Endogenous Credit and Investment Cycles with Asset Price Volatility

⇤

Francesco Carli1 and Leonor Modesto2 1 2

Católica Lisbon School of Business and Economics

Católica Lisbon School of Business and Economics, and IZA

December 17, 2014

Abstract It is commonly accepted that credit market frictions are an important source of macroeconomic fluctuations. But what is the link between the two? And what is the driving factor of asset prices volatility? To answer these questions we have introduced a specific credit friction, limited commitment, in a general equilibrium model with production and investment in productive capital, where agents can trade bonds. The model always displays a stationary equilibrium where bonds are traded. In addition, a stationary equilibrium with no bonds may also exist. More importantly, limited commitment may generate stochastic endogenous fluctuations driven by self-fulfilling volatile expectations (sunspots), yielding credit and investment cycles and bond price volatility consistent with data.

Keywords: Credit Frictions, Limited Commitment, Indeterminacy, Sunspots. JEL classification: E32, E44 ⇤

We would like to thank Rui Albuquerque, Teresa Lloyd-Braga, Xavier Raurich, Catarina Reis,

Pedro Teles, and Alain Venditti for helpful discussions, as well as seminar participants at CatólicaLisbon, the V IIBEO Workshop, the 2014 PET Conference, the 3rd LuBraMacro Meeting, and the 2014 ASSET Annual Meeting.

1

Introduction

The recent financial turmoil has shown that macroeconomic fluctuations are tied to the functioning of credit markets. One possible explanation is that credit market frictions amplify and propagate exogenous shifts in fundamentals.1 This work provides an alternative explanation that does not rely on shifts in preferences, technologies, and other payoff relevant fundamentals. The key credit market friction considered is limited commitment, which by introducing endogenous borrowing constraints, generates stochastic endogenous fluctuations driven by self-fulfilling volatile expectations (sunspots), yielding credit and investment cycles and asset price volatility consistent with data.2 This means that animal spirits in credit markets are an important source of macroeconomic fluctuations.3 A similar conclusion is reached by Gu et al. (2013b), that also introduce limited commitment into a very stylized model that can be reinterpreted as a Kehoe and Levine (1993) pure endowment economy. Even though the authors characterize the properties of the global dynamics of their model, they do not consider a productive asset such as capital, and therefore they are unable to study the consequences of limited commitment on asset prices. To address this issue, we introduce limited commitment in a general equilibrium model with production and capital accumulation where agents can trade bonds. In this way we obtain a framework better suited to quantitatively assess fluctuations in standard macroeconomic variables as well as credit and investment cycles and asset prices fluctuations. More precisely, we depart from the standard one good representative agent macroeconomic model by introducing two types of goods, consumption and capital, and two infinitely lived types of agents, entrepreneurs (agents that need to borrow to finance their investment) and workers (agents that want to transfer consumption inter-temporally). The consumption good is produced by firms using capital and labor in a linearly 1

Bernanke and Gertler (1989), Kiyotaki and Moore (1997) are two seminal papers, Gertler and Kiyotaki (2010) an excellent survey. More recently Cooley et al. (2004) study amplification effects within long term lending relationships. 2 Note that in our model expectation shocks, by relaxing and tightening endogenous borrowing constraints, function in a similar way to the exogenous financial shocks considered by Jermann and Quadrini (2012). 3 Angeletos and La’O (2013) also conclude that shocks akin to sunspots may generate aggregate fluctuations in the presence of trading frictions and imperfect communication.

1

homogeneous technology. Entrepreneurs are more impatient and have access to a technology that transforms the consumption good into next period capital. Workers are more patient, work every period, but have no technology to transfer consumption inter-temporally. Finally, entrepreneurs have limited commitment and can choose to default. If entrepreneurs could commit to repay, we would recover the main features of the Ramsey model: workers buy claims on future capital (bonds) to transfer consumption across time, entrepreneurs operate their technology but do not save, and the price of new capital is pinned down by the properties of the capital producing technology.4 The economy exhibits one unique stationary first best equilibrium with positive capital, which is a saddle, one state variable being predetermined and the other non predetermined. Therefore indeterminacy is not possible and endogenous fluctuations (deterministic or stochastic) do not emerge. The introduction of entrepreneurs’s limited commitment may change significantly the properties of equilibria. Cooperative behavior is sustained by the threat of future exclusion from credit markets, which defines a no-default constraint:5 entrepreneurs’ willingness to honor their promises depends on the continuation value of their business, which in turn depends on the future price of bonds. In contrast with Gu et al. (2013b) a stationary equilibrium where bonds are not traded may or may not exist.6 If this steady state is not possible, we have a unique stationary equilibrium with positive capital, and depending on parameter values three different cases may emerge. In the first case entrepreneurs make significant rents by operating their technology, and the threat of exclusion is enough to sustain the first best allocation: the no-default constraint is slack and entrepreneurs do not save. When entrepreneurs rents are not as large, the model delivers a unique steady state with an endogenous borrowing constraint, where investment is below the first best level, but entrepreneurs are still not saving.7 Finally, when rents are very small, 4

Note that with a linear technology this price equals 1, so that capital and consumption are perfect substitutes, as in the one good Ramsey model. 5 The no-default constraint imposes borrowing limits that prevent default. 6 In Gu et al. (2013b) the no credit equilibrium always exists. This shows that the distinction between productive capital and credit is important, since in our model the no credit equilibrium does not coincide with the no capital equilibrium. Note that this last equilibrium always exists in our case. 7 It is still too costly for them to delay consumption.

2

the unique stationary equilibrium still features an endogenous borrowing constraint. However now, to relax the credit constraint (i.e. their temptation to default) entrepreneurs save and invest in their technology. We can interpret this investment as collateral. If instead the equilibrium with no bonds is possible, then it always coexists with one of the three steady states described above, and we have multiplicity of equilibria with positive capital. More importantly, when the no-default constraint binds, and even in the absence of collateral, we now obtain stochastic endogenous fluctuations (sunspots). The price of new capital is no longer pinned down by the properties of the capital producing technology, but can now jump in order to accommodate changes in expectations, with investment reacting accordingly. Therefore, the two state variables are now non predetermined and, although the steady state is still a saddle, indeterminacy (and therefore sunspots) emerge. This, as shown by our simulation exercises, can result in fluctuations that match credit and investment cycles observed in the data. Moreover, the model generates a volatility of the price of bonds which is consistent with the one observed in the data.8 Our work builds on two branches of the literature. From the first one we borrow the use of limited commitment to generate endogenously debt limits. See Gu et al. (2013a). Gu et al. (2013b), Kehoe and Levine (1993), Azariadis and Kaas (2013), Azariadis and Kaas (2007), and Albuquerque and Hopenhayn (2004). However, the setup considered in these studies is different from ours. Either they consider exchange economies without a productive asset, or restrict their analysis to partial equilibrium models. The second branch of the literature investigates, in monetary OLG models, the role of credit market distortions on the emergence of local indeterminacy, bifurcations, and endogenous cycles. Examples are Azariadis and Smith (1996), Azariadis and Smith (1998) and Boyd and Smith (1997), where private information between borrowers and lenders produces multiple stationary equilibria and changes local dynamics, introducing indeterminacy and endogenous fluctuations. Our paper differs from these works in two important dimensions: first, while they use an OLG framework, we consider an economy populated by infinitely lived agents, so that limited commitment is the only friction we introduce. Second, we 8

Differently, Bhamra and Uppal (2014), Adam et al. (2014), and Xiong and Yan (2010) rely on heterogeneous/subjective beliefs to explain excess volatility in bond yields, while Adam et al. (2008) rely on bounded rationality to explain excess stock price volatility.

3

do not impose any restriction on endogenous variables that implicitly determines the pattern of trade. The rest of the paper is organized as follows. In the next section we present the model. In section 3 we obtain the limited commitment solution. In section 4 we study local dynamics and discuss our main results. In section 5 we present a simulation exercise and show that our simulation results match important business cycle features. Finally section 6 concludes. Proofs and computations are relegated to the Appendix.

2

The Model

Time is discrete, lasts forever, and is indexed as t = 0, 1, 2, ... There are two goods, a capital and a consumption good, and we take the consumption good as numeraire. Each period there is a perfectly competitive sector - or a representative firm that produces the consumption good Yt using capital Kt and labor Lt in a linearly homogeneous technology, Yt = F (Kt , Lt ). Both capital and labor are necessary for production, meaning F (0, Lt ) = F (Kt , 0) = 0. We assume that capital used in production fully depreciates, hence consumption goods can only be produced if new capital is provided each period. Furthermore we assume that consumption is non-storable across periods. The economy is populated by a measure 1 of heterogeneous agents: a fraction ↵ of them being workers and a fraction (1

↵) being entrepreneurs. Workers are

endowed with one unit of time every period and can work for the representative firm in exchange for the competitive wage wt = FL (Kt , Lt ). Differently, entrepreneurs are endowed with a technology that transforms the consumption good into the capital good.9 This technology uses as input xt units of the consumption good in period t and returns g(xt ) = x⌫t units of capital at time t + 1. We assume that 0 < ⌫  1.10 Workers can not operate this technology, but can buy in a competitive 9

We follow the usual terminology by calling entrepreneurs the agents that produce capital and need to borrow, and workers the agents that work and lend. See for example the survey by Quadrini (2011). 10 Our results do not depend on the choice of an isoelastic capital producing technology: both the existence of stationary equilibria, their properties, and the local dynamics, would have remained the same if we had considered a strictly increasing and concave function g(x),with g(0) = 0, where ⌫ denotes its elasticity evaluated at the steady state.

4

credit market claims on next period capital, that in our stylized economy we call bonds. We denote by qt the price of each bond, which is the amount of consumption goods needed at time t, for a claim on one unit of capital at t + 1. Furthermore, W E we denote by kt+1 and kt+1 respectively workers and entrepreneurs net demand

for bonds at time t, where we use the convention that a positive value stands for purchases, and a negative value represents sales. Trading over these claims is subject to limited commitment: entrepreneurs can divert resources from their investment technology and repudiate their promises. They will only chose to repay if it is in their interest. This means that credit contracts must adjust to be self-enforcing. Workers have no disutility from working and are risk-averse. For the sake of simplicity we consider a log utility function: Ut =

1 X

⌧ t

log(cW ⌧ )

⌧ =t

where cW ⌧ denotes a worker’s consumption in time ⌧ . Differently, entrepreneurs are risk-neutral, and their preferences are defined over consumption according with the utility function Vt =

1 X

⌧ t E c⌧

⌧ =t

where cE ⌧ denotes the consumption of an entrepreneur in time ⌧ . The parameters and

are the discount factors of workers and entrepreneurs, respectively. We

assume that:11 Assumption 2.1. Entrepreneurs are more impatient than workers:

> .

The timing is the following: at the beginning of period t, after the investment technology returns x⌫t

1

units of capital, the entrepreneur repays/redeems his

claims ktE . Therefore, his wealth consists in x⌫t

1

+ ktE units of capital. This capital

is supplied to the representative firm, in exchange for rt (x⌫t

1

+ ktE ) units of con-

sumption, where rt is time t rental rate of capital.12 Perfect competition ensures that rt = FK (Kt , Lt ).13 A worker, instead, enters time t with claims on ktW units 11

This is a standard assumption to motivate trade, see Quadrini (2011) An entrepreneur supplies all his capital to firms, not storing any capital for next period because it is optimal to do so. The proof is available upon request. 13 Note that the representative firm profits are zero. 12

5

of capital. After redeeming these claims, he supplies one unit of time and capital ktW to the representative firm, earning wt + rt ktW . Capital then fully depreciates, and both workers and entrepreneurs have to choose how much to consume and how much to save. A worker can only use his savings to purchase capital for the next period. Therefore it is not credible for a worker to sell claims on future capital, so that ktW

0 holds for all t. Differently, an entrepreneur can use his savings either

to buy or sell claims on future capital or to invest in his technology. We can then write a worker budget constraint as W W cW t + qt kt+1  wt + rt kt

and an entrepreneur budget constraint as E E E ⌫ cE t + qt kt+1 + xt  rt [kt + xt 1 ] E xt + qt kt+1

0

where the second inequality implies that entrepreneurs can only borrow to invest in their technology, i.e. they can not borrow to consume in the current period.

3 3.1

Equilibrium with Limited Commitment The workers’ problem

A worker that enters time t with ktW bonds, solves the following problem: Ut (ktW ) =

(

max

cW t ,

W kt+1

)

s.t.

(1a)

W log(cW t ) + Ut+1 (kt+1 ) W W cW t + qt kt+1  wt + rt kt W kt+1

0,

cW t

0

(1b) (1c)

Due to log preferences cW t > 0. Then, from the first order conditions Et (cW t+1 ) W ct W cW t + qt kt+1

Et (rt+1 ) qt wt 6

rt ktW = 0

(2) (3)

W kt+1

(4)

0

W where (2) has to hold with equality if kt+1 > 0. The right-hand side of equation (2)

gives us the expected discounted real return of a bond, and can also be interpreted as the expected Tobin’s q: rt+1 is time t+1 value of a unit of capital; by multiplying by

3.2

we discount it by one period, whereas qt is today’s cost of replacing capital.

The entrepreneurs’ problem

Entrepreneurs can not commit to honor their debts and may chose to divert resources from the capital producing technology, consuming them and enjoying in this way an immediate private benefit. Specifically, an entrepreneur saves rt (x⌫t

1+

ktE ) cE t and receives

E qt kt+1 units of consumption goods from workers. Afterwards, eE , and entrepreneurs can choose to consume an extra amount, that we denote by C t

repudiate the credit contract in the beginning of the following period. This behavior gives entrepreneurs utility (1

✓) for each unit of consumption, where ✓ 2 (0, 1) is

a parameter that measures the cost of hiding resources. Upon repudiation, workers eE )⌫ . Eascan seize entrepreneurs capital, given by (rt (x⌫ + k E ) cE qt k E C t 1

ily, entrepreneurs best deviation is to consume

t

eE C t

=

t rt (x⌫t 1

+

t+1 ktE )

t E ct

E qt kt+1

investing nothing in their technology. Assuming that the punishment for deviating is (perpetual) exclusion from credit markets, entrepreneurs deviation payoff is: Vet+1 = (1

h ✓) rt (x⌫t

E 1 + kt )

cE t

E qt kt+1

i

In order to avoid this problem, credit contracts must satisfy the no default constraint (5) below, i.e. expected future discounted utility (the continuation payoff) must exceed the deviation payoff: ⇥

E Et Vt+1 (kt+1 , xt )

⇤

h

Vet+1 = (1

✓) rt (xt

1

+

ktE )

cE t

E qt kt+1

i

(5)

This constraint, that ensures that entrepreneurs do not deviate and default on their promises, must be explicitly introduced in their problem. Therefore, an entrepreneur that enters period t with x⌫t

1

+ ktE units of capital, solves the following

7

problem: Vt (ktE , xt 1 ) =

max

E , x (cEt , kt+1 t)

s.t.

⇥ ⇤ E cE t + Et Vt+1 (kt+1 , xt ) cE t

0

xt

(6a)

0

(6b)

E E ⌫ cE t + qt kt+1 + xt  rt [kt + xt 1 ] E xt + qt kt+1 0 ⇥ ⇤ E Et Vt+1 (kt+1 , xt )

Let

t

(1

h

(6c)

✓) rt (xt cE t

1

E qt kt+1

i

+ ktE ) (6d)

and ⌘t be the multipliers associated with (6b) and (6c) respectively, while µt is

the multiplier associated with the no-default constraint (6d).14 Because constraint (6b) binds, we can rewrite constraint (6d) as ⇥ ⇤ E Et Vt+1 (kt+1 , xt )

(1

(7)

✓)xt

The first order, complementary slackness, and envelope conditions are 1

t

(8)

0

with equality if cE t > 0 E cE rt [x⌫t 1 + ktE ] = 0 t + qt kt+1 + xt ⇥ ⇤ E ⌘t xt + qt kt+1 =0 ⇥ ⇤ E µt [ Et Vt+1 (kt+1 , xt ) xt (1 ✓)] = 0 ( ) (1 + µt )rt+1 t+1 qt = E t (⌫ 1) (1 + µt ) t+1 rt+1 ⌫xt (1 ✓)µt n (⌫ 1) (⌫ ⌘t = E t t µt [ t+1 rt+1 ⌫xt t+1 rt+1 ⌫xt

(9a) (9b) (9c) (9d) 1)

1 + ✓]

o

(9e)

Definition 3.1. An equilibrium with limited commitment consists in sequences W 1 E E 1 1 1 (cW t , kt+1 )t=1 , (ct , xt , kt+1 )t=1 , (qt , rt , wt )t=1 , ( t , µt , ⌘t )t=1 , aggregate capital Kt and 14

Since we want to focus on equilibria with capital accumulation, in what follows we assume xt > 0.

8

an initial condition x0 such that: 1. Taking prices as given, workers maximize (1a)-(1c) so that (2)-(4) hold; entrepreneurs maximize (6a)-(6d) so that conditions (8), (9a)-(9e) hold. 2. Capital and labor earn their marginal product, rt = Fk (Kt , Lt )

(10a)

wt = FL (Kt , Lt )

(10b)

3. Credit market clears W ↵kt+1 + (1

E ↵)kt+1 =0

(11)

↵)x⌫t

(12)

4. Capital evolves according with Kt+1 = (1

3.3

Stationary equilibria

In this subsection we characterize non-trivial stationary equilibria with positive capital, K > 0.15 We start addressing equilibria where the no-default constraint is slack. Then we move to equilibria where the no-default constraint binds.

3.4

Stationary equilibria with a slack no-default constraint

When the no-default constraint is slack, the threat of future exclusion provides entrepreneurs with the incentives to repay. As a result, in equilibrium, workers’ purchases of bonds from entrepreneurs are strictly positive. Lemma 3.2. If the no-default constraint (7) is slack, we have k W > 0. Proof. See Appendix 7.1. Since k W > 0, from equation (2) the price of bonds is q= r 15

(13)

Clearly K = 0 is a stationary equilibrium: it is a trivial one, in which production never takes place, and all decision variables are equal to zero, k E = k W = cE = cW = x = 0.

9

so that at the steady state the Tobin’s q,

r , q

equals 1. From equation (3) we obtain

cW = w + (1

)rk W

(14)

Moreover, as in equation (9d) µ = 0, we have q=

1 ⌫x(⌫

(15)

1)

We can also prove that entrepreneurs are strictly better consuming all their income and saving nothing for the future. Lemma 3.3. If the no-default constraint (7) is slack, entrepreneurs savings equal zero, i.e. x + qk E = 0. Proof. See Appendix 7.2. Furthermore, as entrepreneurs savings are equal to zero, from (9b) and (13) we obtain kE =

x = q

x r

(16)

Replacing (16) in the market clearing condition (11) gives 1

kW =

↵ ↵

kE =

1

↵ x ↵ r

(17)

And finally replacing (16) in the budget constraint of entrepreneurs (9a) we obtain E



c = r x⌫

x r

(18)

Lemma 3.4. If the no-default constraint (7) is slack, entrepreneurs consumption is positive, cE > 0, hence

= 1.

Proof. See Appendix 7.3 If we let z =

K ↵

be capital per worker, from (10a)-(10b) we have r = f 0 (z) w = f (z) 10

zf 0 (z)

(19)

Then, using the law of motion of capital (12) we can write the rental rate of capital as a function of investment x: r=f

0

✓

↵)x⌫ ↵

(1

◆

(20)

Therefore, the only thing left to determine, is the level of investment x. Combining (20) with (13) and (15) we obtain f

0

✓

(1

↵)x⌫ ↵

◆

=

x(1

⌫)

(21)

⌫

Equation (21) closes the model, uniquely determining entrepreneurs’ investment in a stationary equilibrium with a slack no-default constraint. We can state the following: Proposition 3.5. An equilibrium with a slack no-default constraint exists if and only if ⌫  ⌫(✓) ⌘

+ (1 ✓)(1

)

.

Proof. See Appendix 7.4. When this equilibrium exists, the level of investment is the same as if entrepreneurs had access to a commitment technology. Accordingly, we denote the level of investment that solves (21) by xF B , where F B stands for first best. The condition ⌫  ⌫(✓) guarantees that entrepreneurs earn enough rents, so that future credit market exclusion is a sufficiently severe punishment to overcome the one period temptation to deviate.16

3.5

Stationary equilibria with a binding no-default constraint

When the no-default constraint binds, from (9c) we have µ (1

✓)x. Using stationarity, from (6a) we obtain V (k E , x) =

0 and V (k E , x) = cE 1

. Combining the

two last expressions we obtain cE =

(1

✓)(1

16

)x

>0

(22)

If entrepreneurs could credibly commit to repay we would not have to guarantee that ⌫  ⌫(✓), because the first best equilibrium is defined by (21) for any ⌫ 2 (0, 1].

11

Therefore, we can state the following Lemma: Lemma 3.6. In a stationary equilibrium with production where the no-default constraint (7) binds, entrepreneurs consumption is strictly positive, i.e. cE > 0, and = 1. When (7) binds, entrepreneurs and workers may or may not trade bonds. This contrasts with what happens when the no-default constraint (7) is slack, where by Lemma (3.2), bonds are always traded. Furthermore, when (7) binds and bonds are traded, entrepreneurs may or may not save to invest in their technology. Note that this is not possible when the no-default constraint (7) is slack, where bonds are always traded and entrepreneurs do not save, as proved in Lemma (3.3). As explained below, we interpret entrepreneurs own investment as collateral. In the remaining part of the section, we present these three equilibria separately. 3.5.1

Stationary equilibria with bonds and no collateral

If entrepreneurs collateral equals zero, then equations (13), (14), and (16)-(19) hold. Equating (18) and (22), and using (19), we obtain f

0

✓

(1

↵)x⌫ ↵

◆

=x

1 ⌫



+ (1

✓)(1

)

(23)

Equation (23) closes the model, determining the steady state investment level x, when (7) binds, bonds are traded, and entrepreneurs do not save. It is easy to prove that (23) has a unique solution. To prove existence, we have to guarantee that indeed the no-default constraint is binding and entrepreneurs savings are equal to zero. Let xBN C denote the level of investment x that solves (23), where BN C stands for "bonds and no collateral". We can state the following: Proposition 3.7. A stationary equilibrium with a binding no-default constraint, bonds, and no collateral exists if and only if ⌫(✓)  ⌫  ⌫(✓) ⌘

+(1 ✓)( + (1 ✓)(1

) , )

where

⌫(✓) is given in Proposition 3.5. Finally, investment is below the first best level, xBN C < xF B . Proof. See Appendix 7.5

12

3.5.2

Stationary equilibria with bonds and collateral

If workers purchase bonds from entrepreneurs, equation (13) holds. Solving for µ in (9e), where ⌘ = 0, we obtain 1 r⌫x(⌫ 1) µ= >0 r⌫x(⌫ 1) 1 + ✓ Replacing the last expression and q =

r from (13) into (9d), solving for r, and

using (19) we obtain f

0

✓

(1

↵)x⌫ ↵

◆

=x

1 ⌫



+ (1

✓)( ⌫

)

(24)

Equation (24) closes the model, determining the steady state investment level x, when (7) binds, bonds are traded, and entrepreneurs savings are positive, i.e. loans are collateralized. It is easy to prove that (24) has a unique solution. To prove existence, we have to guarantee that indeed the no-default constraint is binding and colateral is positive. Let xBC denote the level of investment x that solves (24), where BC stands for “bonds and collateral”. We can state the following Proposition 3.8. A stationary equilibrium with a binding no-default constraint, bonds, and collateral, exists if and only if ⌫(✓)  ⌫  1, where ⌫(✓) is given in Proposition 3.7. Finally, investment is below the first best level, xBC < xF B . Proof. See Appendix 7.6. 3.5.3

Stationary equilibria with no bonds

If bonds are not traded, k E = k W = 0. Easily, we obtain that workers consumption equals their wage: cW = w, with w given by (19). Moreover, the no-default constraint should bind. Lemma 3.9. In any non-trivial stationary equilibrium in which bonds are not traded, i.e. k E = k W = 0, we have µ > 0 and (7) must bind. Proof. See Appendix 7.7.

13

As a result, we obtain consumption of entrepreneurs from (22). Combining (22) with (9a), we obtain

(rx⌫ 1

x)

= (1

✓)x

Solving the last expression for r, and combining with (19), we finally obtain f

0

✓

(1

↵) ↵

x

⌫

◆

=x

1 ⌫



+ (1

✓)(1

)

(25)

Equation (25) closes the model, determining the steady state investment level x, when (7) binds and bonds are not traded. It is easy to prove that this solution is unique. To prove existence, we have to guarantee that indeed workers do not purchase bonds. Let xN B denote the level of investment x that solves (25), where N B stands for “no bonds. We can state the following Proposition 3.10. A stationary equilibrium with no bonds exists if and only if ⌫

1 ✓ . 1 ✓+ ✓

Proof. See Appendix 7.8. If this equilibrium does not exist, we have uniqueness of equilibrium, i.e. bonds are traded, and depending on parameter values the no-default constraint is either slack or binding, and in the latter case entrepreneurs savings are either zero or positive. This is illustrated in Figure 1 where we partition the parameter space for stationary equilibria. When the elasticity ⌫ is sufficiently small (below the blue line), entrepreneurs earn significant rents by operating their technology. As a result, permanent exclusion from credit markets is a severe punishment that dissuades them from defaulting, so that the the first best levels of credit and investment are sustained. Workers purchase bonds to transfer consumption inter-temporally, and the rate of return on capital is too low to convince the more impatient entrepreneurs to invest. For intermediate values of ⌫ (above the blue line but below the red), entrepreneurs rents are relatively smaller, and the first best level of credit and investment would violate the no-default constraint. Entrepreneurs are still not investing in their technology, because it is costly for them to postpone consumption. Nevertheless, credit and investment adjust (reduce) to avoid that entrepreneurs choose to default. Finally, when ⌫ is relatively large (above the red line), entrepreneurs 14

rent are almost negligible. For these values of ⌫ entrepreneurs invest a positive fraction of their income in the capital producing technology. This investment works as collateral: by providing entrepreneurs with a stake in the project, collateral relaxes credit constraints. Easily, given a value of ⌫, a larger ✓ increases the cost of deviation, and helps sustaining larger levels of credit and investment. This explains the positive slope of both lines, ⌫(✓) and ⌫(✓). ⌫(✓) ( )

1 ( )

1 ✓

Figure 1: State space (✓, ⌫) and stationary equilibria. In contrast, when the equilibrium with no bonds exists, the economy always exhibits two non trivial stationary equilibria: one in which bonds are traded, where entrepreneurs investment is either given by (21), or (23), or (24), and a second one in which bonds are not traded, and entrepreneurs’ investment is given by (25). This situation is depicted in Figure 1, in the 3 regions above the green line.

4

Local Dynamics

In this section we focus only on equilibria in which workers and entrepreneurs trade bonds. Moreover, we restrict our analysis to the cases in which entrepreneurs do not save, i.e. we assume that ⌫  ⌫(✓). In this way we guarantee that our results are not driven by the existence of colateral.

1

15

4.1

Local dynamics with a slack no-default constraint

Define zt =

Kt ↵

(26)

to be capital per worker.17 From the law of motion of aggregate capital (12), we can write time t capital per worker as zt =

1

↵ ↵

x⌫t

1

⌘ z(xt 1 )

(27)

so that, using (10a) and (10b), the interest rate and the wage can be written as rt = f 0 (z(xt 1 )) ⌘ r(xt 1 ) wt = f (z(xt 1 ))

z(xt 1 )f 0 (z(xt 1 )) ⌘ w(xt 1 ).

(28) (29)

As the no-default constraint is slack, µt = 0, and from (9d) we can write the price of bonds as a function of xt : qt =

1 ⌫x⌫t

1

⌘ q(xt ).

(30)

Also, since entrepreneurs do not save, from (9b), we have E kt+1 =

xt ⌘ kE (xt ) q(xt )

(31)

W and from equations (11) and (9a), we can write workers purchases of bonds, kt+1 ,

and entrepreneurs consumption, cE t , as a function of time t and time t 1 investment, xt and xt 1 : ↵ xt ⌘ kW (xt ) ↵ q(xt ) = f 0 (z(xt 1 )){x⌫t 1 + kE (z(xt 1 ))} ⌘ cE (xt 1 )

W kt+1 =

cE t 17

1

Note that zt 6= ktW .

16

(32) (33)

Substituting the previous expressions in the workers first order condition and budget W constraint (2)-(3) we obtain our two dynamic equations in xt , xt+1 , cW t , and ct+1 .

Et (cW f 0 (z(xt )) t+1 ) = q(xt ) cW t

(34a)

cW t + q(xt )kW (xt ) = f (z(xt 1 ))

z(xt 1 )f 0 (z(xt 1 )) + f 0 (z(xt 1 ))kW (xt 1 ) (34b)

Note that the dynamic system (34a) and (34b) coincides with the dynamic system of the first best equilibrium, where the no-default constraint (6d) does not enter the entrepreneurs problem. It is clear that investment is a predetermined variable, whose behavior is determined by past decisions. However consumption is a non-predetermined variable, whose level is influenced by expectations. Therefore, endogenous fluctuations driven by self-fulfilling volatile expectations (sunspots) may emerge if the steady state is indeterminate. To see whether this is the case we analyze below the local stability properties of the dynamic system in the neighborhood of the steady state. We start by presenting the linearized version of our dynamic system (34a) and (34b). "

e cW t+1 x et

#

= JF B

"

e cW t

x et

1

#

+

"

ut+1 0

#

where a ˜ over a variable denotes percentage deviations from the steady state, e.g. x et =

ut+1 = e cW t+1

dxt , x

JF B is the Jacobian matrix evaluated at the steady state18 and

Et (e cW t+1 ), with Et (ut+1 ) = 0, is a forecast error.

The determinant, DF B , and the trace, TF B , of matrix JF B , whose expressions

are given in Appendix 7.11, correspond respectively to the product and sum of the two roots (eigenvalues) of the associated characteristic polynomial Q( 2 FB

F B TF B

+ DF B .

F B)

⌘

To analyze the local stability properties of our model we use the geometrical method proposed by Grandmont et al. (1998). This method amounts to locate how the trace and determinant of our system evolve in the space (T, D) , when some parameters are made to vary continuously in their admissible range. In Figure 2, we have represented three lines relevant for this purpose: the line AC, defined by 18

The matrix JF B is given in Appendix 7.10

17

D

B

C

T A

Figure 2: Relevant regions in the (T, D) space 1, where a local eigenvalue of J is equal to 1; the line AB, defined by

D = T D =

T

1, where one eigenvalue is equal to -1; and the segment BC, defined

by D = 1 and |T | < 2, where two eigenvalues are complex conjugates of modulus 1. It is easy to verify that when T and D fall in in the interior of triangle ABC,

both eigenvalues have modulus lower than one, so that steady state is a sink. The steady state is a saddle when |T | > |D + 1| (one eigenvalue with modulus higher than one and one eigenvalue with modulus lower than one). In all other regions the steady state is a source (the modulus of both eigenvalues are higher than 1). Also when the number of eigenvalues with modulus less than one is higher than the number of predetermined variables the steady state is indeterminate and sunspots may emerge. Therefore with a slack no-default constraint, with one predetermined variable, we will have indeterminacy only when the steady state is a sink. In that case for a given initial value x0 there will be an infinite number of trajectories along which the system will converge to the steady state. Also, as known, there will be infinitely many stochastic endogenous fluctuations (sunspots) arbitrarily close to the steady state. However, as in this environment 0 < DF B < TF B never happen and we can claim the following:

1, this will

1

Proposition 4.1. If the no default constraint is slack, the stationary equilibrium with bonds is a saddle. Proof. See Appendix 7.11. In this case, there is a unique trajectory, cW t , xt 18

t=,1,...1

with a given x0 suffi-

ciently close to the steady state, that remains close to the steady state. This means that, in the absence of exogenous shocks on fundamentals, the forecast error ut+1 is necessarily zero and there is a unique convergent path to the steady state, so that endogenous fluctuations do not emerge. The dynamic model is locally determinate. Corollary 4.2. If the no default constraint is slack, the stationary equilibrium with bonds is never indeterminate. When the no-default constraint is slack, the local dynamic properties of the stationary equilibrium are identical to those of the standard Ramsey model: the steady state is locally determinate, it is a saddle, and endogenous fluctuations do not emerge. We can therefore conclude that, if commitment problems are not relevant, the Ramsey environment, where one single good is used both as a consumption and capital good constitutes a good approximation, even if the technology that transforms one into the other is not linear.

4.2

Local dynamics with a binding no-default constraint, bonds, and no collateral

From the workers problem we obtain as before (2) and (3). Also expressions (27)(29) still apply. Moreover, since entrepreneurs savings are again zero, using (9a) and (9b), from the entrepreneurs problem we obtain as before: xt q ht = rt x⌫t

(35)

E kt+1 =

cE t

i E + k 1 t .

(36)

However, now the no-default constraint is binding, i.e: (1

cE t+1 + (1

✓)xt = Et

✓)xt+1 .

(37)

Combining the credit market clearing condition (11) with (35) we obtain as before that: W kt+1 =

1

↵ ↵

E kt+1 =

19

1

↵ xt ↵ qt

(38)

Replacing the last expression in the worker’s budget constraint (3), and using also (27)-(29), we can rewrite it as: cW t +

1

↵ ↵

(39)

xt = w(xt 1 ) + r(xt 1 )ktW

Now, if we forward this expression one period, and we rearrange its terms, we obtain cW t+1 +

W kt+1 =

1 ↵ xt+1 ↵

w(xt )

(40)

r(xt )

W Replacing back this expression for kt+1 into (38) we can write qt as a function of xt ,

xt+1 and cW t+1 : qt =

1

↵ ↵

xt

cW t+1

+

r(xt ) 1 ↵ xt+1 ↵

w(xt )

(41)

⌘ qt (cW t+1 , xt , xt+1 )

In the entrepreneur’s problem, if we forward (36) one period, and we replace it into (37) together with (35) and (40), we obtain (1

✓)xt = Et

⇢



r (xt ) x⌫t

cW t+1 +

↵ 1

1 ↵ xt+1 ↵

↵

w (xt )

r (xt )

+ (1

✓)xt+1 (42)

Finally, replacing (41) into (2) we obtain Et

(

cW t+1 cW t

⇥ ↵ cW t+1 +

1 ↵ xt+1 ↵

(1

↵)xt

wt+1 (xt )

⇤)

=0

(43)

W Equations (42) and (43) are our two dynamic equations, in xt , xt+1 , cW t , ct+1 .

It is clear that now, both consumption and investment are non-predetermined variables, whose values are influenced by expectations about the future. Hence, depending on the local stability properties of the steady-state, there is potentially room for the existence of equilibria trajectories that exhibit bounded fluctuations, sufficiently close to the steady state, driven by self-fulfilling changes in expectations unrelated to economic fundamentals. As explained above, this will happen when the steady state is locally indeterminate. With two non predetermined variables, we will have indeterminacy when the steady state is a sink or a saddle. To discuss local dynamics we present below the linear approximation of the non 20

stochastic version of system (42)-(43) in the neighborhood of the steady-state. "

e cw t+1

x et+1

#

= JBN C

"

e cw t x et

#

(44)

where JBN C denotes again the Jacobian matrix evaluated at the steady state, which is given in Appendix 7.12. The determinant, DBN C , and the trace, TBN C , of matrix JBCN , whose expressions are given in Appendix 7.13, correspond respectively to the product and sum of the two roots (eigenvalues) of the associated characteristic polynomial Q(

BN C )

⌘

2 BN C

BN C TBN C

+ DBN C

As in the case with a slack no-default constraint, we always have DBN C < TBN C

1. See Appendix 7.13 for the proof. Moreover, it is easy to prove that:

Proposition 4.3. Assume that

> min{s, (1

✓) }. The steady state with a

binding no default constraint, bonds, and no colateral is a saddle. Proof. See Appendix 7.13

Corollary 4.4. The steady state with a binding no default constraint, bonds, and no collateral is always locally indeterminate. Indeed, since we have two non predetermined variables, when the steady state is a saddle there are an infinite number of combinations of initial values for both state variables, that keep the system in the stable arm, so that it will always converge to the steady state, i.e., there are now infinitely many bounded deterministic equilibrium trajectories {ct , xt }t=,1,...1 converging to the steady state. Also, as proved by Grandmont et al. (1998), there are also infinitely many non-degenerate stochastic equilibria driven by self-fulfilling volatile expectations (stochastic endogenous fluctuations or sunspots equilibria), that stay arbitrarily close to the steady state. 4.2.1

Sunspot equilibria

When the non-stochastic system has a unique steady state that is a saddle, it follows that the model has a unique fundamental equilibrium represented by the stable branch of the saddle (see Farmer et al. (2012)). In our case the stable branch of the saddle is given by the following first order difference equation in x et+1 x et+1 =

21

et 1x

(45)

found by replacing e ct in (44) with the equality where

1

e cW t =

j22

1

j21

(46)

x et

is the eigenvalue of JBN C smaller than one and jij denotes the element ij

of matrix JBN C . The sequence {e xt } that solves (45), and the corresponding sequence e cW where t

e cW t satisfies (46) constitute the fundamental equilibrium of our economy.

Note that investment converges to a unique steady state xBN C given by (23)

and consumption of workers converges to a unique steady state cW given by c

W

=

1 ↵

 ↵ 1

s s

✓

+ (1

✓)(1

)

◆

+1

x

However, in this economy sunspots matter. To capture the effects of these non fundamental shocks we add forecasting errors et+1 = x et+1 Et (e xt+1 ), with Et (et+1 ) = 0, to equation (46)

x et+1 =

et 1x

+ et+1 .

(47)

Local sunspot equilibria can be obtained by considering that et+1 follows an i.i.d. stochastic process of bounded support with sufficiently small variance (see Benhabib and Farmer (1999) for further discussion). To better illustrate this suppose that, starting from the steady state, a forecasting error hits this economy. Since the values of x and c are tied by (46) the economy will come back to the stable arm, which guarantees convergence to the steady state. Forecasting errors now act as independent sources of the business cycles, even in the absence of intrinsic uncertainty affecting fundamentals, and stochastic bounded equilibrium trajectories, driven by expectations shocks, that stay arbitrarily close to the steady state are possible when the steady state is a saddle.

5

Simulation Exercise

Despite being very stylized, our model is still able to capture the main features of credit market observed series. We illustrate this by presenting below the results of a simulation exercise, where we introduce sunspot shocks near the saddle steady 22

state with a binding no-default constraints, bonds, and no collateral.

5.1

Calibration

To simulate the model, we considered a discount factor for workers

= 0.99, in

line with most calibration exercises, and to motivate trade, we assumed

= 0.92.

We set the capital share at s = 0.33, a value consistent with observed data. In the absence of a reliable estimate for the cost ✓ of hiding resources, we experimented with several values, and we chose a value of ✓ = 0.6. Since empirical studies find an elasticity of substitution significantly different from one, we considered a CES production function with an elasticity of substitution

= 2.5. This value is consistent with the estimates obtained by Duffy

and Papageorgiou (2000), who report robust estimates that are contained in the [1.24, 3.24] interval. Concerning the capital production technology we chose the value ⌫ = 0.98 which guarantees that the steady state is a saddle, where workers and entrepreneurs trade credit contracts, the no-default constraint binds, and entrepreneurs do not save, i.e. 0.97 = ⌫ < ⌫ < ⌫ = 0.99. Moreover, this value is sufficiently close to 1, to ensure that the results are not mainly driven by an excessive concavity of the capital production technology. By construction, the sunspot shocks are i.i.d. shocks. We assume a normal distribution N (0,

e ),

where

e

= 0.0185 was chosen to match the standard deviation

of consumption growth in post war U.S. data.

5.2

Results

Using the value of the parameters described above, we simulated 500 draws of series of length T = 100 each. Below we present some of the time series generated by simulating a single draw. In the top-left panel we show the steady state deviations of worker’s consumption and investment. The two variables move together because of equation (46). The variance of consumption (computed averaging over the 500 draws) matches the volatility found in post-war U.S. data and, as in the data, investment is more volatile than consumption. In the top-right panel we show the simulated series for investment and output. The model performs quite well in

23

generating an excess volatility in investment: the standard deviation of investment is indeed 3.1 times larger than the standard deviation of output. The corresponding number in the data is 4.26, as reported in Meh and Moran (2010). In the left-middle panel we report variables related to the production side of the economy. We observe that the interest rate is countercyclical, while real wages are pro-cyclical, as documented in King and Watson (1996). These movements simply reflect changes in marginal productivities along the cycle. In the right-middle panel we plot the evolution of credit contracts (bonds) purchased by workers and investment. As we can see, credit is more volatile than investment: the average ratio (out of the 500 simulations) of the respective standard deviations is 2.65. Even thought this number is larger than the one observed in the data, the model does a good job in capturing the direction of the inequality. Moreover, credit leads investment by one period. Worker's consumption and Investment

0.4

0.4

0.2

0.2 Cw

0.0

X

-0.2 -0.4

0.0

y

-0.2

X

-0.4 0

20

40

60

80

100

0

20

40

60

80

100

1.0

0.10 y

0.05 0.00 -0.05

0.5

r

0.0

w

-0.5

kW+1

X

-0.10 -1.0 0

20

40

60

80

0

100

0.5

20

40

60

80

100

0.5

0.0

0.0

r

-0.5

-0.5

q

0

20

40

60

80

qFB

100

q

0

20

40

60

80

100

Figure 3: Time Series generated simulating 100 periods . Our model is able to generate price volatility in the credit market consistent with the one observed in the data. The model generates (out of 500 simulations) an average ratio of the standard deviation of the price of bonds qt and output equal 24

to 8.05. This number compares with a value of 5.26, if we use the Moody’s BAA Corporate Bond index, which only includes higher rated corporate bonds. As we expect price variability to be higher for lower rated corporate bonds, the standard deviation of the price of all kind of bonds will be higher than the Moody’s index. This suggests that our results reflect the overall volatility in the prices of bonds, which has been difficult to match by previous works. It is also interesting to further discuss credit price volatility within the model. This is performed within the last two panels. In the left panel we plot deviations of the interest rate rt and of the price of bonds qt from their steady state values. It is obvious that the second variable is significantly more volatile that the first one. While the first variable is pinned down by technology, the second one is not, being very responsive to shocks in expectations. Indeed, out of the 500 simulations, we found that the price qt is almost 10 times more volatile than the interest rate rt . Finally, in the right panel we plot deviations of qt and of qtF B = 1/⌫(xt )⌫ 1 , that represents what would have been the price of one bond, if price was determined as in the first best environment. Note that, with an almost linear technology (⌫ close to 1), in the first best capital and consumption would be almost perfect substitutes, so that the price of bonds qtF B would be always close to 1, not changing along the cycle. This is what we can see in the last right-panel of Figure 3, where the price qt generated by the model is dramatically more volatile than qtF B , which does not fluctuate at all. Since all this excess volatility is due to the existence of limited commitment, this suggests this friction qualifies as an important mechanism in explaining the price volatility observed in credit markets.

6

Concluding Remarks

It is commonly accepted that credit market frictions are an important source of macroeconomic fluctuations. But what is the link between the two? And what is the driving factor of asset prices volatility? To answer these questions we have introduced a specific credit friction, limited commitment, in a macroeconomic general equilibrium model with production and investment in productive capital, where agents can trade bonds. This model is well suited to quantitatively address the consequences of limited commitment on macroeconomic and financial variables, in25

cluding asset prices. We show that this economy always displays one equilibrium where bonds are traded. Depending on the severity of the commitment problem, this can be the first best steady state, or a second best with or without collateral. Additionally, for some parameter configurations, a steady state with no bonds also exists. Furthermore, we show that with limited commitment, even in the absence of shocks to fundamentals, indeterminacy and stochastic fluctuations driven by self-fulfilling volatile expectations may emerge. This, as shown by our simulation results, yields credit and investment cycles that match those observed in the data. More importantly, the model generates volatility of the price of bonds consistent with real data. All this suggests that limited commitment may play an important role in explaining volatility in credit markets. It would be interesting to complement this work by studying the dynamic properties of the stationary equilibrium with collateral and by analyzing the global dynamics of our model. We leave this and other questions to further research.

7

Appendix

7.1

Proof of Lemma 3.2

Proof. Suppose by contradiction that k W = 0 and the no-default constraint is slack. Because the no-default constraint is slack, µ = 0, and from equation (9d) we have q =

1 ⌫x⌫

1

, so that from equation (2) we obtain r⌫x⌫

equation (9e) we have ⌘ = [1

1

 1/ . Moreover from

r⌫x⌫ 1 ]. Since k W = 0, (11) gives k E = 0. Because

we are considering stationary equilibria with production, K > 0, and from the law of motion of capital (12), K = (1 ↵)x⌫ , x must be positive. Therefore in (9b) we have ⌘ = 0. Because ⌘ = [1

r⌫x⌫ 1 ], and

1, it follows that 1

Combining this expression with the one obtained above, r⌫x⌫ 0=1

r⌫x

7.2

⌫ 1

1

/ > 0, which is a contradiction.

1

= 0.

 1/ , we have

Proof of Lemma 3.3

Proof. From equation (15) q = r⌫x⌫

1

r⌫x⌫

1

1 ⌫x⌫

1

. Combining this with equation (13) we obtain

= 1 . Finally, from (9e) we obtain ⌘ = (1 26

/ ) > 0. Hence, from (9b) we

obtain x + qk E = 0.

7.3

Proof of Lemma 3.4

Proof. From equations (13) and (15),

1 ⌫x⌫

1

= r. Replacing this in (18) we obtain

cE = rx⌫ (1

⌫) > 0

where the inequality follows from ⌫ < 1. The equality

= 1 is an immediate

consequence of cE > 0.

7.4

Proof of Proposition 3.5

Proof. Consider equation (21). Easily, the left-hand-side is monotone decreasing in x and the right-hand-side is monotone increasing x. Hence, if they cross, they cross only for one value of x. The existence of such an x is guaranteed by the fact that limx!1 x⌫ = +1, limz!0 f 0 (z) > 0 and limz!1 f 0 (z) < +1, which proofs existence.19 However, for this to be an equilibrium, we have to ensure that (7) is indeed slack. Let xF B be the value of x satisfying (21). For (7) to be slack, the following inequality must hold at the first best: cE 1

> (1

✓)x

Substituting equations (13), (15), and (18) in the expression above, we obtain 1

⌫ > (1 ) ⌫

(1 ==> ⌫ 

+ (1

✓)(1

✓) )

which concludes the proof. 19

We are using these properties instead of the Inada conditions because the former are satisfied by any CES production function, while the Inada conditions are not.

27

7.5

Proof of Proposition 3.7

Proof. Uniqueness of x is simple: the left-hand-side of (23) is monotonically decreasing, and the right-hand-side is increasing. Hence, if they are equal to each other, they can be equal only for a unique value of x. Since limx!1 x⌫ = +1, limz!0 f 0 (z) > 0 and limz!1 f 0 (z) < +1, there always exists a unique value of x solving (23). We call this xBN C . To guarantee that xBN C is the equilibrium investment when the no-default constraint binds, bonds are traded, and entrepreneurs do not save, we still have to ensure that (7) is indeed binding (µ binding (⌘

0) and that (6c) is

0). Replacing q = r from (13) into (9d), we obtain [1 r⌫x(⌫

µ=

1)

r⌫x(⌫ (1

1)

] ✓)

Replacing r from (23) into the equation above, we obtain µ=

⌫[ + (1 ✓)(1 ⌫[ + (1 ✓)(1 )]

)] (1

✓)

(48)

Replacing this value in (9e), and using again r from (23), we obtain ✓( ⌘=

)

⌫[ + (1

+⌫

h

✓)(1

Clearly, from (48), µ is positive only if

+ (1

✓)(1

)]

) (1

⌫[ + (1

✓)(1

i

✓)

(49)

)] < 0. Otherwise its

numerator would be positive, but the denominator would be negative. Hence, one condition that must hold for xBN C to be the equilibrium investment is ⌫

+ (1

✓)(1

)

The second condition for µ to be positive is that the denominator of (48) is negative, hence we must have ⌫

+ (1 ✓) + (1 ✓)(1

28

)

Finally, the third condition that has to be met, is ⌘ > 0. Given that the denominator in (49) is negative, its numerator has to be also negative, i.e. ⌫ Because

✓ + (1 ✓) = + (1 ✓)(1 ) +( + (1

)(1 ✓)(1

✓) < )

+( + (1

)(1 ✓)(1

+ (1 ✓) + (1 ✓)(1

✓) )

)

xBN C is the equilibrium investment if and only if +( + (1

)(1 ✓)(1

✓) )

⌫

+ (1

✓)(1

)

which concludes this proof. To see that investment is below the first best level, xF B > xBN C , remember that the first best equilibrium exists for all ⌫ 2 (0, 1]. This means we want to compare the value of x that solves (21) with the value of x that solves (23), guaranteeing that ⌫(✓)

⌫

⌫(✓), so that the equilibrium with a binding no-default constraint

with bonds and no collateral exists. Since the left-hand side of (21) and (23) is the same, it is enough to prove that the right-hand side of (23) exceeds the right-hand side of (21). Since this is true if ⌫ > ⌫(✓), when the equilibrium with a binding no default constraint bond and no collateral exists, we always have xF B > xBN C , which concludes the proof.

7.6

Proof of Proposition 3.8

Proof. In (24) the left-hand-side is monotone decreasing in x and the right-handside is monotone increasing in x. Hence if they are equal to each other, they can be equal only for one value of x. Since limx!1 x⌫ = +1, limx!1 x⌫ = +1, limz!0 f 0 (z) > 0 and limz!1 f 0 (z) < +1, there always exists a unique value of x solving (24). We call this xBC . For this to be an equilibrium, we have to ensure that entrepreneurs bond purchases k E is negative, and that entrepreneurs savings are positive, meaning x + qk E > 0. Using the binding no-default constraint (7) and entrepreneurs budget constraint

29

(6b), we obtain: (1

✓)x = V (k E , x) =

cE =

1

h n o r x⌫ + k E

1

x

rk E

i

Now replacing r from (24) and solving for k E we obtain x⌫

E

k =

1



+ (1 + (1

✓)(1 ✓)(

) ⌫ )

1

Easily k E is negative for all ⌫  1. Then the only condition that needs to be satisfied is x + qk E

0. This is the case if

x⌫

0 0, this is true only if the term in brackets is positive, which implies ⌫

+( + (1

)(1 ✓) (1

✓) )

which concludes the proof. The proof that investment is below the first best level, xBC < xF B , is similar to the proof in the end Appendix of 7.5.

7.7

Proof of Lemma 3.9

Proof. Suppose by contradiction that k W = k E = 0 and µ = 0. Since we are considering non-trivial stationary equilibria, ⌘ = 0 in (9b). From (9d) and (9e) we obtain q = 1/⌫x(⌫ 1

7.8

1)

and r = 1/[ ⌫x(⌫

1)

]. Replacing both in (2) we obtain

r/q = / > 1, a contradiction. Proof of Proposition 3.10

Proof. In (25) the left-hand-side is monotone decreasing in x and the right-handside is monotone increasing x: hence if they are equal to each other, they can be equal only for one value of x. Since limx!1 x⌫ = +1, limz!0 f 0 (z) > 0 and 30

limz!1 f 0 (z) < +1, there always exists a unique value of x solving (25). We call this xN B . For this to be an equilibrium, we have to ensure that bonds are not traded, i.e. q

r, and that the no-default constraint binds, i.e. µ

0.

Since ⌘ = 0, from equations (9e) and (9d) we obtain 1 r⌫x(⌫ 1) r⌫x(⌫ 1) 1 + ✓ ✓ q= r (⌫ 1) r⌫x 1+✓ µ=

µ is positive only if its numerator and its denominator are positive: it is impossible for both to be negative. Indeed, if the numerator is negative, the denominator is positive, and µ < 0. This gives us the following conditions: r⌫x(⌫

1 1

1)

r⌫x(⌫

✓

0 1)



1 + (1 ✓)(1 1 ✓ + (1 ✓)(1

) )

Notice that the first condition is always satisfied for any ⌫ < 1. Finally, we have to check that q r

r: ✓ r⌫x(⌫ 1)

r

1+✓

Replacing r from (25) we obtain ⌫

(1 ✓) + ✓ (1 ✓) + ✓

which is also satisfied for all ⌫  1. Therefore, the only condition that needs to be satisfied is

⌫>

1 + (1

which completes the proof. 31

✓ ✓)(1

)

7.9

Linearized first best dynamic system  1 s W W e ct+1 e ct = ⌫ + (⌫ (1 ⌫)(1 s) + ⌫ x et [ 1s 1 + ⌫(1 )]

e cW t = ⌫

7.10

JF B Matrix "

1

[ 1s

⌫ 1 + ⌫(1

1 s

(1 ⌫)(1 s)+ ⌫

JF B =

1 s

⌫ + (⌫

1)

(1 ⌫)(1 s)+ ⌫

1 s

1

(50)

1) x et

)]

1+⌫(1 ⌫

⌫ + (⌫

(51)

x et )

1)

1 s

1+⌫(1 ⌫

)

#

where s and , whose expressions are given below, denote respectively the income share of capital and the elasticity of substitution between capital and labor in production, both evaluated at the steady state under analysis: s= 1

7.11

f 0 (z)z f (z) f (z)f 00 (z)z = f 0 (z)[f (z) f 0 (z)z]

=

f 00 (z)z f 0 (z)(1 s)

Proof of Proposition 4.1

Proof. The determinant and the trace of the matrix JF B are given by the following expressions: DF B =

(1

TF B = 1 + Since 1

⌫ > 0, 1

⌫)(1 ✓

s) + ⌫

(52) ◆

1

s

s

0, we have

DF B > 0 TF B = 1 +

⌫+1

✓

1

s

⌫

⌫+1

1 s

1 + ⌫(1

⌫

◆

> 1 + DF B

32

1 s

)

+

1 + ⌫(1 ⌫

(1

⌫)(1

)

+ DF B

s) + ⌫

(53)

Hence the stationary equilibrium is a saddle (locally determinate).

7.12

Linearized second best dynamic system with bonds and no collateral and Matrix JBN C

2 4

h

h

1

1 s s

⇣

1 s s 1

+

⇣

1

(1 ✓)(1

+

(1 ✓)(1

)

⌘

)

+

⌘i

1

i

3" 5

✓

JBN C =

a1,1 a1,2

#

x et+1

"

s

=

"

e cW t+1

#

a2,1 a2,2

s + (1 ✓)(1

⌫1

1

1

0 ⌫

+ (1 ✓)(1

)

(1

)

✓)

#"

where a1,1 = ✓ h

a1,2 = a2,1 = a2,2 =

1

✓

+ (1 ✓

)(1

1

+ (1

0

1 + @1

=

7.13

n

(1

(1

i

✓) ⌫

(1

s)[ + (1 s

h s) 1

✓@

)(1

s) [ + (1 )(1 s h (1 s) + (1 + s

✓)(1

✓)

0

!

✓)]

✓)]

◆ h

)(1

(1

✓)

(1

s)

h

+ (1

◆ + (1

)(1

i1

s)

A

(1

i h )✓ + s + (1

h

i ✓) ⌫

)(1

i

✓) ⌫

(1

+ (1

)(1

✓)(1

s

1

+ A

✓)

!

i ✓) ⌫

)

io

+

!

>0

Proof of Proposition 4.3

Proof. The determinant, DBN C , and the trace, TBN C , of matrix JBCN are given by the following expressions: DBN C =

s h ⇣ v + (1

✓)(1

33

)

⌘

(1

✓)

i

(54)

e cW t x et

#

TBN C =

0

(

s

✓+@

(1

0

@1

+

s)

(1

⇣

1

s)

+

(1

+ s ⇣

1

+

)(1 ✓)

(1

⌘

1"

)(1 ✓)

1A

⌘1

✓

(1

A 1+

s

✓) (1

⌫[ + (1 s)[ + (1

)(1 )(1

✓)] ✓)]⌫

#

◆

(55)

Lemma 7.1. The trace and the determinant in equations (55) and (54) satisfy DBN C < TBN C

1.

0 Proof. Because TBN C ( ) < 0, we know that for all

TBN C ( ) > lim TBN C ( ) = TBN C1 !1 h + (1 = DBN C + 1 + h ⇣ 1

)(1 (1

✓) ✓)

ih

2 (0, +1),

(1

(1

> DBN C + 1

) + (1 s)[1 ⌘ s)✓ + (1 ✓)(1

i (1 )✓] (1 ⌫) ⇣ ⌘i s) 1 (1 )✓

Both the numerator and the denominator of the fraction above are positive, hence DBN C


> min{ (1

1

✓), s}, for all ⌫ 2 [⌫, ⌫], we have

Proof. Easily, DBN C is increasing in ⌫. For ⌫ = ⌫, we have

DBN C (⌫) = = Since

s[

1+

> 0, if

(1 n

✓)] (1

(1

✓)(1

s)[1

(1

)✓] + [s + (1

✓), we have DBN C (⌫) > 0 >

34

✓)(1

s

1. If instead (1

o )] + s[

(1

✓) > , since

✓)]

we assumed

DBN C (⌫)

✓), s}, it must be that

> min{ (1

> |{z}

n

1+

1

(1

s)(1

> s. Then, we obtain

)✓ + (1

< (1 ✓)

= if s

(1

(1

s+1

)

s(

"

)+✓

1+ s)(1

)

DBN C (⌫) >

1+

=

1+

If instead s

(1

(1

s+1

[1

(1

s)(1

1+ n

DBN C (⌫)

)

s(

s)(1

)

(1

(1

✓)(1

s

=

1+ n

1+ n

Therefore, when

(1

(1

)+

)] + s

(1

)

o

s[ (1

s)(1

)

✓)

(1

]

s

)

#

(1

s+1 (1 (1

s)[1 s)[1

> min{s, (1 DBN C

s

(1

s)(1

)

(1

s

)

#

1

)

(1 (1

s(

)

)✓] + [s + (1 s+1

s(1 ✓)(1

"

0, we obtain

)

s)[1 "

✓)(1

>

s

s

|{z}

s

s

) < 0, we obtain

s

(1

✓)(1

)

+s

)✓] + [s + (1

✓)(1 #

s

✓)(1

)]

s

)]

)(s + ) )✓] + [s + (1

✓) }, it follows that DBN C (⌫) >

✓)(1

s

o

)]

o

o>

1

1

and the result follows. Combining the two lemmas above, we obtain DBN C >

1, and DBN C