Borrowing and Lending without Commitment and with Finite Life Luisa Lambertini
Department of Economics University of California, Los Angeles email:
[email protected]
Preliminary and Incomplete: December 1998
Abstract
This paper studies borrowing and lending in an overlapping generation model of threeperiod lived individuals who cannot commit to repay their debts and where loan contracts are imperfectly enforceable. The contribution of the paper is to draw a mapping between the income pro le over the life cycle and borrowing in an environment without commitment and to quantify such relationship with numerical simulations. One of the results of the paper is that, if the income pro le is humpshaped, individuals are borrowing constrained early in life and consumption smoothing fails. This model therefore generates endogenous debt limits for young individuals and this is consistent with recent empirical ndings. The equilibria, the dynamics and the eciency of economies where individuals can commit to repay their debts and of economies where they cannot are characterized. It is found that, when commitment is feasible, the equilibrium is unique and the convergence toward it is non monotone; when commitment is not feasible, multiple equilibria may arise and, if the equilibrium is unique, it is one where borrowing and lending are zero. JEL Classi cation Number:
1
1 Introduction Models of borrowing and lending often assume that loan contracts can be enforced or, alternatively, that borrowers can commit to repay their loans. A vast evidence, however, suggests that loan contract enforcement problems should not be ignored. In fact, a large fraction of loans is supported by substantial collateral, which can be appropriated by lenders in the event of a default; consumption and income are positively correlated for individuals, which is evidence that consumption risk is imperfectly diversi ed across agents; and individuals default on their loans. The implications of imperfectly enforceable loan contracts and of the inability to commit to repay are important: borrowing is limited and risk sharing is imperfect. As a result, agents cannot achieve the desired degree of consumption smoothing over time or states. This paper studies borrowing and lending in a pure exchange overlapping generation model of threeperiod lived individuals who cannot commit to repay their debts and where loan contracts are imperfectly enforceable. In such environment, borrowing is endogenously limited because lenders rationally anticipate the borrower's incentive to default and lend only up to the point where the borrower still has the incentive to repay. Unlike earlier contributions that are based on the in nitelived representative agent model, this paper assumes an OLG structure where individuals live for three periods. A model where individuals live for two periods is not adequate to study borrowing and lending for the reason that any two consecutive generations would overlap for only one period and this is not enough time to engage in a borrowinglending relationship. A life of at least three periods is therefore the minimum necessary to address the issue of borrowing and lending and I have chosen a threeperiod model to keep the analysis as simple as possible. The contribution of this paper is to draw a mapping between the income pro le over the life cycle and the ability of an individual to borrow in an enviroment where commitment is not feasible. More precisely, this paper investigates how borrowing limits are aected by the shape of the income pro le over the life cycle and the other parameters of the model, such as the elasticity of intertemporal substitution and the discount factor. Several numerical simulations quantify such eects. One of the results of the paper is that, if the income pro le is humpshaped (as it is in reality), individuals are borrowing constrained early in life and consumption smoothing fails. This model therefore generates endogenous debt limits for young individuals; this result is consistent with several recent empirical ndings, as in Carroll and Summers [4] and Gourinchas and Parker [8], which have found evidence that consumption tracks income especially for younger individuals and for individuals with low assets, which by far and large, are the younger ones. The rst part of the paper characterizes the equilibria, the dynamics and the e2
ciency of economies where individuals can commit to repay their debts and where they cannot. The paper nds a number of technical results that can be summarized as follows. When commitment is feasible, there is a unique equilibrium and the convergence toward it is non monotonic. When commitment is not feasible, multiple equilibria may arise; when the equilibrium is unique, it is one where borrowing and lending are zero and individuals consume their endowments. Since this is an OLG economy, the steady state interest rate may be above or below its golden rule level and, as a consequence, an equilibrium may be Pareto inecient. I measure the eciency of an allocation by using Debreu's coecient of resource utilization; this allows me to Pareto rank the equilibria and to measure the welfare loss caused by inability to commit and/or the imperfect enforceability of loan contracts. It is found that the equilibria with commitment, even if Pareto inecient, are generally Pareto superior to their counterpart without commitment; several numerical simulations indicate that the welfare loss caused by the inability to commit can be fairly large. The second part of the paper uses numerical simulations to investigate how changes in the parameters of the model, such as the shape of the income pro le over the life cycle, the elasticity of intertemporal substitution and the discount factor, aect the incentive to default and the equilibria and to quantify such eects. More precisely, I study for what parameter space individuals have no incentive to default, so that the equilibria with and without commitment coincide. Humpshaped endowment pro les, low intertemporal elasticity of substitution in consumption, high discount factor and the ability to con scate endowments following a default eliminate the incentive to default. Several other papers have looked at environments without commitment. Eaton and Gersovitz [7] were the rst to use such concept: in the context of sovereign debt, they consider a model in which the threat of exclusion from the international capital market provides a borrowing country with an incentive to repay. Within the same context, Bulow and Rogo [3] explicitly study the role of direct sanctions. In the context of individual insurance, Thomas and Worral [13] characterize ecient allocations when one agent is riskaverse while the other is riskneutral and Taub [12]and Coate and Ravaillon [5] characterize the conditions under which rstbest allocations are subgame perfect. Kocherlakota [10] studies ecient allocations in a symmetric information environment without commitment and emphasizes the dierence between such allocations and ecient allocations with asymmetric information and full commitment. Kehoe and Levine [9] develop a general equilibrium theory of asset trades and prices in the absence of commitment and Alvarez and Jermann [1] study asset pricing and the consequences for the equity premium in such context. Finally, Levine and Zame [11] study whether market incompleteness in the form of lack of commitment has substantial eects on social welfare. All these contributions are based on models where agents are in nitely 3
lived. The theory underlying this paper is related to the contributions above; this paper, however, is dierent from those above because it is based on an OLG model with nitely lived individualsand OLG economies are very dierent from representative agent economies, both when commitment is feasible and when it is not, but especially because its goal is to characterize endogenous debt limits over the life cycle. The paper is organized as follows: Section 2 introduces the model and section 3 characterizes the equilibria with and without commitment. Some numerical simulations are presented in 4. Section 5 studies and quanti es the eect of changes in the parameters on the incentive to default and on the equilibria and section 6 investigates how the ability to con scate the defaulter's endowments aect the incentive to default. Section 7 concludes.
2 The model Consider a pureexchange threeperiod overlapping generations closedeconomy model. For simplicity, the model abstracts from uncertainty and production. Individuals live for three periods: they are young in the rst period of life, middleaged in the second period of life, and old in the last period of life; the generation born at time t will be referred to as generation t; I assume that the generation t is of size Nt = (1 + n)t ; t = 0; 1; : : : : (1) Population grows if n is positive and shrinks if n is negative. There is one consumption good that cannot be stored. For simplicity, all individuals belonging to the same generation have the same endowment. Let superscripts refer to generation and subscripts refer to time. A generation t individual is endowed with an amount ytt 0 of the consumption good when young, ytt+1 0 when middleaged and ytt+2 0 when old. The endowment stream of a generation t individual can be written as the vector yt = fytt; ytt+1; ytt+2g 2 R3+: (2) Individuals' endowment vectors are common knowledge. Generation t total endowment at t + s is Ytt+s = Nt ytt+s; s = 0; 1; 2; (3) and the economy's total endowment at time t is the sum of the three alive generations' total endowments: Yt = Ytt + Ytt,1 + Ytt,2 : (4) A generation t individual maximizes the following lifetime utility function when young Vtt =
2
X
s=0
su(ctt+s );
4
> 0;
(5)
where c is private consumption, which I restrict to be non negative, and is the subjective discount factor. The function u : R+ ! R is strictly increasing, concave in its argument and dierentiable. Lifetime utility is maximized subject to the constraint ctt+2 ct ctt + t+1 + Rt+1 Rt+2 Rt+1
t
t
ytt + Ryt+1 + R yt+2 t+1 t+2 Rt+1
(6)
or, equivalently, the set of oneperiod constraints btt = 0; (7) ytt = ctt + btt+1 ; (8) ytt+1 + Rt+1 btt+1 = ctt+1 + btt+2 ; (9) ytt+2 + Rt+2 btt+2 = ctt+2 ; (10) where btt+1 is private lending at t (borrowing if negative) of a generation t individual, exchanging one unit of consumption at t for Rt+1 units of consumption at t + 1. The gross yield R (1 + r); where r is the interest rate, is taken as given by each individual. Constraint (8), (9) and (10) are, respectively, the budget constraint when young, middleaged and old. I restrict my attention to the case where borrowing is individually feasible, namely it satis es (8) to (10). In words, an individual cannot borrow more than he can actually repay.
3 The economic equilibrium
3.1 The equilibrium with commitment
Consider rst the economic equilibrium when commitment is feasible, namely when an individual can commit to repay his debts. In this economy, individuals never default on their contracts.1 The borrowing decisions of a generation t individual are the solution to the following problem (C )
n
btt+1 ; btt+2
o
= arg max
2
X
s=0
s u(ctt+s)
(11)
subject to (7), (8), (9) and (10) and nonnegative consumption; I label this problem as (C), which stands for \commitment". The rstorder conditions with respect to btt+1 and btt+2 are, respectively u0 (ctt ) = Rt+1 u0 (ctt+1 ); (12) 1 Commitment can be seen alternatively as the equilibrium outcome if an in nitely large punishment
can be in icted upon defaulting individuals.
5
u0 (ctt+1 ) = Rt+2 u0 (ctt+2 );
(13) which link the marginal utilities of consumption when young and when old as follows u0 (ctt ) = 2 Rt+1 Rt+2 u0 (ctt+2 ):
(14)
In equilibrium, borrowing or lending equalizes the marginal utility of current consumption to the present discounted value of future consumption. If the period utility function is isoelastic, namely 1
(ct )1, u(ctt+s) = t+s 1 ; 1,
> 0;
(15)
where is the intertemporal rate of substitution in consumption, the optimal lending decision for a generation t individual when young is: ytt ( Rt+1 ) [Rt+2 + ( Rt+2 ) ] , ytt+1 Rt+2 , ytt+2 btt+1 = (16) [Rt+1 + ( Rt+1 ) ] [Rt+2 + ( Rt+2 ) ] , Rt+1 ( Rt+2) and the optimal lending decision when middleaged is:
( Rt+1 ) ( Rt+2 ) yttRt+1 + ytt+1 , ytt+2 [Rt+1 + ( Rt+1) ] t bt+2 = (17) [Rt+1 + ( Rt+1 ) ] [Rt+2 + ( Rt+2 ) ] , Rt+1 ( Rt+2 ) : Despite looking complicated, the expressions above permit some simple inferences. Higher income in youth, ytt, raises both lending when young and when middle aged; Higher income when middle aged, ytt+1 , reduces lending in youth but raises lending in mid life; Higher income when old, ytt+2 , lowers both lending when young and when middleaged. These eects are consistent with the simple principle  a direct consequence of consumption smoothing  that any increase in future income has a negative eect on current lending and any increase in past and current income has a positive eect on current lending. The optimal lending decisions for a generation t individual also depend on the interest rates Rt+1 and Rt+2 , which are the relevant equilibrium rates over the individual's life cycle. I will say more about this when I analyze the dynamics of the model. The optimal consumption decisions of a generation t individual when young, middle age and old are given by ytt Rt+1 Rt+2 + ytt+1 Rt+2 + ytt+2 ctt = (18) [Rt+1 + ( Rt+1 ) ] [Rt+2 + ( Rt+2) ] , Rt+1 ( Rt+2 ) ; (yttRt+1 Rt+2 + ytt+1 Rt+2 + ytt+2)( Rt+1 ) ctt+1 = (19) [Rt+1 + ( Rt+1 ) ] [Rt+2 + ( Rt+2) ] , Rt+1 ( Rt+2) ; 6
( yttRt+1 Rt+2 + ytt+1 Rt+2 + ytt+2 )( Rt+1 ) ( Rt+2 ) (20) = [R + ( R ) ] [R + ( R ) ] , R ( R ) ; t+1 t+1 t+2 t+2 t+1 t+2 An increase in endowment, no matter at what age, raises consumption in every period of life; the elasticities of consumption when young, middle age and old with respect to income in a certain period of life are the same. The economic equilibrium at t is the sequence of interest rates fRt+sg1 s=1 that implies zero net supply of lending contracts in each period, starting from t. Let net aggregate lending at time t be bt ; zero net supply of lending contracts at time t is de ned as ctt+2
bt+1 (Rt ; Rt+1 ; Rt+2 ) Nt btt+1 (Rt+1 ; Rt+2 ) + Nt,1 btt,1 +1 (Rt ; Rt+1 ) = 0:
(21)
Every period, the young and the middle aged enter the lending market with an optimal lending decision, which depends on past, current and future interest rates, as the notation in (21) shows. The equilibrium interest rate is the one that clears the market from today on. To characterize the steady state, the optimal lending decisions for the young and the middle aged need to be analyzed in more detail. Figure 1 shows the optimal consumption and lending decision for a young individual at time t. The horizontal axis reports consumption in young age, ctt; the vertical axis reports the present value of middleage and old consumption, Ctt+1 ctt+1 + ctt+2 =Rt+2. The indierence curve shows the combinations of (ctt ; Ctt+1) that give the same utility; the budget line shows the combinations of (ctt; Ctt+1 ) that are feasible and its slope is ,Rt+1 . Suppose the individual endowment point is E = (ytt; ytt+1 + ytt+2=Rt+2 ). With interest rate R00 , the young individual is happy to be autarkic, he lends by = 0 and simply consumes his endowment. For any interest rate lower than R00, the young individual's preferred consumption bundle shifts to the right, say to point D; his consumption raises above his endowment, so that he must borrow, by < 0; vice versa, for any interest rate higher than R00 , the optimal consumption bundle shifts to the left and the young individual lends, by > 0. The optimal lending decision for a middleaged individual at t is qualitatively similar; the relevant endowment point is shifted to the left (right) by the amount borrowed (lent) in youth, multiplied by the interest factor. There is an interest rate, R0 , at which the middle aged individual chooses to be autarkic and bm = 0; for R < R0 , the middleaged individual borrows, bm < 0, and for R > R0 he lends and bm > 0. Figure 2 shows the optimal lending schedules when commitment is feasible for the young and the middle age individual in the steady state, where Rt+s = R; 8s 1; to more easily identify the equilibrium, the negative of the young's lending schedule, ,by , is plotted. The formal proofs of the following results are detailed in appendix A. Lending by the young, by (R), is continuous and increasing in R; it intersects the horizontal axis at R00, is negative for R < R00 and positive for R > R00; lending by the 7
t
Ct+1
B t ytt+yt+1 /Rt+2
E D
R’’ ( ytt
t
ct
Figure 1: Optimal youth lending with commitment; Ctt+1 ctt+1 + ctt+2 =R. middle aged, bm (R), is also continuous and increasing in R; it intersects the horizontal axis at R0, is negative for R < R0 and positive for R > R0. Proposition 1 The number of equilibria is always odd. The equilibrium is unique if consumption of each dated good are strict gross substitutes. A sucient condition for strict gross substitutability is that < 1. Proof: see appendix A Aggregate lending in the steady state is b(R) = by (R) + bm (R), which is a continuous and increasing function of R; since b(R) < 0 for R < minfR0 ; R00g and b(R) > 0 for R > maxfR0; R00g, the number of equilibria is odd. If consumption when young, middle age and old are strict gross substitutes, the equilibria is certainly unique. When consumption of each dated good is an increasing function of the price of the others, the optimal lending decisions by young and middle age individuals are increasing in R, so is aggregate lending. A sucient condition for strict gross substitutability of consumption is that < 1. Intuitively, when the intertemporal elasticity of substitution in consumption is suciently high, the individual is willing to transfer consumption from one period to the next in response to a price change. Notice that < 1 is a sucient condition for strict gross substitutability; the equilibrium may well be unique for > 1. In gure 2, there is a unique equilibrium that occurs at point A. Aggregate lending at t depends on Rt ; Rt+1 and Rt+2 , as (21) shows. Therefore, the dynamics of the economy is characterized by a secondorder dierence equation. If consumption when young, middle age and old are strict gross substitutes, one eigen8

by bm A
R’
R’’
Figure 2: Lending schedules with commitment value is negative and stable and the other is unstable and the economy is characterized by non monotonic convergence to the equilibrium.2
3.2 The equilibrium without commitment
Suppose now that individuals cannot commit to repay their loans. After explicitly modeling the punishment, this section solves for the equilibrium. Default is often modeled in a setting with uncertainty: one borrows to invest in a risky project and defaults if the project realization is bad. In this model, default occurs when an individual borrows in excess of what he has an incentive to repay, given the punishment in icted upon default. In a sense, default is the subgame perfect action of a borrower that has borrowed \too much"; a rational lender with symmetric information, however, anticipates the borrower's incentive to default and lends only up to the point where the borrower will repay. As a result, debt limits arise endogenously. As in Kehoe and Levine [9], initially I assume that private endowments cannot be physically dissociated from individuals and therefore cannot be seized in the event of a default; this assumption is relaxed in section 6, where I generalize the results of the paper to the case where private endowments can be partially con scated following a default. The assumption that endowments, such as labor and human capital, can be partially con scated following a default is justi ed by the fact that their supply would be strongly discouraged. More precisely, if wages above a certain minimum are 2 For a formal proof of this result, see Azariadis et al.[2].
9
seized for repayment, the individual has no incentive to supply labor in excess of that minimum. On the other hand, assets, such as stocks and capital, can easily change ownership and can be seized in case of default. For the moment, the only punishment in icted upon defaulters is that they cannot access the intertemporal market after the default. Intertemporal contracts, such as borrowing and lending, require that the two involved parties identify themselves; since this makes it possible to con scate the proceeds of the contracts, it is the subgame perfect decision for defaulting individuals not to enter the intertemporal market after their default. The individual endowments fytt; ytt+1; ytt+2g should be thought of as income from wages and, for the moment, I am going to assume that they cannot be con scated in the event of a default. Since there are no assets in this model, the cost of default is the loss of the welfare gains from intertemporal trade. Since the lender anticipates the borrower's incentive to default, he will only lend up to the where the borrower still has the incentive to repay; as a result, borrowing is constrained. More precisely, the optimal lending decision for a generation t individual when commitment to repay is not feasible is the solution to the following problem: (NC )
n
o
btt+1 ; btt+2 = arg max
2
X
s=0
s u(ctt+s);
(22)
where NC stands for "no commitment", subject to (8) to (10) and the individually rational (IR) constraints: u(ctt+1 ) + u(ctt+2 ) u(ytt+1 ) + u(ytt+2) (23) and u(ctt+2 ) u(ytt+2 ): (24) Constraint (24) simply says the generation t individual prefers to honor his borrowing contract when old rather than to default on it. Since agents default only on their borrowing, this constraint is binding, if any, when income is Ushaped over the life cycle; from (10) it is easy to see that (23) implies btt+2 0, i.e. an individual cannot borrow at all when middle aged. The intuition is simple: old individuals have no incentive to repay any debt because they will not be around the following period and therefore cannot be punished for defaulting. Similarly, (23) says that the generation t individual is better o honoring his borrowing contract when middle aged and old rather than defaulting and being autarkic. Constraint (23) is binding, if any, when income is humpshaped over the life cycle; if binding, it eectively puts a ceiling on youth borrowing. Formally, the rst order conditions with respect to btt+1 and btt+2 in the maximization problem (NC) are (25) ,u0(ctt) + Rt+1 u0(ctt+1 ) + 1Rt+1 u0(ctt+1) = 0; 10
, u0(ctt+1 ) + Rt+2u0(ctt+2 ) , 1u0(ctt+1) + (1 + 2)Rt+2 u0(ctt+2 ) = 0
(26) where 1 is the multiplier on constraint (23) and 2 is the multiplier on constraint (24). Four cases can arise. Case 1: both multipliers are equal to zero and the constraints (23) and (24) are not binding; then, the solution to the problem (NC) is identical to the solution to the problem (C) of section 3.1 and lending is given by (16) and (17). Case 2: 1 > 0 but 2 = 0. Here, constraint (23) is binding but (24) is not; the rst order condition with respect to btt+2 is (13) but the rst order condition with respect to btt+1 is u0 (ctt ) = ( + 1 )Rt+1 u0 (ctt+1 ): Since 1 > 0, consumption when young is lower than it is in problem (C). In words, individuals are borrowing constrained when young and their consumption and income pro les are both rising. For the CES case, lending when young and middle aged is, respectively, btt+1 =
8 >
1 t t 4 ; , R y , y + t +2 1 t +1 t +2 > Rt+2 Rt+1 > ; : [1 + Rt+2 ( Rt+2 ), ]
btt+2 =
8 >
: t+1 > Rt+2 > [1 + Rt+2 ( Rt+2 ), ] ; :
(28)
This notation will be useful later on in the paper. Let S be the space of all possible values of fytt; ytt+1; ytt+2; ; g and let S1 S be the set of values for which (23) is binding. Case 3: 2 > 0 but 1 = 0, namely constraint (24) is binding but (23) is not. In this case, the rst order condition (25) is given by (12), but the rst order condition (7) becomes u0(ctt+1 ) = Rt+2 u0 (ctt+2 )(1 + 22 ):
Since 2 > 0, consumption when old is lower than under problem (C) (see the rst order condition (13)). The lender's anticipation that no old individual will ever repay his debts prevents any borrowing by middle age individuals. Agents would like to raise their middleage consumption when income is Ushaped over life, but their inability to commit to repay prevents borrowing in middle age. For the CES case, the solution is y t ( R ) , ytt+1 btt+1 = t t+1 ; Rt+1 + ( Rt+1 )
(29)
btt+2 = 0:
(30)
11
Let S2 S be the set of values for which (24) is binding. Case 4: both IR constraints are binding; then the solution to problem (NC) is given by (25) and (26), where the latter can be rewritten as u0 (ctt+1 ) = Rt+2 u0(ctt+2 )(1 + 2 2 ): + 1
Hence, both consumption when young, ctt, and when middle aged, ctt+1 , are lower than under problem (C). For any period utility function and therefore for the CES case, btt+1 = 0
(31)
btt+2 = 0:
(32)
In words, autarky is the unique solution if both constraints are binding. Figure 3 gives a graphical intuition for the constraint (23), i.e. for the case where income is humpshaped over the life cycle, as it is in reality. The horizontal axis depicts consumption when middleaged and the vertical axis depicts consumption when old for a generation t individual; suppose the endowment point is E and suppose that unconstrained borrowing when young is btt+1 . After repaying his debt, the individual is at point C and, by lending btt+2 , he achieves the consumption allocation at A. This consumption allocation is feasible only if commitment to repay is feasible. If commitment is unfeasible, any borrowing requiring a gross repayment exceeding R~t+1~btt+1 the thick horizontal line connecting point E and Bis defaulted on. The IR constraint (23) restricts borrowing in youth not to exceed ~btt+1 , so that the endowment point after repaying the borrowing done in youth is B. By lending ~btt+2 , the agent achieves the constrained consumption bundle D.3 Notice that ~btt+2 btt+2 : since the individual can borrow less, his middle age income after debt repayment is higher. The economic equilibrium at time t is the sequence of interest rates that implies a zero net aggregate supply of contracts starting from t, as de ned in (21). Now I study the constrained lending decisions in the steady state. By constrained lending (and/or borrowing), I refer to the lending decision when one or both the IR constraints are binding; the results reported here are for CES utility. Of the four possible cases, case 1 has been studied in section 3.1 and case 4 is trivial. Case 3 is straightforward. Proposition 2 When the endowment pro le is Ushaped over the life cycle, there is a unique constrained equilibrium in which borrowing and lending are zero. 3 Despite the fact that point D is on a higher indierence curve than A, the individual's lifetime
utility is higher in the unconstrained equilibrium A because unconstrained borrowing allows a smoother time pro le of consumption.
12
c t+2
~ ~
D
R t+1b t+1
A
^
c t+2 y
t+2
C
E
B
~ b t+2
b t+2
{
{ c t+1 ^ y
+R
t+1
^
b
t+1
t+1
y y
c t+1
t+1
~
~
+ R t+1b t+1
t+1
Figure 3: Consumption when middle aged and old without commitment Proof: see appendix B The unique constrained equilibrium under case 3 is one where individuals are autarkic. Middle age individuals would like to borrow but cannot do so when commitment is not feasible because lenders anticipate their incentive to default. Lenders therefore cannot nd borrowers to do business with and the equilibrium is therefore one where autarky prevails. Now consider case 2, namely the case where income is humpshaped over the life cycle and the IR constraint (23) is binding. The results are summarized in the following proposition. Proposition 3 If the endowment pro le is humpshaped over the life cycle, there is at least one constrained equilibrium and the number of equilibria is odd, except in the degenerate case where two equilibria coincide. The equilibrium, when unique, is one where borrowing and lending are zero. In the multiple equilibria case, there is at least one stable equilibrium with borrowing and lending. Proof: see appendix C Two scenarios may arise in this economy. In the rst scenario, there is a unique equilibrium where the credit market is closed. In the second scenario, there are multiple equilibria. I could never generate more than three equilibria in my numerical simulations, but I could not prove that there can be at most three equilibria. When there are three equilibria, only one of them is stable and there is active borrowing and lending 13
by by bm B A
C bm
A
~
~
R’
R’
Figure 4: Lending without commitment: unique and multiple equilibria in it. Figure 4 shows (negative) lending by the young (27), labeled ,by , and lending by the middle aged (28), labeled as bm ; the case where there is a unique equilibrium is depicted on the left and the case with three equilibria is depicted on the right. All formal proofs are in the appendix. In this economy, there is always at least one equilibrium and this is equilibrium A in gure 4. This is an autarkic equilibrium that occurs at point E (the endowment point) of gure 3. The interest rate R~ 0, namely the negative of the slope of the budget line tangent to the indierence curve at E, is so low that the individual is perfectly happy to consume his endowment. As the interest rate rises, the middle age individual wants to lend some of his current endowment, which explains why bm is positive and increasing for interest factors above, but not too much, R~0 ; the young individual borrows but only up to the point where he will not default, i.e. up to the horizontal distance between E and the budget line tangent to the isoquant thru E in gure 3, and this explains why ,by is also positive and increasing for interest rates above, but close enough, to R~0 . For interest rates well above R~ 0, youth borrowing begins to fall and approaches zero in the limit. The intuition is simple. The interest payment becomes a large fraction of total debt repayment, driving borrowing to zero as the interest rate rises, and total debt repayment is limited by the incentive to default (again, the thick line in gure 3 is bounded from above). For interest rates well above R~ 0, middle age lending begins falling and approaches zero in the limit, too. This happens because total debt repayment grows with higher interest rates, thereby driving the endowmentafterrepayment and lending down. Both ,by and bm approach zero as the interest rate goes to in nity but, since the former goes to zero faster than the latter, there are: 1) either no equilibria other than A (left panel); 2) or two equilibria, B and C, in addition to A (right panel). Notice that B and C are equilibria where there is active borrowing and lending. The condition under which multiple equilibria arise is that ,~by (R) , ~bm (R) > 0 14
for R > R~ 0; in words, youth borrowing must increase more than middle age lending as the interest rate rises above R~ 0.4 This condition is rather messy and cannot be solve analytically; the following fact, however, was found with the help of numerical simulations. Fact 1 If the endowment pro le is humpshaped over the life cycle, multiple equilibria arise as becomes small. Intuitively, total debt repayment consistent with no default rises fast if the indierence curves are more sharply curved, as it is the case with low intertemporal rates of substitution. In a model with in nitely lived agents, the equilibrium interest rate is lower in an economy where commitment is not feasible than in one where commitment is feasible, everything else being equal. The comparison of equilibrium interest rates is more complicated in this setting because multiple equilibria can arise. Nevertheless, it is possible to compare the unique equilibrium interest rate in the economy with commitment with the autarkic equilibrium in the economy without commitment, i.e. the equilibrium that occurs at R = R~ 0 . Proposition 4 Let Rc be the unique equilibrium interest rate in the economy with commitment; let R~ 0 be the interest rate in the autarkic equilibrium in the economy without commitment. Then, Rc > R~ 0 if ytt=ytt+1 < 1 and ytt+2 =ytt+1 < 1, i.e. if the endowment pattern is humpshaped over the life cycle. Proof: see appendix D. This proposition is quite intuitive. The interest rate must be low for a middleaged individual to be happy in autarky (see again gure 3) when his income pro le is humpshaped. The dynamics of the IR constrained economy for case 2 and 3 is characterized by a rstorder dierence equation. Intuitively, one of the two lending decisions is a corner solution and therefore does not depend on any interest rate. The unique autarkic equilibrium is always unstable. When multiple equilibria arise, the odd equilibria are unstable and the even equilibrium is stable. Figure 5 shows the phase diagram for the same economies without commitment depicted in gure 4. The phase diagram for the economy with the unique equilibrium A is depicted on the left; notice that the equilibrium A is unstable because the curve5 is atter than the 45degree line when it cuts it. On the right, the phase diagram for the economy with multiple equilibria is shown: equilibria A and C are unstable and equilibrium B is stable. 4 The curve , y is atter than the curve m around ~ , which ensures that equilibrium B exists if , y rises above m. See appendix C. b
b
b
R
b
0
5 Notice that the curve is drawn in the (Rt+1 ; Rt ) space.
15
C
A
B AA
Figure 5: Phase diagrams for economies without commitment: unique and multiple equilibria
4 Simulations This section presents a quantitative assessment of some economies with and without commitment using numerical simulations. I consider the economy described in section 2 where individuals have the isoelastic period utility function of (15), population is constant, i.e. n = 0, and the size of each generation is normalized to one; there are no outside assets. For convenience, I am going to label the endowment vector of the typical individual as y = fy0; y1; y2g: The length of each period is set to 20 years in the simulation; with a superscript a I refer to annual values of parameters and/or variables: annual consumption, for example, is period consumption divided by 20, a = 1=20 , Ra = R1=20 and the annual endowment stream is humpshaped over the life cycle: (y0a; y1a; y2a) = (1; 2; 1). Consider rst the case where = 1, i.e. the period utility is logarithmic, and the discount rate is = 1, whose results are presented in table 1. Consider rst the economy with commitment, which is reported on the left of the table. The rst row reports the gross annual interest rate, Ra , which is equal to 1, i.e. the net interest rate is 0. This result is due to the symmetry of the endowment pattern over the life cycle: an endowment patter with higher income in youth than in old age generates a lower equilibrium interest rate and vice versa if the endowment is higher in old rather than in young age. In the economy 16
= 1, a =1, y0a=1, y1a=2, y2a =1
Variable With commitment Without commitment unstable Ra 1 0.966 t bt+1 6.6667 0 t bt+2 6.6667 0 btt+2 =Yt 8.3% 0% ca0 1.3333 1 ca1 1.3333 2 a c2 1.3333 1 t Vt 9.85024 9.68034 0.21 1.4 1015
100 0 5.8 default yes
Table 1: Equilibria with and without commitment of table 1, the typical individual borrows when young and lends when middleaged; the second row reports youth lending btt+1 and the third row reports middleage lending btt+2 . The fourth row shows aggregate lending as a percentage of the economy's current aggregate endowment, which amounts to about 7 percent in our economy. Rows ve to seven show annual consumption, which is completely at over the life cycle. The third to the last row reports lifetime utility; the parameter is the eigenvalue of the dynamic system calculated around the equilibrium and it tells how the economy approaches its steady state: if negative, the economy converges cyclically; if positive, it converges monotonically. For the economy with commitment, the eigenvalue reported is the stable one (there is second but unstable eigenvalue, not reported here); the value of = ,0:21 indicates that the economy converges to its steady state non monotonically. The parameter is Debreu's coecient of resource utilization and it is used to compare welfare in the current equilibrium against welfare in the Pareto optimal equilibrium (the golden rule) of an economy where individuals have the same total endowment over their life. More precisely, OLG endowment economies can be Pareto inecient if the endowment pro le is not symmetric so that the equilibrium interest rate is above or below its golden rule level.6 The economy studied in table 1 is one where the golden rule interest rate level is simply one; if the equilibrium interest rate is dierent from one, consumption cannot be at over life and welfare is lower. Any allocation with a gross interest rate dierent from one is Pareto inecient in the sense 6 The golden rule level of the interest rate is, as in an OLG economy with capital, one plus the rate
of growth of the population plus the rate of growth of the economy.
17
that a central planner who maximizes an intertemporal social welfare function that weights equally the utility of current and future generations can make everyone better o by redistributing individuals' endowments in an actuarially fair way, i.e. leaving an individual's total endowment over the lifecycle unchanged, so as to bring the equilibrium gross interest rate to one. The parameter measures by how much individual total life endowment at the golden rule equilibrium must be scaled down to obtain the same utility as under the current economy. More precisely, let V^ be the utility under the current economy and let V (y) be the utility under the golden rule economy with the endowment vector y. The parameter is such that V^ = V (y), where each endowment is given by yj = (1 , )yj ; j = 0; 1; 2. Notice that the parameter measures the welfare dierence between two equilibria, but it does not take into account the transition from one equilibrium to the other. The economy considered in table 1 is Pareto ecient and = 0. Suppose now that commitment to repay debts is not feasible. The allocation described on the left column of table 1 is not individually rational because individuals would default on their borrowingas the last row of the table points out. The equilibrium of the economy without commitment is described on the right column. The equilibrium is unique and unstable  it is point A in gure 5a  which implies that individuals are autarkic and the credit market is not active. Intuitively, borrowing that is not defaulted upon cannot be sustained here because autarky is not costly enough for individuals. A lower elasticity of intertemporal substitution as well as a lower endowment in old age, for example, would make default more costly. Since individuals cannot borrow, consumption equals endowment; the lack of consumption smoothing generates a loss in social welfare with respect to the Pareto ecient allocation of the economy with commitment equivalent to a 5.8% reduction in aggregate income. In other words, the social welfare costs of endogenous market incompleteness is about 5.8% of aggregate income. This is a rather large number and it is not a realistic quantitative assessment of the cost of market incompleteness because of the strong and simplifying assumptions made here strictly on tractability grounds. A longer life and the possibility to at least partially con scate the endowment of a defaulting individual contribute to make some borrowing without commitment sustainable. The presence of assets, such as capital, may also help to make borrowing without commitment feasible provided individuals live more than three periods; however, assets will not reduce the borrowing limits in the rst period of life unless an agent is born with them (i.e. inherits them). Table 2 summarizes the simulation for an economy identical to that of table 1 except that individuals now have a lower intertemporal elasticity of substitution than before, i.e. now = 0:5. Individuals do not default on their youth borrowing and the equilibrium without commitment is the same as in an economy where commitment is feasible. Due to the symmetry of the endowment pro le, the economy is Pareto ecient. 18
= 0.5, a =1, y0a =1,y1a=2, y2a =1
Variable With and without commitment Ra 1 btt+1 6.66667 btt+2 6.66667 t bt+2 =Yt 8.3% ca0 1.33333 ca1 1.33333 ca2 1.33333 t Vt 0.1125 0.146
100 0 default no
Table 2: Equilibria with and without commitment, =0.5 At last, table 3 shows the numerical simulation for an economy characterized by multiple equilibria when commitment is not feasible. In this economy, individuals are impatient (the period discount factor, , is equal to 0.3), the elasticity of intertemporal substitution is 0.5 and the endowment pro le is strongly humpshaped and skewed toward the rst period of life. Since individuals want to consume most of their endowment early on in life, the equilibrium interest rate in the economy with commitment is well above one; the consumption pro le is rising over the life cycle because R is also above one in equilibrium. The economy with commitment is Pareto inecient: a change in the endowment pro le that leaves total endowment constant but brings the interest rate down to one by raising youth and lowering middle age endowment improves social welfare by almost 50%. The equilibrium in the economy with commitment is not individually rational and individuals default on their borrowing. The economy without commitment has three equilibria, which correspond to the equilibria A, B and C of gure 4; equilibria A and C are unstable and equilibrium B is stable, as shown by the value of their eigenvalue, . Individuals are in autarky at equilibrium A and the welfare loss of the complete inability to access the credit market is very high, about 70% of aggregate endowment. Equilibria B and C are equivalent in terms of utility; since the interest rate is higher in C than in B, consumption when middle age is lower in C than in B and consumption when old is higher in C than in B. The stable equilibrium B in the economy without commitment is Pareto superior to the equilibrium in the economy with commitment. In other words, in this setting an IR constrained equilibrium may be welfare superior to an unconstrained equilibrium. In an in nitely lived representative agent model, the equilibrium when commitment is 19
= 0.5, a =0.94, y0a =1,y1a =3, y2a =0.8
Variable With commitment Ra btt+1 btt+2 t bt+2 =Yt ca0 ca1 ca2 Vtt
100
default
1.089 4.412 4.412 4.6% 1.221 1.567 2.012 0.05277 0.331 48.3 yes
Without commitment unstable (A) stable (B) unstable (C) 0.931 0.98 1.023 0 7.529 7.529 7.529 7.529 0% 7.84% 7.84% 1 1.376 1.376 3 2.387 2.025 0.8 1.037 1.398 0.06063 0.04695 0.04695 1.9 106 0.8145 1.29485 70.47 32 32
Table 3: Equilibria with and without commitment; multiple equilibria feasible is always ecient and the IR constrained equilibrium when commitment is not feasible is welfare inferior to it. This may not be the case in an OLG setting, where the allocations with commitment may be Pareto inecient to start with. In the economy of table 3, the IR constraints improve welfare because, by limiting borrowing, the interest rate falls closer to its golden rule level. In fact, borrowing is higher in equilibrium B than it is in the equilibrium with commitment.
5 Comparative statics This section studies how changes in the parameters of the model aect the individuals' incentive to default, the equilibrium and the eciency of an economy. Consider rst the role of old age income, y2. A low endowment in old age makes autarky costly, everything else being equal. As endowment in old age becomes smaller, the incentive to default also falls and eventually disappears so that the allocations with and without commitment coincide. This is particularly true when the elasticity of intertemporal substitution is low and the discount parameter is high; for example, when < 1, the period utility function satis es the Inada condition so that low values of y2 certainly eliminate the incentive to default. Figure 6 shows the results of a comparative statics exercise for y2 for the economy of table 1. Figure 6a shows the incentive to default of middle aged individuals as a function of y2; the incentive to 20
default here is expressed as:
U d , U nd jU nd j
where, as in (23), U d = u(ytt+1 ) + u(ytt+2) and U nd = u(ctt+1 ) + u(ctt+2) with ctt+1 and ctt+2 are consumption in middle and old age conditional on repayment of debt.The incentive to default is negative when y2a 0:5 and positive when y2a > 0:5: the equilibrium with and without commitment therefore coincide for y2a 0:5. Figure 6b shows the parameter , the measure of Pareto ineciency. The curve c is for the economy with commitment, which is Pareto ecient when y2a = 1; as the endowment in old age moves away from 1, the economy becomes inecient because the equilibrium interest rate is driven away from its golden rule level of one. The curve nc is for the economy without commitment; for y2a 0:5, it coincides with c because the two economies coincide, but for y2a > 0:5 the two economies are characterized by dierent equilibria. The economy without commitment is more inecient: the dierence in eciency is large for low values of old age endowment it is equivalent to a 14% aggregate endowment reduction for y2a = 0:5but it gets smaller as old age endowment increases. In fact, the two economies get closer and closer as old age endowment increases because so does the equilibrium interest rate. Figure 6c shows the equilibrium interest rate in the economy; as soon as the debt limits become binding, the interest rate falls to R~0 and borrowing goes to zero. Further increases in y2a push the interest rate up (see appendix C) because middle age lending decreases, but the equilibrium remains an autarkic one, as shown in gure 6d, which depicts youth borrowing as a function of y2a . Figure 7 shows the comparative statics for y1a, the endowment of a middle age individual. An increase in middle age income, everything else being the same, makes the endowment pro le more humpshaped; since individuals want to smooth consumption, borrowing and lending become more important and we should therefore expect that, as y1a increases, the incentive to default is reduced and eventually becomes negative. For the economy of table 1, the incentive to default becomes negative for y1a 4 (see gure 7a); therefore, the economies with and without commitment coincide for middle age endowments above 4. The economy with commitment remains Pareto ecient as y1a changes because the equilibrium interest rate stays equal to one, due to the symmetry of the endowment pro le (see gure 7c and 7b); for the economy without commitment, however, the interest rate is well below its golden rule level of 1 and the degree of ineciency raises to more than 25% of total endowment for middle age endowments close to four. Ineciency becomes larger as y1a approaches four because individuals are autarkic (see gure 7d) and a more humpshaped endowment pro le implies a more humpshaped consumption pro le. 21
γc γnc
(a)
(b)
(c)
(d)
Figure 6: Comparative statics: y2a The elasticity of intertemporal substitution in consumption, , is the individuals' willingness to substitute consumption across dierent periods; a high value of means that individuals are more willing to shift consumption from one period to another and their indierence curves are more gently sloped; a low value of indicates that individuals care a lot to smooth consumption over time. Therefore, higher elasticities of intertemporal substitution raise the incentive to default because the welfare cost of a ragged consumption pro le is not high enough to prevent default. I let vary over the interval 0.1 to 5 in the economy of table 1; the incentive to default, depicted 22
γnc
γc
(a)
(b)
(c)
(d)
Figure 7: Comparative statics: y1a in 8a, is negative for 0:5 but positive after that; in other words, borrowing is constrained for 0:5. When commitment is feasible, the economy is Pareto ecient as shown by the fact that c = 0 in gure b; however, the economy without commitment is inecient for 0:5 and, as expected, the welfare losses from ineciency get smaller as the elasticity of intertemporal substitution increases. When approaches (positive) in nity, the economy with and without commitment tend to coincide. The unique equilibrium in the economy without commitment is one where borrowing and lending are zero (see gure d); as the individual becomes more willing to substitute consumption across periods, the autarky interest rate (see gure c) must raise because, 23
γnc
γc (a) (b)
(c)
(d)
Figure 8: Comparative statics: if the interest rate remained constant, he would be better o borrowing and raising his current consumption. As tends to positive in nity, the autarky interest rate approaches 1= . The discount parameter aects the incentive to default in an important manner: by raising the utility stemming from future consumption, higher s raise the welfare cost of autarky when endowment is humpshaped over the life cycle and therefore reduce the incentive to default. For the economy parameterized in table 1, agents 24
γnc γc
(a)
(b)
(c)
(d)
Figure 9: Comparative statics: would default even for > 1; in fact, the discount parameter has to raise above 2 to make agents care enough about the future not to renege their debt contracts (see gure 9a). More patience makes individual lend in order to shift consumption toward the end of life and, by doing so, lowers the equilibrium interest rate, as shown in gure 9c. The economy without commitment is more Pareto inecient than the economy with commitment and the dierence is greatest when = 1.
25
6 Con scable endowments Insofar I have assumed that private endowments cannot be seized in the event of a default. In many countries, however, wages and salaries can be partially seized following a default in order to reimburse the lender. This section studies and evaluates numerically the welfare implications of the assumption that private endowments can be at least partially con scated following the default on a borrowing contract. The main result is rather intuitive: the larger the fraction of private endowments that can be seized, the better o individuals because the equilibrium gets closer to the one with commitment. This result is not surprising because the punishment becomes more severe and the incentive to default disappears as a larger fraction of endowments can be seized; in fact, commitment is equivalent to a situation where commitment is not feasible but the (credible) punishment for default is so severe that individuals would always choose to repay their debts. Suppose that current and future private endowment of a defaulting individual can be partially expropriated in order to repay the lender. More precisely, if a generation t individual defaults at time t + 1, the lender can seize at most ytt+1 at time t + 1 and ytt+2 at time t + 2 from him to recuperate the amount Rt+1 btt+1 owed to him, with 0 1. In every period following a default, any amount still owed by the defaulter is repossessed by the lender as long as the latter is still alive; in principle, since individuals live for three periods, the lender may be unable to recover during his life time the full amount owed by the defaulter. Of course, default is not subgame perfect in this setting and therefore never takes place in equilibrium. Under the assumption that endowments are partially con scable, the optimal constrained lending decision for a generation t individual is the solution to the NC problem (22), subject to (8) to (10) and the IR constraints: u(ctt+1 ) + u(ctt+2 ) u(ytt+1 , min[ytt+1 ; ,Rt+1 btt+1 ]) + u(ytt+2 , )
(33)
u(ctt+2 ) u(ytt+2 , min[ytt+2 ; ,Rt+2 btt+2 ]);
(34)
and where
t t t lender is alive = min[(yt+2; ,(Rt+10bt+1 + yt+1)Rt+2 ] ifotherwise The individual now takes into consideration that, in addition to being unable to access the intertemporal market, his current and future endowments will be partially or completely seized if he decides to default; this reduces the utility of being autarkic and makes him less willing to default. More precisely, the constraint (33) shows that, in case of default by a middle age individual, his current endowment available for consumption is reduced by the largest amount between what he owes to the lender and what can (
26
(a)
(b)
(c)
(d)
Figure 10: Con scable endowments be seized; in the latter case, the lender can seize the borrower's old age endowment, up to a fraction , for what the borrower still owes plus the interest on this amount matured since last period. If the endowment pro le is humpshaped over the life cycle, individuals borrow when young and default when middle aged, when their lenders are old; since lenders will not be alive the following period, this implies that the borrower's endowment will be partially seized only in the period in which default takes place. The constraint (34) has a similar interpretation. Also, if ytt+1 ,Rt+1 btt+1 , an individual will never choose to default when middle aged because the equilibrium allocation is at best the same as full repayment; the same holds true and the individual will never default when old if ytt+2 ,Rt+2 btt+2 . The rst order conditions for this problem and their interpretation are identical to those of section 3.2; however, the constraints are binding for a smaller set of values of the parameter space than for the case where endowments cannot be expropriated at all. Formally, let S1s 2 S be the subset of the values for which (33) is binding and let S2s 2 S be the subset of the values for which (34) is binding; then S1s S1 and S1s S2 . The implications for welfare are that, as long as the equilibria with commitment are Pareto superior to those without commitment, legal institutions that make private 27
endowments more easily con scable after a default improve social welfare. Figure 10 shows the eect of con scable endowments as modelled above on the incentive to default. I use the economy of table 1 as benchmark and let one parameter change at a time keeping the other parameters constant and nd the value of 2 [0; 1] that makes the individual indierent between defaulting and not defaulting the contract. The points above the curve in each graph are combinations such that the IR constraints are not binding and vice versa for the points below the curve. Notice that values of < 0:4 ensure that an individual is never tempted to default for the parameter values considered in this exercise. Graph (a) shows that even if only 12.5% of private endowments can be seized after a default, this is sucient to eliminate the incentive to default as old age endowment rises. Graph (b) tells that a similar story for the case where middle age endowment is allowed to vary and notice that the incentive to default is eliminated if only a small fraction of the endowments is con scable, even when the endowment pattern is relatively at over the life cycle. Interestingly, even when individuals are impatient, namely as approaches zero, endowments con scable up to 40% are sucient to eliminate default.
7 Conclusions This paper has studied how much borrowing and lending takes place in a threeperiod OLG endowment economy where individuals cannot commit to repay their debts and debt repayment is the subgame perfect action chosen by the borrower facing a certain punishment for default. When debt repayment is not subgame perfect, borrowing and lending are limited; in many cases, the unique equilibrium of the economy is one where borrowing and lending are zero and the welfare losses from such market incompleteness are large. The incentive to default on a debt is a function of the parameters of the model and I have studied under which parameter con guration the incentive to default is binding. The OLG setting used in this paper has allowed me to focus on how the shape of the income pro le over the life cycle aects the incentive to default. An interesting result of this work is that a humpshaped income pro le generates debt limits early on in life; this result is consistent with the empirical nding that young rather than older individuals are borrowing constrained. The result that subgame perfect borrowing and lending may be limited also emerges from an in nitely lived representative agent model; the other results speci c to this OLG model are that: 1) multiple equilibria may arise; 2) most equilibria with commitment are Pareto superior to those without commitment, but the opposite may also occur; 3) most equilibria where debt repayment is not subgame perfect are autarkic equilibria, i.e. even small and limited borrowing without commitment cannot be sustained. 28
This paper is a rst step in the study of individual borrowing, lending and consumption decisions that explicitly incorporates individually rational lending contracts. The model studied here, even in its simplicity, suggests that he role of the incentive to default and their eect on borrowing and on consumption are important and this is consistent with recent ndings based on individual consumption data. Future extensions of this work may extend the model to a more realistic number of periods and introduce assets to assess quantitatively the importance of individually rational lending contracts. This setting could also be used to study the welfare consequences of intergenerational transfer programs, such as spending on education and social security.
Appendix A Proof of Proposition 1 Aggregate lending at t is de ned in (21); in the steady state, Rt = Rt+1 = Rt+2 = R and aggregate lending in the steady state is b(R) bt+1 (R) = btt+1 (R) + btt,1 +1 (R). b(R) is a continuous function of R because by (R) and bm (R) are continuous functions of R (see (16) and (17)). One can easily see that 8 > < +1 if > 0:5 lim b(R) = ,1; lim b(R) = > y0 if = 0:5 R!+1 R!0+ : 0+ if < 0:5 Since b(R) approaches 0 from above when < 0:5, the number of equilibria is odd. The equilibrium is unique if b(R) is monotonic increasing in R. A sucient condition for b(R) to be monotonic increasing is that consumption in dierent ages are strict gross substitutes in the steady state: the demand for consumption at each age is an increasing function of the price of the others. More precisely, for any generation s; 8s, consumption when young, middle aged and old are strict gross substitutes if @bss+1 @css 2 = Rt+1 > 0; @ (1=Rs+1 ) @Rs+1
@css+1 @bs @bs = bss+1 +Rs+1 s+1 , s+2 > 0; @Rs+1 @Rs+1 @Rs+1 @css+2 @bss+2 = > 0; @ (Rs+1 Rs+2 ) @Rs+1
The conditions above imply @bss+1 > 0; @Rs+1
@bss+1 > 0; @Rs+2
@bss+2 @css = > 0; @ (1=Rs+1 Rs+2 ) @Rs+1 ( ) @css+1 @bss+2 @bss+1 2 = Rs+2 @R , @R >0 @ (1=Rs+2 ) s+2 s+2 @css+2 @bs = bss+2 + Rs+2 s+2 > 0: @Rs+2 @Rs+2
@bss+2 > 0; @Rs+1
29
@bss+2 >0 @Rs+2
)
@bss+2 >0 @R
@bss+1 > 0; @R
@b(R) > 0: @R
)
For reason of space, the conditions for strict gross substitutability are not reported here; they are relatively straightforward but long and messy and are available from the author upon request. A sucient condition for all of them to be satis ed is that 1.
2
B Proof of Proposition 2
Consider case 3. The unique equilibrium in case 3 is at, for generation s; 8s 1 yss+1 R = R~ 00 = s "
ys
#
1
;
and steady state lending by generation s; 8s, is bss+1(R~ 00) = bss+2(R~ 00) = 0.
C Proof of Proposition 3 Consider now case 2. From (28) bss+2 (R) < 0
s if R < R~0 ; where R~ 0 = 1 yyss+2 s+1
From (27),
!
1
:
bss+1 (R~ 0 ) = 0:
Hence, R = R~ 0 is always an equilibrium of the constrained economy. One can calculate that @bss+1 @bss+2 > 0 and = 0; @R @R R~0
R~0
namely, the constrained optimal lending when young is at at R = R~0 while the constrained optimal lending when middle aged is increasing at R = R~0 , and lim ,bs = 0+; R!+1 s+1
lim bs = 0+; R!+1 s+2
@ , bs+1 = 0+; lim R!+1 s
@R
s
@bs+2 lim = 0+ R!+1 @R
i.e. both optimal lending when middle aged and when young tend to zero as the interest factor grows large. However, since ,bss+1 goes to zero faster than bss+2, this proves that the number of equilibria is odd. 2 30
The condition for multiple equilibria is that ,bss+1 (R) , bss+2(R) > 0; for some R > R~ 0, which can be rewritten as 3
2
1 1 i ( yss+1)1, + (yss+2 )1, 5 ,1 h s s , > 0: 4 2Rys+1 + ys+2 , 1 + R + R ( R ) [1 + R( R), ] This condition cannot be solved analytically; however, numerical simulations show it is satis ed as becomes small.
D Proof of Proposition 4 Consider the economy where commitment is feasible and let b(R) be aggregate total lending, as de ned in (21), in the steady state; we know that b(R) is monotone increasing (see appendix A). Let R~ 0 be the interest rate in the autarkic equilibrium of the economy without commitment. b(R~0 ) < 0 can be rewritten as ytt+2 ytt+1
!
1
1 ytt ytt+2 , ytt+1 < 1 , ytt+2 ; yt yt yt yt "
#
t+1 t+1 t+2 ytt+2 ytt ytt+1 < 1; ytt+1 < 1, namely
which is certainly satis ed if humpshaped over the life cycle.2
t+1
if individual endowment is
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