamson (1996).

3.1

Equilibrium

In equilibrium, the money and credit markets have to clear in each island. , therefore the market clearing The real money supply in each period is M pt condition on the money market is given by ¡ ¢ M = γ 1t + γ 2t d, pt

(3)

where superscripts indicate the demand for real balances by banks in region 1 and 2 respectively. Equation (3) implies that ¢ ¡ 1 γ t+1 + γ 2t+1 pt = . pt+1 (γ 1t + γ 2t ) Credit markets in region 1 and 2 also must clear, hence demand and supply of credit must be equal, that is in region j = 1, 2 we must have ¡ ¢ ¡ ¢ ψ Rtj = 1 − γ jt

β x 1+β

(4)

Under the assumption that f (k) = kα , 0 < α < 1 for all j, we have 1 ¡ ¢ ³ ´ 1−α ψ Rtj = Rαj . Hence in equilibrium: t

µ

1 ¶ 1−α

¡ ¢ = 1 − γ jt

β x. 1+β ¡ ¢ ¢α ¡ Letting φ ≡ β α 1−α , it follows that 1 − γ jt Rt = φ 1 − γ jt . Therefore [ 1+β x] in equilibrium Z 1 j j γt γ t = 1 − η (sj ) F (π) dπ. (5) (γ 1t+1 +γ2t+1 ) γ 1 +γ 2 α Rtj

j γt 2 γ1 +γ t t

t

t

α

(γ 1t+1 +γ2t+1 )+φ(1−γjt )

When both regions are considered, the resulting two-dimensional, first-order system defines the equilibrium law of motions for the economy. 11

3.1.1

Steady State analysis

We concentrate our analysis on steady-state equilibria. In steady state equation (5) becomes Z 1 j γ = 1 − η (sj ) F (π) dπ j γ α γ j +Φ(1−γ j )

for j = 1, 2. Notice that in the steady state each equation becomes independent from the other. We state the following Proposition 1 There exists a unique (γ 1∗ , γ 2∗ ) ∈ (0, 1)2 which satisfies both equilibrium equations, hence the steady-state equilibrium is unique. Proof. See appendix. It is worth noting that the equilibrium is never Pareto Optimal. (For a proof see Balasko and Shell (1980).) Intuitively, optimal risk-sharing dictates that a bank equalize the rate of return for both movers and non-movers. In fact, the economy does not face a random amount of resources relative to the liquidity shock. However, banks cannot adjust the amount of currency holdings after observing the liquidity shock. A bank must choose monetary reserves before observing the liquidity shock, even though ex-post, in the state of nature in which borrowers repay their loans with interest, the bank would be able to borrow money, for example, from a central bank. This is the role that the central bank will play. Before introducing the central bank, however, we allow banks to trade contingent claims to trade credit risk on a given island.

4

The model with asset markets but no central bank

In this section we consider the problem of the bank and the equilibrium of the economy when banks can trade credit risk on asset markets. We model asset markets by opening a market for Arrow securities in each island in which banks can trade immediately after young agents make their deposits. Arrow securities of course are not a perfect representation of asset markets. For example, Arrow securities markets are self-financing. However, we believe that they are good representation in that they allow the trade of goods across 12

states of nature; Allen and Gale (2003) follow the same approach in modeling asset markets. Let θj1 and θj2 denote the Arrow securities of a bank in region j which pay one unit of the consumption good when the state is s = s1 and s = s2 respectively. Arrow securities are traded before the observation of the shocks, and the determination of γ 0 s and θ0 s is simultaneous. As in the previous sections, the fraction of real balance reserves used to repay movers is determined after the observation of the shocks. The constraints that the bank faces in this case are: pt πrm (s, π) = γ jt αjt (s, π) (6) pt+1 and (1 − π) r (s, π) = γ jt (1 − αt (s, π))

¡ ¢ pt + 1 − γ jt Rtj (s) + θj (s) pt+1

(7)

The diﬀerence from the previous section is the presence of the term representing the Arrow security obligation of the bank in state s. Intuitively, the bank now has an additional tool that can be used to transfer consumption between states of nature for depositors who are not relocated. In this sense the value of monetary reserves is diminished because the bank can cover some of the credit risk it is facing through asset markets. The problem of the bank in this case is Z 1 X ¤ £ mj j − − max ln (x d) + η (s) (s, π) d + (1 π) ln r (s, π) d f (π) dπ π ln r t t m,j j r

,r

s

0

subject to (6) and (7), the usual non-negativity constraints, and q1 θj (s1 ) + q2 θj (s2 ) = 0 where q1 and q2 are the prices of the Arrow securities that pay in state s1 and s2 respectively. Normalizing, θj (s1 ) + qθj (s2 ) = 0,

(8)

where q ≡ qq12 is the relative price of Arrow securities. Equation (8) is the self-financing constraint typical of Arrow securities trading. The solution to 13

this problem sets

αj (s, π) =

h i p π γ jt p t +(1−γ jt )Rtj (s)+θj (s) t+1

p γ jt p t t+1

¢ ¡ π ∗j γ jt , s ≤ π < 1

1;

;

¢ ¡ π < π ∗j γ jt , s

(9)

where the critical value of the liquidity shocks, depending the state s, are given by π ∗j

(γ t , s) = h pt γ jt pt+1

pt γ jt pt+1 i. ¡ ¢ + 1 − γ jt R (s) + θj (s)

(10)

Having determined the liquidation policy of the bank once the state of the economy is realized, we need to determine the (ex-ante) choices of γ jt , θj (s1 ) , and θj (s2 ) . Let us solve the case of bank j = 1. The case of bank 2 is symmetric. Using the optimal values for α0 s we can formulate this problem as max η

γ,θ1 ,θ2

∗ (s ) πZ 1

0

+η

Z1

π ∗ (s1 )

· ¸ ¡ ¢ 1 1 pt 1 log γ t + 1 − γ t R (s1 ) + θ (s1 ) f (π) dπ + pt+1

£¡ ¤ª © ¢ π log γ 1t + (1 − π) log 1 − γ 1t R (s1 ) + θ1 (s1 ) f (π) dπ+

(1 − η)

∗ (s ) πZ 2

0

+ (1 − η)

· ¸ 1 1 pt log γ t + θ (s2 ) f (π) dπ pt+1

Z1

π ∗ (s2 )

¤ £ π log γ 1t + (1 − π) log θ1 (s2 ) f (π) dπ

subject to: q1 θ11 + q2 θ12 = 0. 14

where θ1i ≡ θ1 (si ) , i = 1, 2. The solution to the problem of the bank in region 1 gives γ 1t

Z1

=1−η

π ∗1 (s1 )

Z1

F (π) dπ − (1 − η)

F (π) dπ,

π ∗1 (s2 )

and Z1

R1 (s1 ) (1 − η) θ1t (s2 ) = t qt

F (π) dπ.

π ∗1 (s2 )

. where qt ≡ qq2t 1t The solution to the problem of the bank in region 2 gives γ 2t

=1−η

Z1

π ∗2 (s1 )

and

F (π) dπ − (1 − η)

Z1

F (π) dπ

π ∗2 (s2 )

θ2t (s1 ) = qt R2 (s2 ) η

Z1

π ∗(s

1)

F (π) dπ

Two observations are important about the solution to the bank’s problem. First, the presence of asset markets aﬀects the demand for real balance reserves of the bank. Asset markets give the bank a new tool for transferring real risk (i.e. credit risk generated by the productivity shocks aﬀecting entrepreneurs). The bank still insures depositors against liquidity risk, for which it needs currency, but it has another tool that allows the provision of resources to repay non-relocated depositors when borrowers do not pay back their loans. In general, asset markets will in part substitute for real balances. However, and this is the second important remark, the presence of the liquidity shock will aﬀect the demands (and prices) for Arrow securities. Specifically, the amount of Arrow security that pays oﬀ in state s2 for bank in region 1 depends on the ratio of the interest rate on loans and the price of the Arrow security, which is an indication of the relative cost of obtaining consumption in state of nature s2 , multiplied by the probability of the event that s2 will occur and the bank will suﬀer a shortage of liquidity. 15

4.1

Steady state equilibrium

In equilibrium, the money market, loans markets, and Arrow securities market must clear. The market clearing conditions are the same as in the previous section, with the addition of the Arrow securities market. In order, money markets clearing requires

which implies that

¡ 1 ¢ M γ t + γ 2t d = pt ¢ ¡ 1 γ t+1 + γ 2t+1 pt = . 1 2 (γ t + γ t ) pt+1

Loan markets clearing conditions imply that ¡ ¢ 1 − γ 1t d = ¡ ¢ 1 − γ 2t d =

µ

α R1

1 ¶ 1−α

µ

α R2

1 ¶ 1−α

Arrow securities markets clear when

q1 θ11 + q2 θ12 = 0 q1 θ21 + q2 θ22 = 0 and θ11 = −θ21 θ12 = −θ22

16

It is easy to show that equilibria always exist. For example, there is an equilibrium where π ∗1,1 = π ∗2,2 = 1 and π ∗ij ∈ (0, 1) , i 6= j 3 . This means that a bank in region j does not exhaust the whole amount of real balances held as reserves when the solvency shock j is favorable. That is, bank in region j never liquidates the complete stock of real balances when the productivity shock in region j reaches the high value. Therefore, from the bank’s optimal choice movers and non - movers get the same consumption whenever this happens in this favorable state. Hence there is complete risk sharing region - wise. This was never possible when there were no Arrow securities. Note however that in the region where the solvency shock is not favorable depositors do not get full insurance. Hence risk sharing is not complete island wise. However we have not ensured global uniqueness of the steady state, even 3

There is a technical detail that must be clarified. The objective function for bank 1 includes an expression equal to ¸¾ · Z1 ½ ¢ ¡ q1 π log γ 1 + (1 − π) log 1 − γ 1 R1 − θ21 f (π) dπ q2

π∗ 11

which after replacing in θ12 , R1 and

q1 q2

by the expressions obtained above it is

Ã " Z Z1 ( ¢ ¡ 1 1 α−1 1 π log γ + (1 − π) log φ 1 − γ 1 − γ − (1 − η)

1

F (π) dπ

π∗ 1,2

π∗ 11

But note that when π ∗1,1 = 1 it is because 1 − γ 1 = (1 − η)

R1

!#)

f (π) dπ

F (π) dπ. This expression

π∗ 1,2 R1

would be equal to an expression which is indeterminate, since

1

(1 − π) f (π) dπ = 0 and

log[0] = −∞. However it can be shown that when γ 1 converges (from above) to 1 − (1 − η) R1 F (π) dπ then the expression π∗ 1,2

"

1

log 1 − γ − (1 − η)

Z

1

F (π) dπ

π∗ 1,2

# Z1 π∗ 11

(1 − π) f (π) dπ

R1 converges to 0 for γ 1 suﬃciently close to 1 − (1 − η) π∗ F (π) dπ. Hence this equilibrium 1,2 exists as long as we define the equilibrium value of the expression in the objective function mentioned above equal to its limit (equal to 0).

17

though in all the examples we produced only one steady-state equilibrium exists. Notice that equilibria are not Pareto optimal in this case. Depositors who are relocated and depositors who are not still face uncertainty about their consumption when old, even though ex-ante a bank no longer faces no uncertainty about the availability of resources in the economy in diﬀerent states of nature. When there is an excess demand for liquidity movers pay a cost in terms of lower return on their deposits.

5

The economy with asset markets and a Central Bank

In this section we complete the financial structure of our simple economy and add a central bank. The central bank operates a discount window to provide short-term (i.e. one-period) loans of currency to banks facing an excessively high amount of withdrawals at the end of period t. Note that these loans are made after shocks are realized, therefore they are pure liquidity loans, the bank will be solvent in period t + 1, when borrowers repay their loans and Arrow securities trades clear. The diﬀerence with the previous case is that ex-ante a bank knows that it will be able to take contingent loans from the central bank. We assume that the central bank charges a zero net nominal (and real in steady state) interest rate on discount window loans. We let δjt (s, π) denote the amount of real balances that a bank in region j borrows from the Central Bank at date t. This amount of currency is used to pay movers in period t and will be repaid in period t + 1 to the central bank. The constraints of the bank are given by the following pt pt + δ jt (s, π) pt+1 pt+1 ¤ pt ¡ £ ¢ pt j j j j − δ t (s, π) (1 − π) rt (s, π) = γ t 1 − αt (s, π) + 1 − γ jt Rtj (s) + θjt (s) pt+1 pt+1 πrtmj (s, π) = γ jt αjt (s, π)

In this case problem of the bank is to choose optimally a liquidation policy αjt (s, π) , a borrowing policy δ jt (s, π) , the amounts of Arrow securities θjt (s) to trade, and the fraction of deposits γ t to hold as reserves. As in the previous

18

sections the bank chooses αjt (s, π) and δ jt (s, π) after observing the shocks: ¶ µ pt pt j j j max + π ln γ t αt (s, π) + δt (s, π) pt+1 pt+1 05αjt (s,π)51, δ jt (s,π) ¶ µ ¤ pt ¡ £ ¢ j pt j j j j j − δ t (s, π) + 1 − γ t Rt (s) + θt (s) (1 − π) ln γ t 1 − αt (s, π) pt+1 pt+1

In the appendix we show that one solution to this problem sets i h j pt j j j ¢ ¡ π γ t pt+1 +(1−γ t )Rt (s)+θt (s) ; π 5 π∗j γ jt , s j j pt γ αt (s, π) = t pt+1 ¢ ¡ ∗ 1; π > π j γ jt , s

and

δjt (s, π) =

(

0; pt+1 pt

¡ j ¢ ∗ γt , s π 5 π j h £¡ i ¤ ¢ pt π 1 − γ jt Rtj (s) + θjt (s) − γ jt pt+1 (1 − π) ;

¢ ¡ π > π ∗j γ jt , s

where π ∗j is given by (10). The bank holds a certain amount of reserves, and ¢ ¡ uses them to pay movers as long as π 5 π∗j γ jt , s . For larger values of the relocation shock, the bank borrows currency for one period from the discount window. This is not the only solution to the bank’s problem. In fact, the liquidation and borrowing policies depend on the total amount of currency reserves the bank decided to acquire before observing the liquidity and productivity shocks, and this amount is indeterminate with zero-nominal-rate discount window lending. If it were not, the bank would hold only currency when currency’s rate of return dominated other rates of return; vice versa, the demand for currency reserves would be zero if money were dominated in rate of return by other portfolios of assets. We can make these statements because in this section, with a central bank operating a discount window, we have a complete set of markets. The easiest way to note this fact is by rewriting the maximization problem of the bank subject to only one budget constraint. Solving for δ jt (s, π) and substituting we obtain: ¡ ¢ pt πrtmj (s, π) + (1 − π) rtj (s, π) = γ jt + 1 − γ jt Rtj (s) + θjt (s) , pt+1

which implies θjt

(s) =

πrtmj

(s, π) + (1 −

π) rtj

· ¸ ¡ ¢ j j pt j (s, π) − γ t + 1 − γ t Rt (s) . pt+1 19

(11)

Recall that the self-financing condition at the beginning of date t is given by: θjt (s1 ) + qθjt (s2 ) = 0. Replacing in this equation the expression for θjt (s) gives the sole budget constraint for maximization problem of the bank: · ¸ 2 ¤ X ¡ £ mj ¢ j j j pt j qs πrt (s, π) + (1 − π) rt (s, π) = qs γ t + 1 − γ t Rt (s) pt+1 s=1 s=1 (12)

2 X

which holds for every π. The problem of the bank can then be written as max

2 X s=1

η (s)

·Z

0

1

¢ ¡ ¢¢ ¡ ¡ π ln rtmj (s, π) + (1 − π) ln rtj (s, π) f (π) dπ

¸

subject to (12). In the appendix we show that the first order conditions to this problem imply that rtmj (s, π) = rtj (s, π) for every s and π. The bank in this case is able to oﬀer movers and nonmovers the same rate of return. Note that the rate of return oﬀered is not random, as it would be if there were not asset markets. It is now evident what role banks, the central bank, and asset markets play in this model. Banks provide liquidity insurance to depositors and the central bank allows for the existence of complete markets over liquidity shocks. Asset markets allow banks to trade credit risk. Note that credit risk is not “intertemporal” but “cross-sectional”: that is, uncertainty for which asset markets are used does not concern the intertemporal distribution of resources, but total amount of resources available in a certain period. We show in the appendix that the rates of return oﬀered to movers and non-movers must be equal to ( 2 · ¸) ¡ ¢ η (s) X p t qs γ jt + 1 − γ jt Rtj (s) . (13) qs p t+1 s=1 Equation (13) states that the return that a bank promises to movers and non-movers in state s is equal to the total present value of goods received in 20

period t+1 weighted by the ratio of the probability of s relative to the price of the Arrow security that pays oﬀ in s. This is natural given the completeness of Arrow securities markets. In the appendix we also show that to insure an interior solution for γ t the following condition must hold: pt = q1 Rtj (s1 ) + q2 Rtj (s2 ) . (q1 + q2 ) (14) pt+1 Equation (14) is a no-arbitrage condition stating that the return on money (the inverse of the gross inflation rate) must be a weighted average of the promised returns from entrepreneurs of both regions, where the weights depend upon the prices of Arrow securities Using the normalization adopted so far for the prices of Arrow securities we let q2t qt ≡ . q1t Therefore the no-arbitrage condition can be expressed as µ ¶ pt pt j j − Rt (s1 ) = −qt − Rt (s2 ) pt+1 pt+1 for every j. To analyze the equilibrium of the economy, it is first essential to get the optimal net demand for Arrow securities by each bank type. We use the constraint (11) , and substitute optimal rates of return (13) and the arbitrage condition (14) to obtain ¶ µ ¸ · ¢ ¡ ¢ j η (s) ¡ j j j j pt j θt (s) = q1 Rt (s1 ) + q2 Rt (s2 ) − γ t + 1 − γ t Rt (s) qs pt+1 We show in the appendix that the optimal net demand functions for these securities by each bank type are given by ¶ µ ¤ pt £ 1 1 qt γ t − (1 − η) (1 + qt ) θt (s1 ) = p µ t+1 ¶ £ ¤ pt θ1t (s2 ) = (1 − η) (1 + qt ) − qt γ 1t qt pt+1 ¶ ¤ pt £ η (1 + qt ) − γ 2t (s1 ) = pt+1 µ ¶ ¤ pt £ 2 1 2 θt (s2 ) = γ t − η (1 + qt ) qt pt+1 θ2t

µ

21

5.1

Equilibrium

In equilibrium θ1t (s) + θ2t (s) = 0 for every s, so that the asset market clears This condition implies that qt =

γ 2t + 1 − 2η γ 1t + 2η − 1

(15)

Thus, the relative price of the Arrow securities, which is the relative cost of transferring resources from one solvency state to the other, must be equal in equilibrium to the ratio of two expressions which depend on the fraction of deposits maintained in cash, γ jt . However, equation (15) can also be rewritten as qt =

2 (1 − η) − (1 − γ 2t ) 2η − (1 − γ 1t )

This expression shows that the relative cost of transferring goods across states of nature must be related to the probabilities of success for every entrepreneur type and to the fraction of deposits that banks are willing to lend. Also, note that in equilibrium (1 + qt )

pt = Rt1 (s1 ) + qt Rt1 (s2 ) = Rt2 (s1 ) + qt Rt2 (s2 ) pt+1

holds. Because we assumed that Rt1 (s2 ) = Rt2 (s1 ) = 0, the last equation is equivalent to Rt1 (s1 ) = qt Rt2 (s2 ) Hence, the relative cost of transferring goods between states s1 and s2 must be equal to the relative returns that banks obtain from entrepreneur types when their projects are successful. The money market and the loans markets must also clear. These market clearing conditions remain unchanged:

and

¡ ¢ M = γ 1t + γ 2t d pt Rtj =

α ¡ ¢1−α d1−α 1 − γ jt 22

5.2

Steady state analysis

In the appendix we prove the that the steady state equilibrium is unique, and in addition we show that there are no other equilibria. Specifically,we show the following: o n 1 1 1 α 1−α Proposition 2 Under the condition d < min η , 1−η there exists a

unique steady state (γ 1 , γ 2 ) ∈ π ∗j (γ j , sj ) then dπ, so " Z 1 ¡ ¢ j j γ 1 − γ > 1 − η (sj ) "

1 − η (sj )

=

"

1 − η (sj )

Z

1

R1

π ∗j (γ j ,sj )

Z

F (π) dπ > #

F (π) dπ η (sj )

#

F (π) dπ η (sj )

π∗j (γ j ,sj ) 1

π∗j (γ j ,sj )

F (π) dπ

π ∗j (γ j ,sj )

π∗j (γ j ,sj )

But

1

#

R1

π ∗j (γ j ,sj )

ÃZ

1

π ∗j (γ j ,sj )

ÃZ

1

π∗j (γ j ,sj )

¡ ¢ F π ∗j (γ j , sj )

¡ ¡ ¢¢ F π ∗j γ j , sj dπ

¡ ¡ ¢¢ F π ∗j γ j , sj dπ

!

£ ¡ ¢¤ ¡ ¡ ¢¢ F (π) dπ η (sj ) 1 − π ∗j γ j , sj F π ∗j γ j , sj 25

!

R1 It is also true that for all π < 1 we have F (π) < 1, so π∗ (γ j ,sj ) F (π) j R1 dπ < 1 − π ∗j (γ j , sj ) . Since η (sj ) > 0 then η (sj ) π∗ (γ j ,sj ) F (π) dπ < η (sj ) j ¤ £ ¤ £ R1 1 − π∗j (γ j , sj ) ⇔ −η (sj ) π∗ (γ j ,sj ) F (π) dπ > −η (sj ) 1 − π ∗j (γ j , sj ) ⇔ 1 j £ ¤ R1 − η (sj ) π∗ (γ j ,sj ) F (π) dπ > 1 − η (sj ) 1 − π ∗j (γ j , sj ) . Therefore j

"

1 − η (sj )

Z

1

π ∗ (γ j ,sj )

#

£ ¡ ¢¤ ¡ ¡ ¢¢ F (π) dπ η (sj ) 1 − π ∗j γ j , sj F π ∗j γ j , sj

£ £ j ∗ ¡ j ¢¤¤ £ ¡ ¢¤ ¡ ¡ ¢¢ > 1 − η (sj ) 1 − π j γ , sj η (sj ) 1 − π ∗j γ j , sj F π ∗j γ j , sj £ ¤ £ ¤ But since η (sj ) < 1, then 1 − η (sj ) 1 − π ∗j (γ j , sj ) > 1 − 1 − π ∗j (γ j , sj ) = π ∗j (γ j , sj ) . Thus ¡ ¢¤¤ ¡ ¢¤ ¡ ¡ ¢¢ £ £ £ η (sj ) 1 − π ∗j γ j , sj F π ∗j γ j , sj 1 − η (sj ) 1 − π ∗j γ j , sj £ ¡ ¢ ¡ ¢¤ ¡ ¡ ¢¢ > π ∗j γ j , sj η (sj ) 1 − π∗j γ j , sj F π∗j γ j , sj We thus have shown that " Z ¡ ¢ j j = 1 − η (sj ) γ 1−γ

1

#

F (π) dπ η (sj )

π ∗j (γ j ,sj )

ÃZ

1

F (π) dπ

π ∗j (γ j ,sj )

£ ¡ ¢ ¡ ¢¤ ¡ ¡ ¢¢ > π ∗j γ j , sj η (sj ) 1 − π ∗j γ j , sj F π ∗j γ j , sj

!

which is what we wanted to demonstrate. ¥

A.2

Solving the rest of the bank problem for the case of Central Bank and Arrow Securities

The problem of the bank can be written as 2 X

η (s)

s=1

subject to

·Z

0

1

¢ ¡ ¢¢ ¡ ¡ π ln rtmj (s, π) + (1 − π) ln rtj (s, π) f (π) dπ

¸

· ¸ 2 ¤ X ¡ £ mj ¢ j j j pt j qs πrt (s, π) + (1 − π) rt (s, π) = qs γ t + 1 − γ t Rt (s) pt+1 s=1 s=1

2 X

26

Let φ (π) be the multiplier of this constraint. We solve this backwards. We first take γ jt as given and also we take every possible realization of π as given. The Lagrangian for this problem is L (π) =

2 X s=1

¡ ¡ ¢ ¡ ¢¢ η (s) π ln rtmj (s, π) + (1 − π) ln rtj (s, π)

· ¸ ¡ £ mj ¤ ¢ j j pt j j +φ (π) qs γ t + 1 − γ t Rt (s) − πrt (s, π) + (1 − π) rt (s, π) pt+1 s=1 2 X

The first order conditions are are

η (s) ∂L (π) = 0 ⇔ mj = φ (π) qs jm ∂rt (s, π) rt (s, π) η (s) ∂L (π) = 0⇔ j = φ (π) qs j ∂rt (s, π) rt (s, π) Therefore rtmj (s, π) = rtj (s, π) and so is true that qs rtmj (s, π) = qs rtj (s, π) . Replacing these conditions on the left hand side of the constraint gives 2 X s=1

¤ £ qs πrtmj (s, π) + (1 − π) rtj (s, π) =

(1 − π) 1 π + = φ (π) φ (π) φ (π)

Then, the constraint can be rewritten as · ¸ 2 X ¡ ¢ j 1 j pt j = qs γ t + 1 − γ t Rt (s) φ (π) pt+1 s=1

η(s) =⇒ and so, we had from before that qs rtmj (s, π) = φ(π) ( 2 · ¸) X ¡ ¢ p η (s) t rtmj (s, π) = qs γ jt + 1 − γ jt Rtj (s) qs p t+1 s=1

but we also had rtmj (s, π) = rtj (s, π) . Therefore, replacing ¡ mj this¢ in the objective function, for every π and s we get that π ln rt (s, π) + (1 − π) ¡ ¢ ln rtj (s, π) is equal to ( ( 2 · ¸)) ¡ ¢ j η (s) X j pt j ln qs γ t + 1 − γ t Rt (s) qs pt+1 s=1 27

Therefore the ex-ante utility is ·Z 1 ¸ 2 X ¢ ¡ j ¢¢ ¡ ¡ mj η (s) π ln rt (s, π) + (1 − π) ln rt (s, π) f (π) dπ s=1 2 X

0

" 2 X

· ¸# ¢ ¡ p t + ln qs γ jt + 1 − γ jt Rtj (s) p t+1 s=1 hP h ³ ´ ¢ j ii ¡ P2 2 η(s) j j pt j So then, choosing γ t to maximize s=1 η (s) ln qs +ln s=1 qs γ t pt+1 + 1 − γ t Rt (s) h hP ¢ j ii ¡ 2 j pt j − γ q + 1 γ , is the same as choosing γ jt that maximizes ln t pt+1 t Rt (s) s=1 s i h ¡ ¢ P pt + 1 − γ jt Rtj (s) , subject to γ jt ∈ [0, 1] . Therefore, equal to 2s=1 qs γ jt pt+1 µ

η (s) = η (s) ln qs s=1

¶

we will have that γ jt ∈ (0, 1) if pt = q1 Rtj (s1 ) + q2 Rtj (s2 ) (q1 + q2 ) pt+1

for every j. Otherwise we have a corner solution. Now, if this is the case, we have now that · ¸ 2 X ¡ ¢ j j pt j qs γ t + 1 − γ t Rt (s) = q1 Rtj (s1 ) + q2 Rtj (s2 ) pt+1 s=1 h ¡ ¢i pt (q1 + q2 ) − q1 Rtj (s1 ) + q2 Rtj (s2 ) = 0. Let qt ≡ qq2t . Therefore since pt+1 1t

the no-arbitrage condition can be expressed as (1 + qt ) Rtj (s2 ) . Let us see the problem for each region.

pt pt+1

= Rtj (s1 ) + qt

• Region 1 Bank According to our derivation above, in this case we have · ¸ 2 X ¡ ¢ 1 1 pt 1 qs γ t + 1 − γ t Rt (s) = q1 Rt1 + 0 p t+1 s=1 Recall that for every j, θjt (s) = π rtmj (s, π) + (1 and also recall that rtmj (s, π) = rtj (s, π) and: rtmj

η (s) (s, π) = qs

( 2 X

− π)

rtj

h ¡ ¢ j i j pt j (s, π) − γ t pt+1 + 1 − γ t Rt (s)

· ¸) ¡ ¢ j j pt j qs γ t + 1 − γ t Rt (s) pt+1 s=1 28

so for j = 1 : ¸ (X · · ¸) · ¸ 2 ¡ ¡ ¢ ¢ 1 η (s) p t j 1 1 1 1 pt 1 − γt θt (s) = qs γ t + 1 − γ t Rt (s) + 1 − γ t Rt (s) qs pt+1 pt+1 s=1 ¸ ¸ · · ¡ ¢ 1 η (s) 1 1 pt 1 q1 Rt − γ t + 1 − γ t Rt (s) = qs pt+1 So let us compute

θ1t (s1 ) = ηRt1 − γ 1t

¢ ¡ pt − 1 − γ 1t Rt1 pt+1

pt = Rt1 (s1 ) + qt Rt1 (s2 ) = But from the no-arbitrage condition (1 + qt ) pt+1 pt Rt1 , so Rt1 = (1 + qt ) pt+1 . Replacing this in the above expression for θ1t (s1 ) we get: ¶ µ ¤ pt £ 1 1 θt (s1 ) = qt γ t − (1 − η) (1 + qt ) pt+1

We can now solve for θ1t (s2 ) using identical arguments. We know that θ1t (s2 ) =

1−η pt q1 Rt1 − γ 1t q2 pt+1

pt Recalling that Rt1 = (1 + qt ) pt+1 then ¶ µ £ ¤ pt 1 (1 − η) (1 + qt ) − qt γ 1t θt (s2 ) = qt pt+1

• Region 2 Bank

Following the same steps we obtain:

On the other hand θ2t

pt θ2t (s1 ) = ηqt Rt2 − γ 2t pt+1 ¶ µ ¤ pt £ η (1 + qt ) − γ 2t = pt+1

¸ ¸ · ¡ ¢ 2 1−η 2 2 pt 2 q2 Rt − γ t (s2 ) = + 1 − γ t Rt q2 pt+1 µ ¶ ¤ pt £ 2 1 γ t − η (1 + qt ) = qt pt+1 ·

29

A.3

Proof of proposition 2

We first show that there exists a unique R1 , R2 satisfying the stationary βx . Note that the first condition can equilibrium equations. Recall that d ≡ 2+β R1 +R2 be written as R2 = R1 , which simplifies to R2 =

R1 R1 − 1

The second equation can be rewritten as 1

1

−α −α α 1−α ¡ 1 ¢ 1−α α 1−α ¡ 2 ¢ 1−α 2ηR − R R = 2 (1 − η) R2 − d d

1

Hence we have two equations in two unknowns. The first equation defines a curve on the plane (R1 , R2 ) which is strictly decreasing with asymptotes in (1, 1) . The second equation also defines implicitly a curve on the plane (R1 , R2 ). To get the derivative we apply the implicit function theorem to the map: ! Ã 1 1 −α −α 1−α ¡ ¡ 1 2¢ ¢ α α 1−α ¡ 1 ¢ 1−α 2 2 1−α 1 Φ R , R ≡ − 2 (1 − η) R − + 2ηR − R R d d ΦR1 (R1 ,R2 ) 2 − = . at the point where Φ (R1 , R2 ) = 0. In general we have that dR 1 ΦR2 (R1 ,R2 ) dR In this case we have µ ¶ 1 −α ¡ 1 ¢ 1−α ¡ 1 2¢ α α 1−α −1 R ΦR1 R , R = 2η + d 1−α

so

Ã ! 1 −α ¡ 1 2¢ α 1−α α ¡ 2 ¢ 1−α −1 R ΦR2 R , R = − 2 (1 − η) + d 1−α 1

2

2η +

α 1−α d

dR =µ dR1 2 (1 − η) +

¡

α 1−α 1

¢

α 1−α α d 1−α

30

−α

(R1 ) 1−α

−1

−α

(R2 ) 1−α −1

¶ >0

It remains to show that for low R1 the first curve is above the second one and for large R1 the reverse is true. Clearly, according to the first curve, R2 approaches infinity when R1 ↓ 1. According to this curve it is also true that as R1 ↑ ∞ then R2 ↓ 1. For the second curve behavior it is convenient to write down the condition 1

1

−α −α α 1−α ¡ 2 ¢ 1−α α 1−α ¡ 1 ¢ 1−α 2 (1 − η) R − R R = 2ηR1 − d d

2

Suppose first that R1 ↓ 0. Therefore 2ηR1 − α 1 α 1−α

−α 1−α

1 1−α

d

−α

(R1 ) 1−α ↓ −∞ since

−α 1−α

0 and small the first curve is strictly above the second curve. Also, when R1 ↑ +∞ we will have that R2 ↑ +∞ . Otherwise, if R2 ↑ R∗∗ < +∞, the expression 1 1−α

2 (1 − η) R2 − α d 1 1−α

−α

(R2 ) 1−α goes towards a finite number, but the expression

−α

2ηR1 − α d (R1 ) 1−α ↑ +∞ when R1 does, therefore the equality should not hold for suﬃciently big R1 . Because the curve is strictly increasing, it cannot happen that while R1 ↑ +∞ then R2 ↓ −∞ . Therefore, for suﬃciently large R1 the value of R2 must also be suﬃciently large. Therefore the second curve is strictly above the first curve for R1 big. Because of the monotonicity properties and continuity, there exists a unique R1 , R2 where the two curves intersect. Because of the properties of the first curve, both R1 and R2 must be strictly greater than one. Now we show that this pair of interest rates is a stationary equilibrium. To do this we show that there is an equivalence between this pair of interest rates and the pair (γ 1 , γ 2 ) ∈ (0, 1)2 that solves the steady state of the dynamical

31

system in γ jt . Suppose R1 , R2 is a steady state for the system ! µ ¶Ã 1

2−

2−

α 1−α β x 2+β

µ

1

1

α 1−α β x 2+β

(R1t+1 ) ¶Ã

1 1−α

1 1

(Rt1 ) 1−α

+ +

1

2 (Rt+1 )

1 1−α

1 1

(Rt2 ) 1−α

!

R1 R2 = 1 t t 2; Rt + Rt

and that Rj > 1. Hence it must be true that 1 =

Rt1 = Rt2

R1 R2 R1 +R2

1

2 (1 − η) −

α 1−α 1

β x) (Rt2 ) 1−α ( 2+β 1

2η −

α 1−α 1

β x) (Rt1 ) 1−α ( 2+β

and

1

R1 = R2

2 (1 − η) − 2η −

α 1−α 1 (R2 ) 1−α

β x 2+β 1 α 1−α 1 β (R1 ) 1−α 2+β x

(

(

)

)

1

Hence given that 1 − γ j =

α 1−α 1 (Rj ) 1−α

β x 2+β

and since Rj > 1 it is clear that

( ) α γ j < 1. Let φ ≡ β α 1−α = d1−α by definition of d. Since we assumed that x £ 1−α ( 2+β ) 1−α ¤ 1−α φ (2η)φ1−α φ b1 and R2 > 1−α . To show this, it suﬃcient to show that the point R ≡ (2(1−η))

φ , (2η)1−α

b2 ≡ R

φ (2(1−η))1−α

is to the left of the intersection of the two curves. ³ ´ b1 , R b2 it is true that To prove this last fact, note that at R

Ã ! 1 1 −α −α ³ ³ ³ ´ 1−α ´ ´ 1−α 1−α α α 1−α 1 2 2 2 1 1 b b b b b b Φ R ,R + 2η R − R R = − 2 (1 − η) R − d d = − ((2 (1 − η))α φ − φ (2 (1 − η))α ) + ((2η)α φ − φ (2η)α ) = 0 ³ ´ 1 b2 b That is, the point R , R lies on the curve defined by Φ (R1 , R2 ) = 0. Now

b2 with we compare R

b1 R b 1 −1 R

to see whether this point lies above or below the φ b2 = second, downward sloping curve. We know that R and (2(1−η))1−α b1 R

b1 − 1 R

=

(2η)1−α

³

φ

φ (2η)1−α

32

φ ´= φ − (2η)1−α −1

Since φ < (2η)1−α + (2 (1 − η))1−α then φ − (2η)1−α < (2 (1 − η))³1−α . Thus, ´ b1 φ φ R 2 1 b2 b b < . R < , R R is Therefore and so the point b 1 −1 (2(1−η))1−α φ−(2η)1−α R

below the downward sloping curve. But then the pair (R1 , R2 ) where both b1 and R2 > R b2 . This shows the claim. As curves intersect must imply R1 > R 1

1

α 1−α

a consequence, recalling that 1 − γ j = 1

claim, it is clear that

1

(R1 ) 1−α

2η − 1, and therefore γ j > 0 for both j. With this in mind, recall then that 1

R1 = R2

2 (1 − η) −

α 1−α 1 (R2 ) 1−α

β x) ( 2+β

1

2η −

α 1−α 1 (R1 ) 1−α

β x) ( 2+β

=

2 (1 − η) − (1 − γ 2 ) 2η − (1 − γ 1 )

1

> 0. Therefore either 2 (1 − η) − (1 − γ 2 ) > 0 and 2η − (1 − γ 1 ) Clearly R R2 > 0 or 2 (1 − η) − (1 − γ 2 ) < 0 and 2η − (1 − γ 1 ) < 0. If the first two set of inequalities hold, then we have at the same time that γ 2 > 2η − 1 and γ 1 > 1 − 2η. If the first two set of inequalities hold then we have at the same time γ 2 < 2η − 1 and γ 1 < 1 − 2η, but this second case must be ruled out since this implies that at least one γ j is strictly negative. So the first set of inequalities α holds. Given the definition of γ j we have that Rj = 1−α for β (1−γ j )1−α ( 2+β x) 1−α 1 (1−γ 2 ) = every j. Hence R and so: 1−α R2 (1−γ 1 ) 1−α

or

1 − 2η + γ 2 (1 − γ 2 ) 2 (1 − η) − (1 − γ 2 ) = = 2η − (1 − γ 1 ) 2η − 1 + γ 1 (1 − γ 1 )1−α ¡ ¢1−α ¡ ¢ ¡ ¢¡ ¢1−α 1 − γ2 2η − 1 + γ 1 = 1 − 2η + γ 2 1 − γ 1

which is one of the two equations of the dynamical system in γ jt (in steady state). 33

On the other hand, we also had R1 + R2 = R1 R2 , equivalent to = R1 . Replacing we have

R1 R2

+1

1−α

(1 − γ 2 ) α 1+ ³ ´1−α 1−α = 1 β (1 − γ ) (1 − γ 1 )1−α 2+β x

1−α

But

(1−γ 2 )

(1−γ 1 )1−α

=

2(1−η)−(1−γ 2 ) 2η−(1−γ 1 )

2

= 1−2η+γ so after some algebra we get that 2η−1+γ 1

γ1 + γ2 =

α (2η − 1 + γ 1 ) ³ ´1−α β (1 − γ 1 )1−α 2+β x

Hence the values of γ j defined above satisfies the two equations that must hold at the steady state equilibrium for the system with (γ 1t , γ 2t ) . Since Rj determines uniquely γ j and by the uniqueness of (R1 , R2 ) then the pair of (γ 1 , γ 2 ) that satisfies both equations are also unique. The converse is also straightforward to show. Suppose there exists a unique pair of γ 1 , γ 2 where γ 1 > 1 − 2η and γ 2 > 2η − 1 and satisfying γ1 + γ2 = and

α (2η − 1 + γ 1 ) (2η − 1 + γ 1 ) φ ³ ´1−α = β (1 − γ 1 )1−α (1 − γ 1 )1−α 2+β x

¡ ¢1−α ¡ ¢ ¡ ¢¡ ¢1−α 1 − γ2 2η − 1 + γ 1 = 1 − 2η + γ 2 1 − γ 1

Therefore, define Rj ≡ satisfies both equations

φ . (1−γ j )1−α

We need to show that the pair (R1 , R2 )

1=

R1 R2 R1 + R2

and ¢ −α ¢ −α 1 ¡ 1 ¡ 2 (1 − η) R2 − φ 1−α R2 1−α = 2ηR1 − φ 1−α R1 1−α

equivalent to

"

R2 2 (1 − η) −

µ

φ R2

# 1 ¶ 1−α 34

"

= R1 2η −

µ

φ R1

# 1 ¶ 1−α

We basically work backwards relative to the first part of the proof. We know that γ1 + γ2 = but

γ 1 +γ 2 2η−1+γ 1

γ1 + γ2 φ (2η − 1 + γ 1 ) φ = =⇒ 1−α 1 (2η − 1 + γ ) (1 − γ 1 ) (1 − γ 1 )1−α

γ 1 +2η−1+1−2η+γ 2 2η−1+γ 1

=

=1+

1−2η+γ 2 . 2η−1+γ 1 2 1−α

But from the second equation (1 − γ ) 1−α (1 − γ 1 ) , so

φ (1−γ 1 )1−α 1

Hence

=1+

1−2η+γ 2 . 2η−1+γ 1 2

(2η − 1 + γ ) = (1 − 2η + γ ) 1−α

(1 − γ 2 ) 1 − 2η + γ 2 = 1 + 1+ 2η − 1 + γ 1 (1 − γ 1 )1−α But 1 + to 1 +

1−α 1−γ 2

(

)

(1−γ 1 )1−α

R1 . R2

is equal to 1 +

Therefore the equality

R1 =

µ µ

φ (1−γ 1 )1−α φ (1−γ 2 )1−α

φ (1−γ 1 )1−α

¶ ¶

, which in equilibrium is equal

=1+

1−2η+γ 2 2η−1+γ 1

is equivalent to

φ 1 − 2η + γ 2 R1 R2 + R1 = 1 + = 1 + = 2η − 1 + γ 1 R2 R2 (1 − γ 1 )1−α

so R1 R2 = R1 + R2 and the first equation is shown. 1−α (2η − 1 + γ 1 ) >From the second condition in the (γ 1 , γ 2 ) system (1 − γ 2 ) 1−α = (1 − 2η + γ 2 ) (1 − γ 1 ) implies after some algebra that R1 (2η − 1 + γ 1 ) = R2 (1 − 2η + γ 2 ), therefore R1 (2η − (1 − γ 1 )) is equal to R2 (2 (1 − η) − (1 − γ 2 )) . 1

But from the definition of Rj we have 1 − γ j =

φ 1−α

1 j 1−α

. Hence the equality R1

R

(2η − (1 − γ 1 )) = R2 (2 (1 − η) − (1 − γ 2 )) is equivalent to Ã ! Ã ! 1 1 φ 1−α φ 1−α 1 2 R 2η − = R 2 (1 − η) − 1 1 (R1 ) 1−α (R2 ) 1−α which is the second equation we wanted to get. This completes the proof of uniqueness of steady state. To show that the stationary equilibrium is indeed the unique o n equilibrium, 1 implies let φ be defined as above. Note that the condition φ < min η1 , 1−η o n © ª 1−α 1 1 1−α 1−α + (1 − η) η . To show this, note that min η , 1−η = that φ < 2 35

1 1−η

when η

1 . 2

For η

0 (since 21−α > 11−α = 1). Also Φ1 12 = 0 and also we have that £ ¤ 1 Φ01 (η) = 21−α (1 − α) η −α − (1 − η)−α − (1 − η)2 i h 2 −(1+α) ” 1−α −(1+α) − Φ1 (η) = 2 (1 − α) (−α) η + (1 − η) 0. Hence, ¡the ¢function Φ1 (η) attains a strictly positive value at its maximum in 0, 12 and the function is strictly in the whole interval. ¡ concave ¢ 1 This implies that Φ1 (η) > 0 for all η ∈ 0, 2 . Hence for η ∈ [0, 12 ) then © ª 1 21−α η 1−α + (1 − η)1−α > 1−η . For η > 12 consider © ª 1 Φ2 (η) ≡ 21−α η 1−α + (1 − η)1−α − η

¡ ¢ Clearly Φ2 12 = 0 and Φ2 (1) = 21−α − 1 > 0. Computing ¡ 1 ¢ the derivatives, it 0 easy to show that Φ2 (η) = 0 for some η = η 2 ∈ 2 , 1 . Since Φ2 is strictly concave then at η 2 the function Φ2 attains a strictly positive value at η 2 and all η ∈ ( 12 , 1]. These two arguments state then that therefore n Φ2o(η) > 0 for £ ¤ 1 min η1 , 1−η 5 21−α η 1−α + (1 − η)1−α , where the equality only holds at n o 1 1 1 η = 2 . This shows then that φ < min η , 1−η implies that the condition in the statement of the proposition is satisfied. We now undertake the proof of uniqueness of equilibrium in two steps, one corresponding to the case η < 12 and the other to the case η > 12 (the case η = 12 is trivial). γ 1 > 1 − 2η, γ 2 > 2η − 1. • Case 1: η > 12 .

This implies that 2η − 1 > 0 and so γ 2 > 0. Hence it remains to show in this case that γ 1 > 0 > 1 − 2η. To do this, recall that the dynamical system in (γ 1t , γ 2t ) can be written as: γ 1t+1 + γ 2t+1 =

φ (γ 1t + 2η − 1) 1−α (1 − γ 1t ) 36

¡ ¢1−α ¡ 2 ¢1−α ¡ 1 ¢ ¡ ¢ 1 − γ 1t γ t + 1 − 2η = 1 − γ 2t γ t + 2η − 1

The second equation can also be re-expressed as ¢ ¡ ¢ ¢1−α ¡ 2 ¢1−α ¡ 1 ¡ γ t+1 + 1 − 2η = 1 − γ 2t+1 γ t+1 + 2η − 1 1 − γ 1t+1 and from the first equation we get both γ 2t+1 =

γ 2t+1 = 1 + γ 1t+1 −

φ(γ 1t +2η−1) 1−α

(1−γ 1t )

φ(γ 1t +2η−1) 1−α

(1−γ 1t )

− γ 1t+1 and 1 −

. Replacing these two expressions in the last

equation we get:

µ ¶ ¡ ¢1−α φ (γ 1t + 2η − 1) 1 1 − γ t+1 + 1 − 2η 1 − γ t+1 1−α (1 − γ 1t ) µ ¶ ¢ φ (γ 1t + 2η − 1) ¡ 1 1 − = 1 + γ t+1 − γ + 2η 1 t+1 1−α (1 − γ 1t )

which is a one - dimensional dynamical system. We already know that this system has two steady states. One in γ 1t+1 = 1 − 2η (which is not an equilibrium) and the other one where γ 1t+1 > 1 − 2η obtained ¡ ¢in the last subsection. The system defines implicitly a curve on the γ 1t , γ 1t+1 plane. We first show that this curve is strictly increasing on 0 for γ 1t and γ 1t+1 greater than tion of γ 1t , and it can be shown that dγt+1 1 t or equal to 1 − 2η and strictly less than one. Proof of Lemma. To show that there is an implicit¡ function, ¢ by the 1 1 Implicit Function Theorem it is enough to show that Fγ 1t+1 γ t , γ t+1 6= 0 (we 37

will in fact show that this is strictly negative). To do this, we just compute this derivative: ¡ ¢ Fγ 1t+1 γ 1t+1 , γ 1t µ ¶ ¡ ¢1−α ¢−α ¡ 1 ¢ ¡ α (γ 1t + 2η − 1) 1 1 − − − − − = − 1 − γ t+1 γ 1 + 2η (1 α) 1 γ t+1 t+1 1−α d1−α (1 − γ 1t ) ¶−α µ ¢ ¡ 1 α (γ 1t + 2η − 1) 1 − − (1 − α) 1 + γ t+1 − + 2η 1 γ t+1 1−α d1−α (1 − γ 1t ) ¶1−α µ α (γ 1t + 2η − 1) 1 ·1 − 1 + γ t+1 − 1−α d1−α (1 − γ 1t )

>From this expression it is straightforward to see that for any γ 1t ≥ 1 − 2η, ¡ ¢ γ 1t+1 ≥ 1 − 2η, and less than one, then Fγ 1t+1 γ 1t+1 , γ 1t < 0. This shows that the implicit function is well defined. On the other hand, to sign the implicit ¡ ¢ dγ 1 we need to get Fγ 1t γ 1t+1 , γ 1t . This is equal to: derivative dγt+1 1 t ! Ã 1 (1−α) 1 1 −α ¡ 1 ¢ + (γ t + 2η − 1) (1 − α) (1 − γ t ) (1 − γ t ) · Fγ 1t γ t+1 , γ 1t = φ 2(1−α) (1 − γ 1t ) " ¶−α # µ 1 ¡ ¡ ¢ ¢ (γ + 2η − 1) 1−α 1 − γ 1t+1 + γ 1t+1 + 2η − 1 (1 − α) 1 + γ 1t+1 − φ t 1−α (1 − γ 1t ) ¡ ¢ For any γ 1t ≥ 1−2η, γ 1t+1 ≥ 1−2η, and less than one, then Fγ 1t γ 1t+1 , γ 1t > 0. Therefore, by the Implicit Function Theorem it is obvious that ¡ ¢ Fγ 1t γ 1t+1 , γ 1t dγ 1t+1 ¡ ¢ >0 =− dγ 1t Fγ 1t+1 γ 1t+1 , γ 1t This ends the proof of this lemma. γ , 1) with γ¯ < 1. The second part shows that this map goes through (¯

Lemma 4 When γ 1t+1 → 1 then γ 1t → γ¯ < 1.

Proof. >From the equation defining the dynamical system on γ 1t take limits on both sides where γ 1t+1 → 1 and γ 1t → γ¯ . We must have that µ ¶ φ (¯ γ + 2η − 1) 1−α − 1 + 1 − 2η (1 − 1) (1 − γ¯)1−α ¶ µ φ (¯ γ + 2η − 1) (1 + 2η − 1) = 1+1− (1 − γ¯)1−α 38

³ Given that the left hand side is zero, this is equivalent to 0 = 2η 2 − and given that η > 0, what this implies is

φ(¯ γ +2η−1) (1−¯ γ )1−α

´

,

φ (¯ γ + 2η − 1) =2 (1 − γ¯)1−α To get this equality, it is necessary that γ¯ > 1 − 2η (which is true since the map is strictly increasing) and that (1 − γ¯)1−α > 0, which implies γ¯ < 1 as desired. Given that the curve γ 1t+1 (γ 1t ) pass through the 45o line only through two points, one at (1 − 2η, 1 − 2η) and another one above this, the two lemmas imply that the curve must cut the 45o line at the second steady state (the stationary equilibrium) from below. This shows that this steady state is equilibrium for this economy, unstable and so, if γ 1 > 0, then ¡it is the unique ¢ since any other combination of γ 1t , γ 1t+1 outside the stationary equilibrium leads to either the point (1 − 2η, 1 − 2η) or to some value greater than one. Neither of the two situations cannot be equilibrium cases. It remains to be shown that γ 1 > 0. Given the last lemma, it is suﬃcient to show that when γ 1t+1 = 0 then γ 1t > 0. To prove this, recall that the system can be reduced to: ¶ µ ¡ 1 ¢ ¡ ¢ ¢1−α φ (γ 1t + 2η − 1) ¡ 1 1 1 − − − F γ t+1 , γ t ≡ γ 1 + 2η 1 γ t+1 t+1 1−α (1 − γ 1t ) ¶1−α µ ¢ ¡ 1 φ (γ 1t + 2η − 1) 1 − − 1 + γ t+1 − + 2η 1 =0 γ t+1 1−α (1 − γ 1t ) £ ¤ We evaluate F at (0, 0) , which gives F (0, 0) = (2η − 1) φ − 1 − (1 − φ (2η − 1))1−α . We know that η > 12 so (2η − 1) > 0. Also we know that F is strictly decreasing in γ 1t+1 and strictly increasing in γ 1t . Therefore, it is suﬃcient to show that F (0, 0) < 0. If this is true, then when γ 1t = 0 then the corresponding value of γ 1t+1 must be strictly negative. But F (0, 0) if and only if φ − 1 < (1 − φ (2η − 1))1−α . To show this we proceed by contradiction. Suppose 1−α 1−α . However, since then that φ − 1 ≥ (1 − φ (2η n − 1))o = (φ + 1 − 2φη) 1 1 1 1 1 φ < η (recalling that min η , 1−η = η for η > 2 ) then φη < 1 and so −2φη > −2, and so 1 − 2φη > −1. Since 1 − α > 0 then (φ + 1 − 2φη)1−α > (φ − 1)1−α . Putting things together we get that φ − 1 ≥ (1 − φ (2η − 1))1−α = (φ + 1 − 2φη)1−α > (φ − 1)1−α . But since 1 − α < 1 then this implies that 39

φ − 1 > 1 or φ > 2. But φ < min

n

1 , 1 η 1−η

o

5 2, a contradiction. Thus φ − 1

< (1 − φ (2η − 1))1−α as desired. Hence F (0, 0) < 0 and so, when γ 1t = 0 then γ 1t+1 < 0. • Case 2: η < 12 .

The proof here follows similar arguments, so we just sketch part of it. First note that γ 1 > 1 − 2η > 0. Then it remains to show that γ 2 > 0 > 2η − 1. The diﬀerence in the procedure is that we will work with γ 2t as the variable instead of γ 1t . Recall that the equilibrium conditions were: ¢ ¡ 1 γ t+1 + γ 2t+1 pt γ 2t + 1 − 2η qt Rt2 = ; q = = t pt+1 (γ 1t + γ 2t ) 1 + qt γ 1t + 2η − 1 and ¡ ¢ ¡ ¢ ¢1−α ¡ 2 ¢1−α ¡ 1 1 − γ 1t γ t + 1 − 2η = 1 − γ 2t γ t + 2η − 1

>From the first three equations we get ¡ 1 ¢ µ ¶ µ 2 ¶µ ¶ γ t+1 + γ 2t+1 φ γ t + 1 − 2η φ qt = = 1−α (γ 1t + γ 2t ) 1 + qt (1 − γ 2t )1−α γ 1t + γ 2t (1 − γ 2t ) Therefore the dynamical system describing the equilibrium (which is equivalent to the one presented at the beginning of this proof) is ¡ 1 ¢ φ (γ 2t + 1 − 2η) γ t+1 + γ 2t+1 = 1−α (1 − γ 2t ) ¢ ¡ ¢ ¡ ¢1−α ¡ 2 ¢1−α ¡ 1 1 − γ 1t γ t + 1 − 2η = 1 − γ 2t γ t + 2η − 1

We proceed as before, reducing this system to a one-dimensional system in γ 2t . From the first equation γ 1t+1 =

φ (γ 2t + 1 − 2η) − γ 2t+1 ; 2 1−α (1 − γ t )

1 − γ 1t+1 = 1 + γ 2t+1 −

40

φ (γ 2t + 1 − 2η) 1−α (1 − γ 2t )

and replacing in the second equation forwarded one period gives: µ ¶1−α ¢ ¡ 2 φ (γ 2t + 1 − 2η) 2 − 1 + γ t+1 − + 1 2η γ t+1 1−α (1 − γ 2t ) µ ¶ ¡ ¢1−α φ (γ 2t + 1 − 2η) 2 2 − γ t+1 + 2η − 1 = 1 − γ t+1 1−α (1 − γ 2t )

We then define

¡ ¢ G γ 2t , γ 2t+1 µ ¶ ¡ ¢1−α φ (γ 2t + 1 − 2η) 2 2 ≡ 1 − γ t+1 − γ t+1 + 2η − 1 1−α (1 − γ 2t ) ¶1−α µ ¡ 2 ¢ φ (γ 2t + 1 − 2η) 2 − 1 + γ t+1 − γ t+1 + 1 − 2η 2 1−α (1 − γ t ) ¢ ¡ Therefore an equilibrium path is characterized by G γ 2t , γ 2t+1 = 0, which implicitly defines a function γ 2t+1 (γ 2t ) provided that the conditions for the Implicit Function Theorem hold. Following ¡ 2identical ¢ arguments ¡as2 in 2the¢ 2 lemma before, it can be shown that Gγ 2t+1 γ t , γ t+1 < 0 and Gγ 2t γ t , γ t+1 > 0 for all γ 2t and γ 2t+1 less than one and strictly greater than 2η − 1. Hence dγ 2

γ 2t+1 (γ 2t ) is well defined and dγt+1 > 0. In a similar fashion as in case 1, it 2 t γ < 1, which implies that the is easy to prove that if γ 2t+1 → 1 then γ 2t → b steady state γ 2 > 2η − 1 is locally unstable. This concludes the proof of the proposition.¥

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42