Changing Credit Limits,
Changing Business Cycles
Henrik Jensen
University of Copenhagen and CEPR
ySøren Hove Ravn
University of Copenhagen
Emiliano Santoro
University of Copenhagen
First version: September 2013
This version: February 2015
Abstract
In the last decades, capital markets across the industrialized world have undergone massive deregulation, involving increases in the loan-to-value (LTV) ratios of house-holds and …rms. We study the business-cycle implications of this phenomenon in a dynamic general equilibrium model with multiple credit-constrained agents. Start-ing from low LTV ratios, a progressive relaxation of credit constraints leads to both higher macroeconomic volatility and stronger comovement between debt and real variables. This pattern reverses at LTV ratios not far from those currently ob-served in many advanced economies, since credit constraints become non-binding more often. As expansionary shocks may make credit constraints non-binding, while contractionary shocks cannot, recessions become deeper than expansions. The non-monotonic relationship between credit market conditions and macroeconomic ‡uc-tuations poses a serious challenge for regulatory and macroprudential policies.
Keywords: Occasionally non-binding credit constraints, business cycles, capital-market liberalization, capital-capital-market regulation.
JEL: E32, E44.
We thank Fernando Broner, Gauti Eggertsson, Martin Ellison, Luca Guerrieri, Tom Holden, Dirk Niepelt, Ivan Petrella, Gitte Yding Salmansen, Andrea Tambalotti, and seminar participants at Bank of England, Copenhagen Business School, Danmarks Nationalbank, DIW Berlin, University of Copenhagen, University of Lancaster, University of Oxford, the “Workshop on Macroeconomics, Financial Frictions and Asset Prices” at the University of Pavia, the 2014 SCE Conference on “Computing in Economics and Finance” in Oslo, the “8th Nordic Summer Symposium in Macroeconomics” in Laugarvatn, and the “9th Anglo-French-Italian Workshop” at the Université of Aix-en-Marseille for helpful comments and suggestions. The usual disclaimer applies.
yCorresponding author. Postal address: Department of Economics, University of Copenhagen, Øster
1
Introduction
Credit ‡ows are crucial for the functioning of an economy where inhabitants want to alter the pro…le of purchases over time. Consumers may want to smooth consumption and …nance their purchases of durable goods. Likewise, …rms may desire to obtain funds for investment projects that only pay o¤ later. Such intertemporal trades are typically plagued by informational problems leading to a multitude of …nancial market imperfections. One implication is that households and …rms become credit constrained, and often have to provide collateral to obtain loans. Under such circumstances, the degree to which credit constraints bind is in‡uential for the economy’s response to various disturbances. The main purpose of this paper is to study the business-cycle properties of a conventional DSGE model with credit constraints á la Kiyotaki and Moore (1997) under di¤erent credit conditions. Since the limits to credit acquisition are largely determined by how much an agent can borrow against his collateral— the loan-to-value (LTV) ratio— we model changes in credit conditions as changing LTV ratios.
Capital markets have undergone massive deregulation across the industrialized world in the past decades. Figure 1 shows loans relative to assets for households and …rms in the US for the post-war period. The observed secular increases are consistent with increased credit availability through higher LTV ratios.1 The literature on credit limits and business cycles, however, is largely silent on the business cycle implications of substantial changes in the LTV ratios. We therefore contribute to …lling this void, as much recent policy debate following the …nancial crisis has focused on limiting credit availability in general, or imposing countercyclical credit measures, e.g., IMF (2013).
We incorporate collateral constraints into a real business cycle model with heteroge-neous agents in the vein of Iacoviello (2005), Liu et al. (2013), Justiniano et al. (2015);
inter alia. A durable good, land, is used for both consumption purposes and production.
In addition, land serves as collateral for “impatient”, credit-constrained households as well as for entrepreneurs. Entrepreneurs also invest in physical capital, which is used as collat-eral. The lenders in the economy are “patient”, non-constrained households. In contrast to most of the existing literature, we explore the implications of credit constraints not 1As we discuss in Appendix A, the aggregate ratios of loans to assets reported in Figure 1 are likely to
understate the actual LTV requirements faced by the marginal borrower. However, while alternative mea-sures may yield higherlevels of LTV ratios, they give rise to the same conclusions about the development over time of these ratios.
1950 1960 1970 1980 1990 2000 20100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L o a n to A sse t R a tio Households 1950 1960 1970 1980 1990 2000 20100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 L o a n to A sse t R a tio Firms
Figure 1: The ratio of liabilities to assets for households and …rms in the United States, 1949Q4–2014Q3.
Source: See Appendix A.
binding at all points in time.
Our main …ndings show that macroeconomic volatility and comovement between debt and real variables display a hump-shaped pattern in response to changes in the LTV ratio. Over most of the range of LTV ratios, higher credit limits allow …nancially constrained agents to succumb to their relative impatience and engage in debt-…nanced consumption and investment. This reinforces the macroeconomic repercussions of shocks that a¤ect the borrowing capacity of these agents. As a result, output ‡uctuations become larger and credit issuance becomes more procyclical with a deepening of …nancial markets.2
Eventu-ally, a further increase in the LTV ratio reverses this pattern— volatility and comovement, however, never return to the levels observed at low credit limits. Higher LTV ratios increase the likelihood that credit constraints become non-binding in the face of expan-sionary shocks. In such cases, the consumption and investment decisions of households and entrepreneurs are delinked from changes in the value of the collateral assets. As a result, positive shocks that bring about an expansion have a weaker impact on the economy than similar-sized negative shocks, leading to lower volatility and a negatively skewed business cycle.
2These …ndings contrast with a strand of the literature that has attributed the observed drop in output
volatility and in the comovement between real activity and debt since the mid-1980s to the wave of …nancial liberalization unraveling over the same period; see, e.g., Jermann and Quadrini (2009) and Campbell and Hercowitz (2009).
The non-monotonic relationship between credit-market conditions and macroeconomic volatility creates a trade-o¤ for macroprudential policy. Lower credit limits may succeed in dampening the asset price sensitivity of intramarginal borrowers, who remain credit constrained before and after the intervention. Yet, on the margin, lower credit limits will increase the frequency by which credit constraints bind, subjecting borrowers more heavily to ‡uctuations in credit availability. For values of the LTV ratio in line with those currently observed in many advanced economies, the latter e¤ect may well dominate, in which case a tightening of credit conditions will have the opposite e¤ect on output volatility than intended.
A key feature of our model is that technology shocks are substantially ampli…ed by the presence of collateral constraints. This is noteworthy, since previous studies have reported that collateral constraints have little or no role in amplifying and propagating shocks to …rm productivity, see, e.g., Cordoba and Ripoll (2004) and Liuet al. (2013). The ampli…cation relies on two main features. The …rst is the presence of land in the production technology, which ensures that a positive technology shock drives up the current and future marginal productivity of land.3 The second is the presence of two types of
credit-constrained agents— impatient households and entrepreneurs— whose borrowing activity features a strategic complementarity: After a positive technology shock, the increase in the marginal productivity of land induces entrepreneurs to demand more of it. The resulting upward pressure on the land price leads to an increase in the value of both impatient households’and entrepreneurs’collateral assets, causing an endogenous relaxation of their credit constraints. Both agents therefore demand more land, causing further upward movements in expected future credit availability, and so on. This two-sided externality, which is stronger at relatively higher LTV ratios, ensures that more land ends up in the hands of the entrepreneurs, and thus in productive use.
At the same time, we show that shocks to land demand play only a minor role for output ‡uctuations, although they remain the main drivers of land prices. This contrasts
with the …ndings of Liu et al. (2013), where entrepreneurial collateralized borrowing
ampli…es shocks to patient households’land demand. In their setup these shocks emerge as a key driver of investment and, in turn, output ‡uctuations. In our model, instead, 3Without land in the production function entrepreneurs would optimally hold zero land, so that their
shocks to land demand shift both patient and impatient households’preferences, so that land tends to be ‘tied up’away from productive use. This greatly reduces the impact of this shock on output. To that e¤ect, the presence of a …nancially constrained household is particularly important, as a positive shock to impatient households’preference for land services reinforces their attitude to accumulate land in order to access credit. Finally, it is important to stress that …nancial shocks in the form of exogenous variations in the LTV ratios faced by households and entrepreneurs account for a substantial share of output ‡uctuations under high LTV ratios in our setup. This is in line with the evidence of Jermann and Quadrini (2012), who report that …nancial shocks in the form of changes to the recovery ratio of assets in liquidation are important drivers of output ‡uctuations.
Our work is related to Guerrieri and Iacoviello (2014), who also exploit the asymmetric implications of occasionally binding credit constraints. They demonstrate that the macro-economic sensitivity to house price changes is much smaller during booms— when house prices are high and collateral constraints therefore likely to be non-binding— than during recessions, when credit constraints typically bind. In the model of Guerrieri and Iacoviello (2014) only households are credit constrained, as they focus on the role of the housing market in the US, whereas we consider a setup with credit constrained households and …rms. In two recent contributions, Justiniano et al. (2014, 2015) have shown that higher credit limits have a modest impact on the macroeconomy, and thus are unlikely to explain the pre-crisis boom in the US economy. However, they only consider increases in the LTV ratio from an already high level (above0:8), in which case our model also produces limited changes in output volatility.4
The rest of the paper is organized as follows. Section 2 presents our model, and the equilibrium and its computation are described in Section 3. Section 4 contains the presentation and discussion of our results. Finally, Section 5 concludes and o¤ers some policy implications as well as directions for further research.
4Walentin (2014) studies an increase in the LTV ratio from0:85to0:95in a model featuring
2
The model
We consider a real business cycle model with heterogeneous agents and credit limits in the vein of Iacoviello (2005), Liu et al. (2013), Justiniano et al. (2015); inter alia. The economy is populated by entrepreneurs and two types of households. Agents are di¤er-entiated by their discount factors. So-called patient households have the highest discount factor, e¤ectively making them lenders in the economy. Impatient households and entre-preneurs, instead have lower discount factors, and can only borrow up to some proportion of the present value of their assets. Both patient and impatient households work, consume non-durable goods and (durable) land, where the latter can be interpreted as related to housing services. Entrepreneurs only consume non-durable goods, and accumulate both land and physical capital, which they rent to producers. These are modelled by a perfectly competitive sector, where …rms of unit mass combine labor from both types of households as well as capital and land from entrepreneurs, so as to produce non-durable consumption goods and new capital goods. Agents are in total of unit mass, with impatient households and entrepreneurs being of mass 0< nI <1 and 0< nE <1, respectively.
2.1
Patient and impatient households
The preferences of the households are de…ned over non-durable consumption,Cti, the stock of land,Hi
t, and the fraction of time devoted to labor,Nti, wherei2 fP; Igrefers to patient
and impatient households, respectively. Household i maximizes
E0 ( 1 X t=0 i t 1 1 i C Cti 1 iC + "t 1 i H Hti 1 iH + i 1 i N 1 Nti 1 iN ) ; i >0; (1) where "t is a land-preference shock satisfying
log"t= log"+ "(log"t 1 log") +ut; 0< " <1; (2)
where " > 0 denotes the steady-state value of "t, and where ut N(0; 2"). Moreover,
0 < i < 1 is the discount factor, i
C > 0, iH > 0 and iN > 0 are the coe¢ cients
of relative risk aversion pertaining to non-durable consumption, land services and leisure, respectively. Households’di¤erent impatience is captured by the assumption that P > I.
This ensures that, in steady state, patient and impatient households act as lenders and borrowers, respectively; cf. Woodford (1986). Utility maximization is subject to the sequence of budget constraints
Cti +Qt Hti Hti 1 +Rt 1Bti 1 =Bti+WtiNti; (3)
where Bi
t is the stock of one-period debt held at the end of period t, Rt is the gross real
interest rate on debt, Qt is the price of land in units of consumption goods, and Wti is
the real wage. Moreover, impatient households are subject to a collateral constraint on borrowing:
BtI st
EtfQt+1gHtI
Rt
; (4)
which states that the maximum borrowable resources equal a fraction of the expected present value of durable goods holdings at the end of period t. This constraint can be rationalized in terms of limited enforcement; cf. Kiyotaki and Moore (1997) and Iacoviello (2005). Lenders are assumed to pay a cost (1 st)EtfQt+1gHtI in period t + 1 if they
are to repossess the collateral in case of default; hence, they will not lend more than
stEtfQt+1gHtI=Rt in period t.
The term st, the loan-to-value ratio, is assumed to satisfy
logst= logs+ s(logst 1 logs) +vt; 0< s<1; (5)
wherevt N(0; 2s). As our aim is to examine the implications of institutional changes in
credit conditions, including regulatory measures, we interprets, which denotes the steady-state loan-to-value (LTV) ratio, as a proxy for the average stance of credit availability.
Patient households’ optimal behavior is described by the standard …rst-order condi-tions: CtP P C = Pt ; (6) P 1 NtP P N = Pt WtP; (7) P t = P RtEt Pt+1 ; (8)
Qt ="t HP t P H P t + PEt ( P t+1 P t Qt+1 ) ; (9)
where Pt is the multiplier associated with (3) for i = P. Similarly, optimal behavior of impatient households is described by
CtI I C = It; (10) I 1 NI t I N = ItWtI; (11) I t I t = I RtEt It+1 ; (12) Qt ="t HI t I H I t + IEt ( I t+1 I t Qt+1 ) +st It t EtfQt+1g Rt ; (13)
where It is the multiplier associated with (3) fori=I, and I
t is the multiplier associated
with (4). Additionally, the complementary slackness condition
I t B I t st EtfQt+1gHtI Rt = 0; (14)
must hold along with I
t 0 and (4).
2.2
Entrepreneurs and …rms
The representative entrepreneur has preferences de…ned over non-durables only (cf. Ia-coviello, 2005; Liu et al., 2013), and maximizes
E0 ( 1 X t=0 E t 1 1 E C CtE 1 E C ) ; EC >0; (15) where CE
t is non-durable consumption, and where we assume that
E < P. This ensures
that the entrepreneurs are borrowers in the steady state. Their relevant sequences of budget constraints are
CtE+It+Qt HtE HtE1 +Rt 1BtE1 =BtE +rKt 1Kt 1+rtH1HtE1; (16)
where It denotes investment in physical capital, Kt 1 is the physical capital stock rented
rK
t 1 and rHt 1 are the rental rates on capital and land, respectively. Capital accumulation
is given by the law of motion
Kt= (1 )Kt 1+ " 1 2 It It 1 1 2# It; 1> >0; >0; (17)
where we have assumed quadratic investment adjustment costs.
Like impatient households, entrepreneurs are credit constrained, and they borrow using capital and land as collateral:
BtE stEt QK t+1Kt+Qt+1HtE Rt ; (18) where QK
t denotes the price of installed capital in consumption units. For simplicity, we
assume that households and entrepreneurs are subject to common credit limits.5
The optimal behavior of the entrepreneurs is characterized by the …rst-order conditions
CtE E C = Et ; (19) E t E t = E RtEt Et+1 ; (20) E t + E t " 1 2 It It 1 1 2 It It 1 It It 1 1 # = EEt ( E t+1 It+1 It 2 1 It+1 It ) ; (21) E t = ErK t Et Et+1 + E(1 )E t Et+1 + E t st Et QKt+1 Rt ; (22) Qt = ErtHEt ( E t+1 E t ) + EEt ( E t+1 E t Qt+1 ) +st E t E t EtfQt+1g Rt ; (23) where Et , E t and E
t are the multipliers associated with (16), (18) and (17), respectively.
Moreover, E t B E t stEt QKt+1Kt+Qt+1HtE Rt = 0; (24) 5The ratios of loans to assets in Figure 1 do not suggest large di¤erences between households and
…rms. In Iacoviello (2005), the LTV ratio faced by entrepreneurs (0.89) is much higher than that faced by impatient households (0.55), while the opposite is the case in Gerali et al. (2010), who set 0.35 for entrepreneurs and 0.7 for households. In sum, in lack of conclusive evidence that LTV ratios faced by …rms are systematically higher or lower than those faced by households, we assume that they are equal.
holds along with E
t 0 and (18). Finally, the de…nition ofQKt implies that
QKt = Et = Et: (25)
Firms operate competitively under a constant-returns-to-scale technology. They rent capital and land from entrepreneurs and hire labor from both types of households, so as to maximize pro…ts. The production technology for output, Yt, is given by
Yt=At
h
NtP NtI 1 i h HtE1 Kt1 1 i
1
; 0< ; ; <1; (26)
with total factor productivity At evolving according to
logAt = logA+ A(logAt 1 logA) +zt; 0< A <1; (27)
where A > 0 is the steady-state value of At, andzt N(0; 2A). Optimal factor-demand
relations follow from the …rst-order conditions:6
Yt=NtP = WtP;
(1 ) Yt=NtI = WtI;
(1 ) (1 )EtfYt+1g=Kt = rtK;
(1 ) EtfYt+1g=HtE = rtH:
3
Equilibrium
We consider a competitive equilibrium where the markets for labor, capital and land for production all clear. These conditions can be incorporated conveniently by rewriting the 6The assumption of imperfect substitutability between labor types follows Iacoviello (2005) and
Justini-anoet al. (2015), among others. Iacoviello and Neri (2010) note that perfect substitutability complicates
…rms’…rst-order conditions to account for the di¤erent masses of supplied factors:
nYt= (1 nI nE)NtP = WtP; (28)
(1 ) nYt= nINtI = WtI; (29)
(1 ) (1 )nEtfYt+1g=(nEKt) = rtK; (30)
(1 ) nEtfYt+1g= nEHtE = rtH: (31)
where n (1 nI nE) (nI)1 n1E . Moreover, as the supply of land is held …xed
at H, equilibrium in the the market for land implies that
H = (1 nI nE)HtP +nIHtI+nEHtE: (32)
Also, the economy-wide net …nancial position is zero, such that
(1 nI nE)BtP +nIBtI+nEBtE = 0: (33)
Finally, the aggregate resource constraint can be written as
nYt= (1 nI nE)CtP +nICtI+nECtE+nEIt: (34)
An equilibrium is sequences of quantities and prices, fYt, CtP, CtI, CtE, It, HtP, HtI, HtE, Kt, NtP, NtI, BtP, BtI, BtEg1t=0 and f P t, I t, E t , It, Et, E t , rtK, rtH, QtK, Qt, WtP, WtI,
Rtg1t=0, respectively, which conditional on sequences of shocks fAt; "t; stg1t=0 and initial
conditions, satisfy the optimality conditions [(6), (7), (8), (9), (10), (11), (12), (13), (19), (20), (21), (22), (23), and (25)], the budget and credit constraints [(3) for i = P; I, (4), and (18)], as well as the technological constraints and market-clearing conditions [(17), (26), (28), (29), (30), (31) (32), (33), and (34)].
3.1
Solution method
In Appendix B we detail the steady state of the model, while Appendix C describes the log-linearized version, which is solved numerically. Due to the assumptions about the discount factors, P < I and P < E, both collateral constraints are binding in steady
state. This follows as (20) pins down the steady-state real interest rate as R = 1= P. As impatient households and entrepreneurs all have a higher subjective real rate of interest, their consumption levels can only be equalized across time when their collateral constraints bind. However, the optimal level of debt of one or both agents may fall short of the credit limit when the model is not at its steady state (say, in case of a large favorable shock), in which case the collateral constraint will be non-binding. In other words, our model features occasionally non-binding constraints, and thus non-linearities.
To account for this, we explicitly treat the collateral constraints as inequalities, and include the complementary slackness conditions, (14) and (24), in the model. In practice, we follow the approach of Holden and Paetz (2012), who develop a solution method for log-linearized DSGE models featuring inequalities. Building on Laséen and Svensson (2011), the central idea is to introduce a set of “shadow price shocks”, which ensures that each of the slackness conditions is satis…ed in each period. If the conditions are violated, the shadow price shocks take on values exactly large enough to make the bounded variables, i.e. the debt levels of credit-constrained agents, equal to their (temporarily) unconstrained value. If the borrowing constraints are already binding, the shadow price shocks are zero. To ensure compatibility with rational expectations, the shocks are added to the model as “news shocks” in the sense of Schmitt-Grohé and Uribe (2012), i.e., they are fully anticipated. For …rst-order perturbations, our solution method yields similar simulated moments as the approach of Guerrieri and Iacoviello (2014); see, Holden and Paetz (2012). We present the details of the solution method in Appendix D.
3.2
Parameterization
We interpret one period as a quarter. The model is calibrated so that, in the steady state, it broadly matches a set of “big ratios”for the US economy as reported, e.g., by Liuet al. (2013). These include the ratios of capital and housing to output, and the share of time devoted to labor. As we describe in the following, this implies a set of relatively standard parameter values.
We let the steady-state LTV ratios faced by households and entrepreneurs, s, be in the range [0.01,0.9] and report standard deviations, correlation coe¢ cients, etc. for 19
di¤erent values within this range.7 In this way, we obtain a comprehensive picture of
the e¤ects of di¤erent LTV ratios on the macroeconomy. Moreover, this approach allows us to uncover potential non-monotonicities and to explore the e¤ects of credit constraints becoming non-binding more often under high average LTV ratios. When we report impulse responses, however, we do so only for two values of s. The …rst is a “high” LTV regime, where s = 0:7. The second, a “low” LTV regime, has s = 0:35.8 The calibration of the
model is in line with s = 0:7, as the ratios reported by Liu et al. (2013) are computed for the sample 1987–2007. These values, however, are not markedly di¤erent when we set
s = 0:35.
We assume that agents are of identical mass; nI = nE = 1=3. As noted, impatient
households and entrepreneurs have a lower discount factor than patient households. We
therefore set I = E = 0:97 and P = 0:99. As for the curvature parameters in the
utility functions, we set i
C = 1 for i=fP; I; Eg, and similarly iH = 1 for i =fP; Ig, so
that all households have log utility in land services and non-durable consumption. The Frisch elasticity of labor supply is given by the inverse of i
N times the steady-state ratio
of leisure to work. Calibrating the latter to around 3 for both types of households, a Frisch elasticity of labor supply of 1=3 implies i
N = 9, i = fP; Ig. We use i = 0:27 for i=fP; Ig, which implies, as desired, that patient households work about 1/4 of their time in steady state, and impatient households slightly more. We calibrate the model so as to obtain a steady-state ratio of residential land to output around 1:45, and of commercial land to output around 0:65, both at the annual level, following values reported by Liu et al. (2013). This requires a value of "= 0:085.
Regarding the parameters describing the technology of the model, we set = 0:7,
implying a non-labor share in the production function close to 1=3. We set the labor income share of patient households to = 0:7, which is in the range of available estimates. For instance, Iacoviello (2005) obtains an estimate of 0:64by matching impulse responses from his model to those from a VAR, while Iacoviello and Neri (2010) …nd a value of 0:79
using Bayesian estimation. The parameter , which multiplied by(1 )measures land’s
share of inputs, is set to 0:13, somewhat higher than the estimated value from Liu et al.
forst, the actual LTV ratios stay inside the interval [0,1] with 95 pct. probability. We consider negative
LTV ratios to be uninteresting, and while LTV ratios above 1 may sometimes occur empirically, we …nd it hard to reconcile with the idea of limited contract enforcement which we follow in this paper.
8These values are within the range of values typically used in the literature; e.g., Mendoza (2010)
reports 0:2–0:3, Calzaet al. (2013) use 0:6, Liu et al. (2013) report 0:75, while Justinianoet al. (2014) set a value of 0:8.
(2013). We assume a capital depreciation rate of = 0:035. The implied annual ratio of capital to output is around1:15, as in Liuet al. (2013). As to the investment adjustment cost parameter, , the empirical estimates range from nearly to zero in Liuet al. (2013) to above 26 in Christiano et al. (2010) in estimated general equilibrium models. We choose an intermediate value of = 4.
For the technology shock, we choose values similar to those applied in most of the real business cycle literature, A = 0:97 and A = 0:005 (see., e.g., Mandelman et al.,
2011). For the land demand shock, we set " = 0:96and " = 0:06, which is close to the
values used by Iacoviello and Neri (2010) and Liu et al. (2013). Finally, for the credit limit shock, we set the persistence parameter s = 0:98, in line with estimated coe¢ cients
from univariate regressions of the LTV-series displayed in the Introduction. We then calibrate s to obtain a standard deviation of the process forlogst logs of around0:06,
as estimated by Liu et al. (2013). This implies s = 0:0119. All parameter values are
shown in Appendix E.
4
Business cycles and credit limits
To examine the business cycle properties of the model, we …rst focus on output volatility. According to Figure 2 this increases in the average LTV ratio, at least up to s 0:8, a value that we have argued to be in line with a …nancially liberalized economy. In fact, this credit limit is close to the typical LTV ratios on mortgage loans in a number of advanced countries, as reported, e.g., by Calza et al. (2013). Initially, an increase in the LTV ratio implies that credit-constrained agents experience a larger increase in their borrowing capacity in the face of a given increase in asset prices, and vice versa. With binding credit constraints, this translates into larger ‡uctuations in debt-…nanced consumption and investment activities by impatient households and entrepreneurs. In other words, these agents fully exploit the changes in their borrowing capacity to satisfy their relative impatience. In turn, this implies an increase in output volatility, as well as a stronger comovement between credit and economic activity. We elaborate further on the latter aspect in subsection 4.5.
Observe that our focus on the possibility of non-binding borrowing constraints becomes particularly important when the average LTV ratio approaches the upper bound of its
sup-0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 S tan da rd d ev iat ion of ou tp ut s
Figure 2: Standard deviation of output for di¤erent LTV ratios.
Note: Numbers are median values from 501 stochastic model simulations of 2000 periods. All time series used to produce business cycle statistics have been preliminarily HP-…ltered, setting = 1600.
port. In fact, beyond s 0:8output volatility falls in the LTV, but remains substantially higher than for economies with low LTV ratios.9 This drop in volatility is due to an in-crease in the frequency of episodes in which …nancial constraints are slack. As illustrated in Figure 3, the entrepreneur …nds himself unconstrained as much as 50% of the time at very high LTV ratios; impatient households, however, only experience this rarely. In con-trast to models with always binding constraints, we thus portray looser credit conditions as causing higher borrowing ability with the same collateral, and also as increasing the probability of not facing any (binding) credit constraints. Under such circumstances col-lateralized borrowing no longer acts as an ampli…er of shocks to the economy. Subsection 4.4 digs deeper into this point.
We now examine how the model’s di¤erent shocks are propagated under di¤erent credit limits. Figure 4 shows the response of output to each of the three shocks under the low and high LTV regime, respectively. To appreciate how the e¤ect of the shocks may be ampli…ed through endogenous collateral values, we also illustrate the e¤ects of each shock in an alternative version of the model, in which both types of borrowers face a …xed credit 9By contrast, Mendicino (2012) …nds that the ampli…cation of technology shocks through collateral
constraints increases in the credit limit, but tends to vanish as the LTV ratio approaches one. This happens as fully e¢ cient debt enforcement is possible in her simple Kiyotaki and Moore (1997) economy.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 10 20 30 40 50 60 70 80 90 100 s F re q u e n c y o f p e ri o d s w ith n o n -b in d in g c re d it c o n s tr a in t, p c t. Impatient Households Entrepreneurs
Figure 3: Frequency of episodes of non-binding constraints for each agent.
Note: See the notes to Figure 2.
limit equal to their steady-state level of debt.
4.1
Technology shocks and ampli…cation
Figure 4 shows that endogenous credit limits are an important source of ampli…cation of technology shocks, especially at high LTV ratios, as compared with the alternative economy where collateral e¤ects are shut o¤.10 This result contrasts with recent business
cycle literature that …nds little or no ampli…cation of shocks to technology through credit constraints, cf. Kocherlakota (2000), Cordoba and Ripoll (2004), Liu et al. (2013), inter alia. The key mechanism behind their lack of ampli…cation is that productivity shocks move future dividends and the riskless rate of interest in the same direction, so that the discounted value of the collateral assets exerts a muted impact on collateral constraints. Our set up mainly di¤ers from these studies by featuring two di¤erent types of credit-constrained agents. Entrepreneurs invest in land and capital goods to be employed for production purposes, while impatient households bene…t from the stream of utility accruing from land holdings. Therefore, we combine the complementarity between land and physical capital in the production function with the e¤ect of strategic complementarities stemming from the demand for land from both types of constrained agents. This demand channel 10See also Figures F.1 and F.2 in Appendix F, which show the response of six key variables to a
0 5 10 15 20 3
4 5 6
x 10-3 Low averag e LTV ratio
0 5 10 15 20
3 4 5 6
x 10-3 Hig h average LTV ratio
0 5 10 15 20
-1 0 1 2x 10
-3 Low averag e LTV ratio
0 5 10 15 20
-1 0 1 2x 10
-3 Hig h average LTV ratio
0 5 10 15 20 0 1 2 3 4 5x 10
-3 Low averag e LTV ratio
0 5 10 15 20 0 1 2 3 4 5x 10
-3 Hig h average LTV ratio
Model
Fixed credit limits
Technology shock
Land demand shock
Credit limit shock Technology shock
Land demand shock
Credit limit shock
Figure 4: Impulse responses of output to a one standard deviation positive technology shock (row 1), a positive land-demand shock (row 2), and a positive credit-limit shock (row 3) for two di¤erent LTV ratios.
is in‡uenced by credit conditions, and opens up the route to a relevant ampli…cation of technology shocks.
To provide an understanding of this result, some analytical insights are helpful. Con-sider …rst patient households’Euler equation for land (9), which is solved forward to yield:
P t Qt=Et ( 1 X i=0 P i" t+i HtP+i P H ) t: (35)
Now consider the case where there are no land demand shocks; e.g., "t = ", 8 t. Since
land does not depreciate, HP
t is e¤ectively an “idealized durable” in the sense of Barsky
et al. (2007). This means that the intertemporal elasticity of substitution in land demand
is close to in…nity. Any short-term movements in HP
t a¤ect the right-hand side of (35)
relatively little, as P is close to one. Hence, we can make the approximation:
P
This condition shows that movements in land prices (in absence of land-demand shocks) mirror movements in Pt , i.e., patient households’marginal utility of consumption. Hence, by (6) and (36), changes in the credit conditions will only a¤ect land-price dynamics to the extent they a¤ect unconstrained households’consumption dynamics:
Qt CtP
P
C : (37)
We then turn to impatient households, whose Euler equation for land demand, (13), is a standard downward-sloping curve, conditional on all other variables. It also follows from (13) that steady-state credit conditions a¤ect impatient households’ land demand, as the expected future price of collateral in present value has greater impact on current
demand the higher is s. This is also the case for the entrepreneurs, to whom we now
turn. Firms’ equilibrium demand for land is given by (31), which equates the marginal productivity of land with the rental rate. Using this condition to eliminate the rental rate in entrepreneurs’Euler equation (23) yields (for the case n=nE):
Qt= EEt ( E t+1 E t Qt+1 ) + E (1 ) EtfYt+1g=HtE Et ( E t+1 E t ) + Etst EtfQt+1g E t Rt : (38)
This is a downward-sloping function, as the marginal product of land decreases in HE
t .
Figure 5 sketches the market for land when hit by a positive technology shock, At
A > 0. The horizontal schedules represent (37), the patient households’ demand. The
downward-sloping schedules to the left represent (38), the entrepreneurs’demand. Finally, the downward-sloping schedules to the right represent (38) and (13) in combination, i.e., entrepreneurs’and impatient households’demand. The steady-state holdings of land are indicated by the points “a” (since the stock of land is …xed, patient households’ land holdings are H HtE HtI).
When a positive technology shock hits, it has a direct e¤ect on the demand of entrepre-neurs because the marginal product of land increases. This is depicted as a movement of demand curves to the right. The increase in production possibilities will increase patient households’consumption, leading to an increase in land prices, cf. (37). This induces a relaxation of entrepreneurs’credit constraints and allows them to demand more land. One can represent this by a movement to the points “b”, where land holdings by entrepreneurs
Figure 5: Land prices and land holdings by di¤erent agents following a positive technology shock. The solid line represents the steady state. The dashed line represents movements in entrepreneurs’ and …rms’ land demand without taking impatient households into account. The dotted line represents the general equilibrium e¤ects when impatient households are taken into account.
have increased slightly, such that the output increase is somewhat ampli…ed the following period.
These arguments, however, ignore important “second-round”e¤ects in our model: The land price increase will also relax impatient households’credit constraints, allowing them to demand more land. The relaxation of both types of agents’credit constraints involves a positive credit externality, which allows both types to further increase their land holdings. This is illustrated by the movement to the points “c”. In this equilibrium, entrepreneurs’ land holdings increase even more (and the associated higher consumption possibilities for patient households drive Qt further up). This increase in entrepreneurial land generates
the ampli…cation of the output response in the following period; cf. Figure 4. Importantly, as seen previously from the entrepreneurs’and households’demand schedules, the sensi-tivity of land demand to changes in credit conditions crucially depends on the LTV ratio. Hence, the higher s, the stronger the ampli…cation of output induced by the simultaneous relaxation of both agents’credit limits. This explains why the ampli…cation of technology shocks is signi…cantly larger under the high LTV ratio, as compared with the low LTV ratio.
4.2
Land-demand and credit-market shocks
We now turn to the output e¤ects of the two other shocks in the economy.11 As shown
in Figure 4, land demand shocks are also ampli…ed in the presence of endogenous credit limits, and in particular when the average LTV ratio is relatively high. The reasons are broadly in line with those outlined for the technology shock: land demand shocks drive up land prices, relaxing impatient households’and entrepreneurs’credit constraints. The agents react by increasing their borrowing, and more so under high LTV ratios.12 Shocks
to credit limits feature a similar transmission, though they have a direct impact on agents’ borrowing capacity, so that they induce an even greater ampli…cation. Once credit limits are relaxed, constrained agents bid up the prices of land and capital, which in turn raises their borrowing capacity even further in the case of endogenous credit limits, giving rise to an endogenous ampli…cation of the initial shock.
It is important to note that, compared with other sources of exogenous perturbation, land-demand shocks have a short-lived impact on aggregate activity under high LTV ratios. This contrasts with Liuet al. (2013), who …nd that in the presence of competing demand for land by patient households and entrepreneurs, the collateral constraint faced by the latter provides a powerful ampli…cation of land demand shocks onto output and investment. This is not the case in our setup, due to the presence of an additional type of constrained agents— impatient households. Following a land demand shock, these agents increase their stock of land in a very persistent manner. As a result, land is eventually diverted away from …rms, and thus from productive use.13 We do therefore not observe any ampli…cation
of investment after a land demand shock. In fact, investment increases by less compared to the case where agents are facing …xed credit limits; cf. …gure F.3.
4.3
Variance decomposition
With the above in mind, it should come as no surprise that land-demand shocks play a minor role for business cycles in our model as compared with technology shocks. Figure 11The e¤ects of these two shocks on six key variables are displayed in Figures F.3-F.6 in Appendix F. 12In fact, in equilibrium the entrepreneurs increase their land holdings on impact, despite the fact that
they are the only agents not directly a¤ected by the shock.
13In a recent paper, Pinter (2014) has questioned the empirical speci…cation of Liuet al. (2013). He
shows that adding credit-constrained households to their model improves the empirical performance and corroborates our model result that land-demand shocks are relatively unimportant for output ‡uctuations.
6 shows the relative importance of di¤erent shocks to output volatility, using a simple measure of our three structural shocks’ relative contribution to the volatility of various aggregates. For a generic variable x, we de…ne the contribution of shock to its variance as
V(x; ) var[x] var[x]
3var[x] P var[x] ; =A; "; s; (39)
where var[x] is the unconditional variance of x, when the structural shock is turned o¤.14 Land demand shocks contribute little to the volatility of output, regardless of the
level of the average LTV ratio.15 Technology shocks appear as the main drivers of real
activity at relatively low values of the average LTV ratio. In addition, …nancial shocks play a signi…cant role for the volatility of real quantities at higher values of the average LTV ratio, as they directly create volatility in the credit constraints that shape the behavior of agents.16
4.4
Business cycle asymmetries
Our approach enables us to explore the asymmetric role played by credit constraints. For high average LTV ratios, episodes of non-binding credit constraints become more frequent. This not only leads to a drop in the volatility of output, as documented above, but also induces negative skewness in the business cycle; see Figure 7. In other words, economic expansions become smaller than recessions.17 This …nding crucially depends on
the presence of occasionally non-binding constraints, and on the fact that only shocks that have a positive impact on asset prices have the potential to make credit constraints non-binding. Figure 7 shows that output skewness is practically zero at low values of the average LTV ratio, becoming increasingly negative as credit limits increase and the collateral constraints become non-binding more often; cf. Figure 3. Eventually, downturns 14We use this measure since our numerical solution method does not allow us to carry out a standard
variance decomposition. However, note that for a linear model, theV(x; )’s are equal to the ratios found by conventional variance decompositions.
15However, as Figure F.8 in Appendix F shows, demand shocks represent the main source of variation
in the land price. This follows immediately by allowing for land demand shocks to a¤ect the right-hand side of (37). The impact of these shocks on land prices will be signi…cant and largely unrelated to credit limits, cf. Figure F.4 where land price dynamics are seen to be relatively independent ofsand of whether credit limits are endogenous or not.
16As expected on a priori grounds, these shocks are the predominant driver of debt ‡uctuations, no
matter the average LTV value, cf. Figure F.8.
17For empirical evidence of asymmetries in the U.S. business cycle see, e.g., Neftci (1984) and Morley
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s R ela tiv e s h oc k c ont ribu tio ns to o u tp u t v o la ti lity σA σε σs
Figure 6: Variance contribution to output ‡uctuations of each of the three shocks in the model for di¤erent LTV ratios.
Note: See the notes to Figure 2.
are substantially deeper than booms.
To gain a better understanding of such asymmetric business cycles, it is useful to study the propagation of “large”shocks, which have the potential to make the borrowing constraints non-binding.18 Figure 8 displays the response of output to a set of large,
positive shocks, as well as the mirror image of the response to equally-sized negative shocks. Under a high average LTV ratio, a positive technological innovation implies that the borrowing constraint of the entrepreneur becomes non-binding on impact, and remains slack for six quarters. Impatient households remain constrained throughout. During the …rst six quarters, entrepreneurs essentially behave like consumption smoothers, in the sense that they optimally choose to borrow less than they are able to. This attenuates the expansionary e¤ects of the technology shock on their demand for land. This, in turn, dampens the boom in aggregate economic activity, as shown in the upper right panel. On the other hand, a negative shock to technology does not lead to a relaxation of either of the borrowing constraints, which therefore remain binding throughout. As a result, impatient households and entrepreneurs are forced to cut back on their borrowing in response to the drop in the value of their collateral assets. This produces a stronger response of output. 18In the stochastic simulations, persistent combinations of positive “normal”shocks will be su¢ cient to
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 s S k e wn es s of ou tpu t
Figure 7: Skewness of output for di¤erent LTV ratios.
Note: See the notes to Figure 2.
In other words, a large, negative technology shock has a larger impact on output than a similar-sized positive shock when occasionally non-binding constraints are taken into account. As already mentioned, this feature is more prevalent under high LTV ratios, giving rise to larger asymmetries in this case.19
The second row of Figure 8 shows that a similar picture emerges for large land demand shocks. In this case, the shock makes the entrepreneurs’collateral constraint non-binding during the …rst 14 quarters after the shock in the high LTV regime, while impatient households remain constrained throughout. As a result, entrepreneurs have no incentive to expand their borrowing capacity by increasing their stock of land. In fact, entrepreneurs lower their land holdings on impact, allowing patient and impatient households to increase their land holdings at the expense of non-durable consumption, which drops. The drop in land available for production leads to a drop in output, as seen from the right panel in the second row of the …gure. On the other hand, there is no attenuation of large negative shocks to the economy. In that case, both collateral constraints remain binding, giving rise to a large drop in output. As for technology shocks, the skewness emerging from large demand shocks is much weaker when the LTV ratio is low, as seen from the left panel in the second row. In this case, the collateral constraint of the entrepreneur becomes non-binding 19Note that this potential asymmetric e¤ect of shocks was already pointed out by Kocherlakota (2000)
0 5 10 15 20 0.08
0.1 0.12
Low average LTV ratio
Positive shock
Mirror image of negative shock
0 5 10 15 20
0.08 0.1 0.12
High average LTV ratio
0 5 10 15 20
-0.02 0 0.02 0.04
Low average LTV ratio
0 5 10 15 20
-0.02 0 0.02 0.04
High average LTV ratio
0 5 10 15 20
0 0.05
0.1
Low average LTV ratio
0 5 10 15 20
0 0.05
0.1
High average LTV ratio Technology
shock
Land demand shock
Credit limit shock
Figure 8: Impulse responses of output to large (20 standard deviations) shocks to
technology (row 1), land demand (row 2), and credit limits (row 3) for two di¤erent LTV ratios.
Notes: Light-grey periods are ones where the entrepreneurs are unconstrained; solid-grey periods are ones where all agents are unconstrained.
for only …ve quarters, while the impatient household again remains constrained.
The bottom row of Figure 8 shows the e¤ects of large credit limit shocks. Under high LTV ratios, the entrepreneurs are unconstrained during the …rst 14 periods after a positive shock, while impatient households become unconstrained for one period. For the reasons discussed above, this leads to a muted response of output. In contrast, a large negative shock forces entrepreneurs into a sizeable reduction of their stock of land, and thus of the amount of land available for production. At the same time, impatient households are also forced to bring down their stock of land, which further depresses the land price, and thus the borrowing capacity of both constrained agents. The result is a large drop in output. For low LTV ratios, credit constraints remain binding throughout.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 s C or rela tio n be tw ee n o ut pu t a nd a ggr eg at e d eb t
Figure 9: Correlation coe¢ cient between output and aggregate debt for di¤erent LTV ratios.
Notes: See the notes to Figure 2.
4.5
Comovement between debt and output
The increase in output volatility observed over most of the range of LTV ratios is, as argued above, a result of the enhanced ability of impatient households and entrepreneurs to engage in debt-…nanced consumption and investment. This also has implications for the comovement between credit extension and real economic activity. Figure 9 plots the correlation between aggregate debt and output. The correlation is increasing for LTV ratios up tos 0:75, eventually inverting its pattern as the average LTV ratio approaches its upper bound and credit constraints become non-binding more frequently.
To understand what determines this pattern, it is useful to study how each type of agent makes consumption and investment decisions under di¤erent credit limits. Starting with the entrepreneurs, Figure 10 shows that the correlation between entrepreneurial con-sumption and debt increases up tos 0:6, declining thereafter. At relatively low values of the average LTV ratio, a marginal relaxation of the collateral constraint allows this type of agent to increase its consumption by accessing more credit. However, as the average LTV ratio increases beyond a certain level, entrepreneurs …nd themselves unconstrained more and more often. This reduces the correlation between entrepreneurial consumption and debt. Similarly, the correlation between debt and investment in capital goods increases over most of the support for the credit limit, and declines when equity requirements
be-0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1
Correlation between aggregate c ons umpti on and aggregate debt
s 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 Correlation betweenCE t andB E t s 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 Correlation betweenI t andB E t s 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 Correlation betweenCI t andB I t s 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 Correlation betweenNI t andB I t s 0 0.2 0.4 0.6 0.8 -1 -0.5 0 0.5 1 Correlation between CP
t and aggregate debt
s
Figure 10: Correlation coe¢ cients between key variables for di¤erent LTV ratios.
Notes: See the notes to Figure 2.
come loose enough. At high levels of the average LTV ratio, occasionally non-binding constraints are crucial in delinking debt and investment dynamics.
Like the entrepreneurs, impatient households also exploit higher credit limits to increase debt-…nanced consumption. However, we have seen that impatient households rarely …nd themselves …nancially unconstrained even at high LTV ratios. As a result, and unlike the entrepreneurs, the correlation between consumption and debt of impatient households increases monotonically with higher credit limits, cf. Figure 10. It is worth noting that impatient households behave in accordance with the “…nancial labor supply accelerator” channel of Campbell and Hercowitz (2009, 2011), according to which positive comovement between debt and labor is most prevalent under low LTV ratios. Tight credit limits force households to work longer hours in order to increase their spending in response to a positive shock. This property is also present in our setting, where looser credit conditions weaken the comovement between BtI and NtI; cf. Figure 10. In our model with multiple agents and input factors, however, this e¤ect does not translate into a weakened comovement between output and aggregate debt.20
20In the model of Campbell and Hercowitz (2009, 2011), labor of credit-constrained households equals
The intertemporal choices of patient households mirror those of their credit-constrained counterparts in the debt market. Indeed, the correlation between their consumption and savings (i.e., aggregate credit) declines over most of the range of average LTV ratios. This indicates that patient households are increasingly willing to postpone consumption, while lending their available resources, at least up to s 0:6. Beyond this point, as down-payment requirements drop further and credit constraints are relaxed, borrowers’demand for additional credit slows down, and so does the propensity of patient households to save. This reverses the correlation pattern.
To sum up, greater credit availability initially allows …nancially constrained agents to give vent to their relatively higher impatience, resulting in a marked increase in the comovement of real activity and debt, which is only reverted when the economy is ‡ooded with credit availability and instances of collateral constraints being non-binding become more frequent. It should be noted that these …ndings are in contrast to theories that advocate the role of …nancial liberalization in promoting a lower procyclicality of credit extension and smoother business cycles. According to Den Haan and Sterk (2010), the correlation between real mortgage debt and real GDP in the US dropped from 0.76 during
the period 1954–1978 to 0.32 in the years 1984–2008.21 Several authors have proposed
…nancial liberalization as one potential explanation for this …nding, and a number of frameworks have been put forward to examine the in‡uence of varying degrees of equity requirements on aggregate volatility and the comovement between real activity and debt.22
The results we have presented do not lend much support to this view.
5
Concluding comments
We have shown that, in a fairly conventional DSGE model with heterogeneous agents and multiple credit constraints, looser credit conditions initially generate increasing business cycle volatility. This pattern reverses at LTV ratios not far from those currently observed
become more lax.
21Campbell and Hercowitz (2011) show that the connection between labor and debt has weakened after
the …nancial reforms of the 1980s. Iacoviello and Pavan (2012) present similar results for the correlation of real mortgage debt with real aggregate consumption, which dropped from 0.72 in the period 1952–1982 to 0.37 in the years 1983–2010.
22See the critical review by Den Haan and Sterk (2010), who report a number of contributions to this
strand of the literature, as well as various citations from policymakers and academics that support this view.
in many advanced economies. The non-monotonic relationship between the LTV ratio and macroeconomic volatility is indeed intertwined with the possibility of credit constraints becoming non-binding the higher are credit limits, which in itself paves the way for asym-metric business cycles.
As much macroprudential policy debate has been focusing on ways of imposing caps on credit availability, such a non-monotonicity questions the adequacy of these proposals. Also, a policy of countercyclical credit availability could produce adverse e¤ects, if imple-mented at an average LTV ratio that is to the right of the maximum-volatility value. As our model is necessarily stylized, quantitative conclusions regarding the value of the LTV ratio at which output volatility reaches its peak should be drawn with care. Nonethe-less, macroprudential regulators should consider that curbing LTV ratios might actually increase macroeconomic volatility in a situation characterized by lax equity requirements. In this case, tighter credit limits will increase the instances of binding collateral con-straints, exposing borrowers to ‡uctuations in the availability of credit. In fact, using an estimated DSGE model, Guerrieri and Iacoviello (2014) demonstrate that non-binding credit constraints for households have played a role in the US housing boom leading up to the Great Recession, while the subsequent credit tightening has been crucial in deepening the contraction. This demonstrates that the macroprudential trade-o¤ we point out is not merely a theoretical possibility.
Our results may also be inferred as general arguments pertaining to …nancial liber-alization measures, but we have deliberately been silent on welfare implications, while focusing on the cyclical implications of di¤erent LTV ratios. However, our model features substantiallevel e¤ects of increasing the LTV ratio: steady-state output in our calibration
is 5:8% higher in the high LTV regime, compared with the low LTV regime. Therefore,
a welfare evaluation will ultimately involve weighing level gains against volatility losses. For this purpose, our model may be too simple and stacked in favor of level gains, even though it features potential welfare losses of negative skewness that are rarely examined in business cycles analyses. A richer model with other realistic frictions is thus needed for a proper evaluation, and is left as a subject for future research.
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Appendices
A
Liabilities and assets in the US
Figure 1 shows the ratio of liabilities to assets for households and …rms in the United States, respectively. All data are taken from FRED (Federal Reserve Economic Data), Federal Reserve Bank of St. Louis. The primary source is Flow of Funds data from the Board of Governors of the Federal Reserve System. For business liabilities we use credit market instruments of non…nancial corporate and noncorporate business. As assets we follow Liu et al. (2013) and use data on both sectors’equipment and software as well as real estate at market value. For households and nonpro…t organizations, we use credit market instruments as data for liabilities and use as assets both groups’real estate at market value and equipment and software of nonpro…t organizations. Further details are available upon request.
The ratios reported in Figure 1 are aggregate measures, and may therefore not re‡ect actual loan-to-value (LTV) requirements for the marginal borrower. Nonetheless, we report these …gures since the ‡ow of funds data delivers a continuous measure of LTV ratios covering the entire period 1949–2014. For households, the aggregate ratio of credit to assets in the economy is likely to understate the actual down-payment requirements faced by households applying for a mortgage loan, since loans and assets are not uniformly distributed across households. In our model, we distinguish between patient and impatient households, and we assume that only the latter group is faced with a collateral constraint. In the data, we have not made this distinction, so that the LTV ratio for households reported in …gure 1 represents an average of the LTV of patient households (savers), who are likely to have many assets and small loans, and that of impatient households (borrowers), who on average have larger loans and fewer assets. Justiniano et al. (2014) use the Survey of Consumer Finances to make this distinction, and identify borrowers as households with liquid assets of a value less than two months of their income. Based on the surveys from 1992, 1995, and 1998, they arrive at an average LTV ratio for this group of around 0.8, while our measure ‡uctuates around 0.5 during the 1990’s. Another approach, following Duca et al. (2011), is to focus on …rst-time home-buyers, who are likely to fully exploit their borrowing capacity. Using data from the American Housing Survey, Duca et al. report LTV ratios approaching 0.9 towards the end of the 1990’s; reaching a peak of almost 0.95 before the onset of the recent crisis. While these alternative approaches are thus likely to result in higher
levels of LTV ratios, we are interested in the development over time of these ratios. While we believe the Flow of Funds data provide the most comprehensive and consistent time series evidence in this respect, substantial increases over time in LTV ratios faced by households have been extensively documented; see, e.g., Campbell and Hercowitz (2009), Duca et al. (2011), Favilukis et al. (2013), and Boz and Mendoza (2014). It should be noted that for households, various government-sponsored programs directed at lowering the down-payment requirements faced by low-income or …rst-time home buyers have been enacted by di¤erent administrations (Chamberset al., 2009). These are likely to have contributed to the increase in the ratio of loans to assets illustrated in the left panel of Figure 1.
Likewise, the aggregate ratio of business loans to assets in the data may cover for a disparate distribution of credit and assets across …rms. In general, the borrowing patterns and conditions of …rms are more di¢ cult to characterize than those of households, as their credit demand is more volatile, and their assets less uniform and often more di¢ cult to assess. Liu et al. (2013) also use Flow of Funds data to calibrate the LTV ratio of entrepreneurs, and arrive at a value of 0.75. This ratio is based on an assumption that commercial real estate enters with a weight of 0.5 in the asset composition of …rms. In contrast, the ratio we report in Figure 1 assigns a weight of 1 to commercial real estate. While the transformation of Liuet al. (2013) would result in higher LTV ratios at any point in time, it would not a¤ect the …nding of rising LTV ratios over time. The secular increase in …rm leverage over the second half of the 20th century has also been documented by Grahamet al. (2014) using data from the Compustat database.23 ,24 These authors report loan-to-asset ratios that are broadly in line with those we present. More generally, an enhanced access of …rms to credit markets over time has been extensively documented in the literature. This involves, for instance, the emergence of a market for high-risk, high-yield bonds (Gertler and Lown, 1999), increased ‡exibility in …rms’ …nancing decisions, and the resulting immoderation in …nancial quantities (Jermann and Quadrini, 2009).
B
The steady state
The deterministic steady state of our model is described in the following, where variables without time subscripts are the steady-state values. We …rst consider the implications of the patient households’optimality conditions. From (6) and (7), we get
CP
P
C = P; (B.1)
and
P 1 NP PN = PWP; (B.2)
respectively. The steady-state gross interest rate on loans is recovered from (8):
PR P = P;
R = 1P; (B.3)
23It should be mentioned that they also show a Flow of Funds-based measure of debt to total assets at
historical cost (or book value) for …rms. The increase over time in this measure is smaller. However, we believe that the ratio of debt to pledgeable assets at market values (as shown in Figure 1) is the relevant measure for …rms’access to collateralized loans, and hence more appropriate for our purposes.
24We emphasize that Figure 1 reports agross measure of …rm leverage. Bateset al. (2009) report that
…rm leveragenet of cash holdings has been declining since 1980, but that this decline is entirely due to a large increase in cash holdings.
emphasizing that it is the time preferences of the most patient individual that determine the steady-state rate of interest. From (9) we …nd
" HP P H + P PQ = PQ; HP P H = " Q P 1 P : (B.4)
Turning to the impatient households, (10) and (11), leads to
CI
I
C = I; (B.5)
and
I 1 NI PI = IWI; (B.6)
respectively. From (12) we obtain the steady-state value of the multiplier on the credit constraint:
I = I 1 IR ;
which by use of (B.3) yields
I= I 1 I
P : (B.7)
From (B.7), we see that in steady state I > 0 since P > I, which proves that the credit constraint (4) is binding in steady state. In a similar fashion, we get from (20):
E = E 1 E
P : (B.8)
Hence, E > 0 implying that the entrepreneurs’ credit constraints, (18), are also binding in steady state. From (13) we get
" HI I H + I IQ+ IsQ R = IQ; HI I H = " Q I 1 I I Is 1 R ; HI I H = " Q I 1 I 1 I P s P ; HI I H = " Q I 1 I P I s ; (B.9)
where the next-to-last line makes use of (B.3) and (B.7).
Turning to the remaining optimality conditions of the entrepreneurs, (19) gives
CE E C = E; (B.10) and (21) implies E " 1 2 I I 1 2# E I I I I 1 + E E I I 2 I I 1 = E
leading to
E = E: (B.11)
This re‡ects that there are no investment adjustment costs in steady state in our no-growth model. Therefore, the shadow value of a unit of capital equals the shadow value of wealth. Combining this with (25), we readily obtain
QK = 1: (B.12) From (22) we obtain ErK E+ E(1 ) E+ EsQK R = E E ErK+ E(1 ) E + E 1 E P s QK R = E 1 +rK = 1 P E sQK E ; (B.13)
where the second line uses (B.8), and the last
