ampliﬁcation and asymmetric effects without collateral...

Amplification and Asymmetric Effects withoutCollateral Constraints∗

Dan Cao

Georgetown University

Guangyu Nie

SHUFE

September 2016

Abstract

The seminal contribution by Kiyotaki and Moore (1997) has spurred a vast litera-

ture on the importance of collateral constraints in propagating and amplifying shocks

to the economy. However, most papers in the literature using collateral constraints

assume non-state contingent debt, i.e., markets are incomplete. To assess the relative

importance of collateral constraints versus market incompleteness, we study a cali-

brated incomplete markets model and solve it with and without collateral constraints.

We find that market incompleteness by itself plays a quantitatively significant role in

the amplified and asymmetric responses of the economy, including land price and

output, to exogenous shocks. In addition, when agents can trade insurance claims

(subject to collateral constraints) against negative shocks, amplification disappears.

Keywords: Incomplete Markets; Collateral Constraint; Amplification; Asymmet-

ric Effects; Dynamic General Equilibrium; Global Nonlinear Methods

∗We wish to thank Daron Acemoglu, James Albrecht, Fernando Alvarez, Behzad Diba, Sebastian DiTella,Luca Guerrieri, Pedro Gete, Mike Golosov, Mark Huggett, Matteo Iacoviello, Urban Jermann, Nobu Kiy-otaki, Arvind Krishnamurthy, Wenlan Luo, Yuliy Sannikov, Alp Simsek, Viktor Tsyrennikov, Susan Vroman,Ivan Werning, and Franco Zecchetto for useful comments and discussions on the paper. We also thank theeditor, Richard Rogerson, and three anonymous referees for very helpful comments and suggestions.

1 Introduction

The mechanisms by which exogenous shocks to the economy propagate and amplify overtime are important in understanding the positive and normative effects of business cycles.The seminal contribution by Kiyotaki and Moore (1997) identifies a prominent mecha-nism - the collateral constraint channel - and has spurred a vast macroeconomic literatureemphasizing collateral constraints.1 However, more recent important papers on the sametopic, such as He and Krishnamurthy (2013) and Brunnermeier and Sannikov (2014), gen-erate propagation and amplification without imposing a collateral constraint. The com-mon feature of all these papers is that agents borrow through non-state contingent debt,i.e., markets are incomplete. One must wonder whether non-state contingent debt by it-self generates most of the amplification effect and whether collateral constraints remaina powerful amplification mechanism if debt becomes state-contingent. In this paper, webuild a simple, calibrated model with heterogeneous agents and aggregate shocks, whichwe solve with and without collateral constraints, in order to understand the qualitativeand quantitative importance of market incompleteness relative to collateral constraints.

The model is a simplified version of Iacoviello (2005). There are two types of risk-averse agents, entrepreneurs and households, in a one-consumption good economy witha fixed supply of an asset: land. There is a land market in which all agents participate. Theentrepreneurs can combine land with labor supplied by the households to produce con-sumption goods. This production process is subject to aggregate stochastic productivityshocks. The households earn wages from working for the entrepreneurs and decide howmuch to consume in consumption goods and in land. We assume that the entrepreneursare less patient than the households, and thus they tend to borrow from the households.

In this economy, we consider equilibria under several alternative financial marketstructures. In the benchmark model, the entrepreneurs can borrow using non-state con-tingent debt subject to a collateral constraint as in Kiyotaki and Moore (1997) and Ia-coviello (2005). The equilibrium of this model features amplified and asymmetric re-sponses of land price and output to (symmetric) exogenous productivity shocks.

To study the importance of non-state contingent debt relative to collateral constraintin the amplified and asymmetric responses, we consider two classes of variations of thebenchmark model. In the first class of models, the entrepreneurs can borrow using non-state contingent debt without being subjected to the collateral constraint (incomplete mar-kets models). On average, the equilibrium in this class of models accounts for two thirds

1Examples include Kocherlakota (2000), Krishnamurthy (2003), Cordoba and Ripoll (2004a,b), Iacoviello(2005), Mendoza (2010), and many other (New Keynesian) models with similar financial channels startingwith Bernanke et al. (1999).

1

of the amplification and asymmetric effects in the benchmark model. In the second class,the entrepreneurs can borrow using financial assets with payoffs contingent on all futurestates (or on the worst future state) subject to collateral constraints. In this class of mod-els, the amplification effect virtually disappears. With fully state contingent assets, theasymmetric effect also disappears.

These findings suggest that market incompleteness by itself plays a relatively moreimportant role in the amplified and asymmetric responses of the economy to exogenousshocks, than collateral constraint does. Under incomplete markets, the effects are dueto the net worth effect as follows:2 an initial negative shock decreases land productiv-ity and consequently land price. Because of market incompleteness, the value of debtholding of the entrepreneurs, which is determined from the previous period, remainsunchanged. Therefore the decrease in land price leads to a decrease in the net worth ofthe entrepreneurs, which in turn reduces their demand for land for production. Sincethe entrepreneurs are more efficient users of land than the households,3 the reduction intheir land demand further depresses the price of land, and further lowers their net worth.This sets off a vicious cycle of falling land price and falling net worth. Thus the effectsof the negative productivity shock on land price and production are amplified.4 More-over, because the utility and production functions are concave, a negative productivityshock has larger effects than a positive productivity shock through the net worth channel(asymmetric effect).

The study of amplification and asymmetric effects requires solving these models usinga global nonlinear method. To this end, we use the Markov equilibrium concept with onlywealth share as the endogenous state variable and global nonlinear method developed inKubler and Schmedders (2003) and Cao (2010) to solve for the equilibrium in each model.But we extend the method to our production economy with elastic labor supply, hous-ing consumption, and a wide range of financial markets structure including incompletemarkets with exogenous borrowing constraints, and collateral constraints with completemarkets. The global nonlinear solution also gives us insights that are absent if we log-linearized the model and allows us to assess the accuracy of the log-linearized solution.

2Brunnermeier and Sannikov (2014) call this channel amplification through prices.3The households derive utility from consuming land, i.e. a form of “home production,” but since the

entrepreneurs combine land and labor to produce output, more land in the hands of the entrepreneurs alsoraise labor demand. The increase in output includes the increase in labor supply from the households, andin total exceeds the decrease in the imputed value of the households’ land consumption. See Appendix Cfor an elaboration of this intuition using a two-period version of our model.

4The amplification through the net worth effect is similar to the fire-sale phenomenon described inShleifer and Vishny (1997). Because the entrepreneurs are more efficient in using land, they are the “naturalbuyers” in Shleifer and Vishny’s language. So when all entrepreneurs reduce their land demand, both landprice and output fall significantly.

2

For example, we find that in the benchmark model, the log-linearized solution (assum-ing that the collateral constraint always binds as in Iacoviello (2005) and Liu et al. (2013))on average overstates the amplification effect by almost 100% under our calibrated shocksize.5 However, under the same shock size, the global nonlinear solution shows that thecollateral constraint binds only 7.2% in the ergodic distribution of the model. The globalnonlinear solution also tells us that the magnitude of the amplification and asymmetriceffects are state-dependent. While on average, market incompleteness alone accounts fortwo thirds of the amplification and asymmetric effects in the benchmark model, the effectsare stronger with collateral constraint than without it when the entrepreneurs’ wealth islow.

The models with incomplete markets with or without collateral constraints imply thatthe responses of land price and output to productivity shocks are not only asymmetricand amplified but also depend on the wealth share of the entrepreneurs. We find that thisimplication is broadly consistent with the data in the U.S. when we take the entrepreneursas corresponding to the non-financial business sector.6 In particular, in the model, the realestate leverage ratio, i.e., the total debt (net of credit) divided by the total value of the landholding of the entrepreneurs, is strictly decreasing in the entrepreneurs’ wealth share.Therefore, we use the real estate leverage ratio of the non-financial business (corporateand non-corporate) sector as a proxy for the entrepreneurs’ wealth share in the model.Using the Flow of Funds data for the U.S., we construct the time series for the real es-tate leverage of the non-financial business sector. One of our empirical findings is thatchanges in TFP have the largest effects on land price and output when these changes arelow and when the real estate leverage ratio of the non-financial business sector is high,which is exactly what our benchmark model predicts. Our empirical findings comple-ment the recent results in Guerrieri and Iacoviello (2015a) who emphasize the asymmetricresponses of economic activities, including consumption and employment, to changes inhouse prices.7

This paper is related to the vast literature on the effects of collateral constraints inaddition to the work cited above. In particular, the benchmark model is a simplifiedversion of Iacoviello (2005). But instead of relying on log-linearization as in Iacoviello(2005), we solve for the global, nonlinear dynamics of the equilibrium with occasionallybinding collateral constraints. Guerrieri and Iacoviello (2015b) suggest a way to adaptthe log-linearization method in a piecewise fashion to handle occasionally binding con-

5By assumption, the log-linearized solution rules out an asymmetric effect.6This connection from model to data is exploited in a recent paper: Liu, Wang, and Zha (2013).7Differently from their analysis, in our models and empirical exercise, we assume that both land price

and output (and consumption) are endogenously determined by exogenous TFP shocks.

3

straints. Mendoza (2010) and Bianchi (2011) also solve for global, nonlinear solution withoccasionally binding constraints in small-open economy models in which interest ratedynamics are exogenous, while they are endogenous in our model.

In a small open economy framework, Mendoza (2010) also compares the equilibriumunder a collateral constraint to the equilibrium under an exogenous borrowing constraintlimit. He finds that imposing an exogenous borrowing limit instead of a collateral con-straint significantly weakens the amplification effect on Tobin’s Q, a measure of assetprice. The difference between our result and his comes from the fact that his result isconditional on the endogenous states in which the collateral constraint is binding, i.e.sudden stop events in his language, while our result is about the averages of the ampli-fication effects over the ergodic distribution of endogenous states. Indeed, as shown inSection 3, conditional on low levels of the entrepreneurs’ wealth, the responses of landprice and output are considerably larger in the benchmark model than in the incompletemarkets model without a collateral constraint. However, the absence of a collateral con-straint also reduces precautionary saving, driving downward the entrepreneurs’ wealthshare. Mendoza (2010)’s comparison does not take this effect into account.8

The importance of market incompleteness shown in this paper also suggests that state-contingent debt can be an important macro-prudential policy tool. Using models withcollateral constraints, Lorenzoni (2008), Geanakoplos (2010), Bianchi (2011), Bianchi andMendoza (2013), and Jeanne and Korinek (2013) argue that leverage should be restrictedand debt should be taxed in booms to avoid the fire-sale externality and financial crisesin subsequent recessions. Given that market incompleteness plays an important role, de-signing debts with some insurance for downturns should also be effective in reducing themagnitude of the subsequent recessions.9 Theoretically, Section 4 shows that completestate-contingent assets or simply some insurance against the worst aggregate shocks,even subject to collateral constraints, can nullify the amplification and asymmetric effects.This finding is similar to the ones in Krishnamurthy (2003) and DiTella (2014) in differentsettings. In practice, for example, Mian and Sufi (2014) advocate for share-responsiblemortgages, i.e., mortgages that reduce principal and mortgage payments upon signifi-cant declines in housing prices, which might have significantly reduced the size of theU.S. financial and economic crisis in 2007-2008.

The paper is organized as follows. Section 2 presents the benchmark model with in-complete markets and a collateral constraint and the solution method, as well as reason-

8See our Figure 7 and the related discussion.9However, the welfare analysis in Online Appendix J suggests that the entrepreneurs might oppose

financial market reforms that complete the markets.

4

able parameters to analyze the solution of this benchmark model. Section 3 studies theincomplete markets model and compares it to the benchmark model. Section 4 studies themodels with complete and partial hedging. Section 5 presents some empirical evidence.Section 6 concludes. Additional proofs and constructions are presented in the Appendix.

2 Benchmark Model

The model is based on Iacoviello (2005). It is a standard model of a production economyin which a durable asset (land) is used as collateral to borrow (with non state contingentdebt) and as an input in production. The model is calibrated to the U.S. economy.10 Wecall this model Model 1a and use it as the benchmark model. Other models presentedlater, Model 2 with incomplete markets and without a collateral constraint and Model 3and Model 4 with complete and partial hedging, will be compared against Model 1a.

2.1 Economic Environment

Consider an economy inhabited by two types of agents: entrepreneurs and householdswho are both infinitely lived and of measure one. There is one consumption good. En-trepreneurs produce the consumption good by hiring household labor and combining itwith land. Households consume the consumption good and land (housing), and supplylabor to the entrepreneurs.

We adopt the standard notation of uncertainty. Time is discrete and runs from 0 toinfinity. In each period, an aggregate shock st is realized. We assume that st followsa finite-state Markov chain. Let st = (s0, s1, ..., st) denote the history of realizations ofshocks until date t. To simplify the notation, for each variable x, we use xt as a shortcutfor xt

(st).

Households maximize a lifetime utility function given by

E0

∞

∑t=0

βt{(c′t)

1−σ2 − 11− σ2

+ j(h′t)

1−σh − 11− σh

− 1η(L′t)

η

}, (1)

where E0 [.] is the expectation operator, β ∈ (0, 1) is the discount factor, c′t is consumptionat time t, h′t is the holding of land. L′t denotes the hours of work. Households can trade in

10Cordoba and Ripoll (2004b) is another extension of Kiyotaki and Moore (1997) to nonlinear productionfunction and concave utility functions. However, their paper only analyzes the case without aggregateshocks. In Online Appendix M, we show that our global nonlinear method applies to their model withaggregate shocks as well. However we choose Iacoviello (2005) to illustrate the importance of land in theeconomy.

5

the market for land as well as a non-state contingent bond market. The budget constraintof the households is

c′t + qt(h′t − h′t−1) + ptb′t ≤ b′t−1 + wtL′t. (2)

Given land in the utility function of the households, implicitly h′t ≥ 0. We impose theno-Ponzi condition on the households,11 i.e., for all t and st:

lim T→∞Et

[(T

∏t′=t

pt′

) (b′T + qT+1h′T

)]≥ 0.

Entrepreneurs use a constant-returns-to-scale technology that uses land and labor asinputs. They produce consumption good Yt according to

Yt = Athυt L1−υ

t , (3)

where At is the aggregate productivity which depends on the aggregate state st, ht is realestate input, and Lt is labor input; υ is the share of land in the production function.

In contrast to Iacoviello (2005), the production function uses the contemporaneousland holding of the entrepreneurs instead of the land holding from the previous period.This minor modification turns out to be crucial to apply the concept of Markov equilib-rium with the reduced set of endogenous state variables, i.e. wealth shares, in Subsection2.2 and the solution method in Appendix B. However this modification does not affectthe key economic forces in the paper.12

We want the entrepreneurs to borrow from the households so we assume that theentrepreneurs discount the future at the rate γ < β. The entrepreneurs maximize

E0

∞

∑t=0

γt (ct)1−σ1 − 11− σ1

(4)

subject to the budget constraint

ct + qt(ht − ht−1) + ptbt ≤ bt−1 + Yt − wtLt. (5)

11As we assume below, the households have a higher discount factor than that of the entrepreneurs. Con-sequently, they tend to lend to the entrepreneurs. Therefore even if we impose tighter borrowing constraintson the households such as a collateral constraint or an exogenous borrowing constraint, these constraintsare not binding in the ergodic distribution of the equilibrium of the economy.

12In a continuous time version of this model presented in Online Appendix L, similar to the modelsin Brunnermeier and Sannikov (2014) and He and Krishnamurthy (2013), the timing assumption becomesimmaterial.

6

Output Yt is produced by combining land and labor using the production function (3).Given the production function of the entrepreneurs, implicitly ht ≥ 0.

As in Kiyotaki and Moore (1997) and Iacoviello (2005), we assume a limit on the obli-gations of the entrepreneurs. Suppose that, if borrowers repudiate their debt obligations,the lenders can repossess the borrower’s assets by paying a proportional transactioncost (1−m)Et [qt+1] ht. In this case the maximum amount that a creditor can borrowis bounded by mEt [qt+1] ht, i.e.

bt + mEt [qt+1] ht ≥ 0, (6)

where bt is the saving (−bt is borrowing) of the entrepreneurs.To finish the description of the model, here we assume that the only source of un-

certainty is the aggregate productivity At. It is straightforward to extend the model toincorporate other sources of uncertainty such as uncertainty in the housing preferenceparameter j.

Lastly, to be consistent with the data, we use a measure of output, called total output,which includes the imputed rental value of land consumed by the households evalu-ated at the marginal rate of substitution between land and non-land consumptions of thehouseholds, i.e.,

Yt = Yt +j (h′t)

−σh

(c′t)−σ2

h′t. (7)

The qualitative and quantitative findings remain similar if we use only output Yt insteadof the total output.

2.2 Equilibrium

The definition of a sequential competitive equilibrium for this economy is standard.

Definition 1. A competitive equilibrium is sequences, which depend on both time t andthe history of the shocks st, of prices {pt, qt, wt}t,st and allocations {ct, ht, bt, Lt, c′t, h′t, b′t, L′t}t,st

such that (i) the allocations {c′t, h′t, b′t, L′t}t,st maximize (1) subject to the budget constraint(2) and the no-Ponzi condition and {ct, ht, bt, Lt}t,st maximizes (4) subject to the bud-get constraint (5) and the collateral constraint (6), and the production technology (3),given {pt, qt, wt}t,st and some initial asset holdings

{h−1, b−1, h′−1, b′−1

}; (ii) land, bond,

labor, and good markets clear: ht(st) + h′t(st) = H, bt(st) + b′t(s

t) = 0, Lt(st) = L′t(st),

ct(st) + c′t(st) = Yt(st) for all t, st.

7

Let ωt denote the normalized financial wealth of the entrepreneurs:

ωt =qtht−1 + bt−1

qtH, (8)

and ω′t denote the normalized financial wealth of the households:

ω′t =qth′t−1 + b′t−1

qtH.

By the land and bond market clearing conditions, ω′t = 1− ωt in any competitive equi-librium. Therefore in order to keep track of the normalized financial wealth distributionbetween the entrepreneurs and the households, (ωt, ω′t), in equilibrium, we only need tokeep track of ωt - the entrepreneurs’ wealth share. To simplify the language, we use theterm wealth distribution for normalized financial wealth distribution.

As in Cao (2010), we define Markov equilibrium as follows.

Definition 2. A Markov equilibrium is a competitive equilibrium in which prices andallocations at time t, as well as the wealth distribution at time t + 1 under different real-izations of the exogenous shocks st+1 depend only on the wealth distribution at time t, ωt

as well as the exogenous state st.

This Markov equilibrium definition features the endogenous state variable ωt thatdepends on land price qt (which by itself depends on the endogenous and exogenousstate variables). This equilibrium was first studied in Duffie et al. (1994). Brunnermeierand Sannikov (2014) and He and Krishnamurthy (2013) use the same type of equilibriumdefinition in their continuous time models.13 We will use the algorithm developed in Cao(2010) to compute this Markov equilibrium.14

13In Online Appendix L, we present a continuous time version of our paper and make this connectionexplicit. However, there are two important differences. First, because of persistent TFP shocks (instead ofI.I.D. depreciation shocks to capital stock as in Brunnermeier and Sannikov (2014) or I.I.D. return shockson dividend as in He and Krishnamurthy (2013)), in our Markov equilibrium, we need to keep track ofboth wealth distribution and the current exogenous shock. Second, we allow for general utility functions,instead of linear or log utility functions as in Brunnermeier and Sannikov (2014) and He and Krishnamurthy(2013).

14We can also define Markov equilibrium using more natural state variables such as ht−1, bt−1, as in mostof the New Keynesian literature using linearization methods, or as in Mendoza (2010) using a nonlinearmethod. Our numerical method applies to such concepts of equilibrium as well. However, our currentMarkov equilibrium concept allows us to summarize the dynamic equilibrium in one endogenous statevariable, ωt, which greatly facilitates numerical computations and offers intuitive representations of howwealth distribution affects land price and output in simple graphs.

8

2.3 Solution

In this subsection, we first show the equations that characterize a competitive equilib-rium. A Markov equilibrium is also a competitive equilibrium, therefore it must alsosatisfy these equations.

Given their land holding at time t, ht, the entrepreneurs choose labor demand Lt tomaximize profit

maxLt{Yt − wtLt}

subject to the production technology given in (3). The first order condition (F.O.C.) in Lt

implieswt = (1− υ)Athυ

t L−υt , (9)

i.e. Lt =((1−υ)At

wt

) 1υ ht and

Yt − wtLt = πtht

where πt = υAt

((1−υ)At

wt

) 1−υυ is profit per unit of land.

Let µt denote the Lagrangian multiplier on the entrepreneurs’ collateral constraint.The F.O.C in the entrepreneurs’ land holding is

(πt − qt)c−σ1t + µtmEt [qt+1] + γEt

[qt+1c−σ1

t+1

]= 0 (10)

and the complementary-slackness condition is

µt (bt + mhtEt [qt+1]) = 0. (11)

The F.O.C for the entrepreneurs with respect to bond holding is

− ptc−σ1t + µt + γEt

[c−σ1

t+1

]= 0. (12)

Similarly, the F.O.C.s for the households are

h′t : −qtc′−σ2t + jh′−σh

t + βEt

[qt+1c′−σ2

t+1

]= 0 (13)

b′t : −ptc

′−σ2t + βEt

[c′−σ2

t+1

]= 0 (14)

L′t : wtc′−σ2t = L′η−1

t . (15)

The first order conditions in ht and h′t shed light on the determinants of land price.

9

Indeed, we can rewrite (10) as

qt = πt + γEt

[qt+1

(ct+1

ct

)−σ1]+ µtmEt [qt+1] cσ1

t .

The first two terms on the right hand side of this equation show that, from the entrepreneurspoint of view, land price is the net present value of present and future profits from pro-duction using land. In addition, the last term on the right hand side shows the collateralvalue of land in the land valuation of the entrepreneurs. Iterating this equation forward,we obtain the expression for land price as the present discounted value of profits, withthe discount factor depending on the marginal utilities of the entrepreneurs as well as onthe multiplier on the collateral constraint:15

qt = πt + Et

[qt+1

{γ

(ct+1

ct

)−σ1

+ µtmcσ1t

}]

= Et

[∞

∑s=0

s−1

∏r=0

{γ

(ct+r+1

ct+r

)−σ1

+ µt+rmcσ1t+r

}πt+s

]. (16)

Similarly, we re-write (13) as

qt =j (h′t)

−σh

c′−σ2t

+ βEt

[qt+1

(c′t+1

c′t

)−σ2]

= Et

[1

c′−σ2t

∞

∑s=0

βs j(h′t+s

)−σh

].

From the point of view of the households, land price is the present discounted value ofcurrent and future marginal utilities from housing consumption.

Becker (1980) shows that in a neoclassical growth model with heterogeneous discountfactors, long run wealth concentrates on the most patient agents, in this case the house-holds. However, in this model, due to the collateral constraint, the entrepreneurs can onlypledge a fraction of their future wealth to borrow. Therefore, despite their lower discountfactor, their wealth does not disappear in the long run. In particular, the model admitsa long run steady state in the absence of uncertainty. In Online Appendix F, we presentthe set of equations that determine this steady-state. In the next subsection, we use thissteady-state solution to calibrate the model.

15This formula is similar to the one in Mendoza (2010). In particular, when the entrepreneurs cannotborrow against their land holding, i.e., m = 0, there will not be any collateral premium in the pricing ofland.

10

Table 1: Baseline Parameter ValuesParameters ValueA 1 mean technologyH 1 total supply of housingβ 0.99 discount factor of householdγ 0.98 discount factor of entrepreneurσ1 1 CRRA coefficient of entrepreneurσ2 1 CRRA coefficient of householdσh 1 CRRA coefficient of household for housingη 1 household labor supply elasticitym 0.89 loan-to-value ratio constraint

Outside of the steady-state, when the variance of the aggregate productivity is strictlypositive, we use the algorithm based on Cao (2011) to solve for the Markov equilibriumof the model. Since a steady state exists, we can also log-linearize around the steady state(assuming the collateral constraint is always binding) as in Iacoviello (2005).16 We canthen compare the accuracy of the two solution methods.

2.4 Parameter Values

Most of the parameter values are taken from Iacoviello (2005) and are given in Table 1.In particular, A and H are normalized to 1. The preference parameters σ1, σ2, σh, η arestandard. We have also done robustness checks and find that these parameters do notaffect the qualitative and quantitative results in this paper.17 The discount factors of thehouseholds and the entrepreneurs, β and γ are set at 0.99 and 0.98.18 Lastly, we set theconstraint on loan-to-value ratio, m = 0.89 as estimated in Iacoviello (2005).19

Two parameters j - households’ weight of housing in utility, and υ - share of land inthe production function are the most important parameters in the model. Housing weightis important since the total output given in equation (7) depends on the marginal utilityfrom housing consumption, and as we show in Online Appendix K, land share in the pro-duction function is a central determinant of the magnitudes of the amplification effects.These two parameters are calibrated using the steady-state equilibrium. We use two mo-

16In a recent paper, Liu, Wang, and Zha (2013) use the same log-linearization method to solve for theequilibrium in an elaborate version of the model in Iacoviello (2005).

17These exact preference parameters also allow us to obtain analytical results in some cases, such as inModel 0 with complete markets and Model 3 with complete hedging or in the two period version of themodel in Appendix C.

18Iacoviello (2005) justifies the choice of the discount factors using existing empirical estimates.19This value of the loan-to-value ratio is also consistent with the value in Liu et al. (2013) who puts a

lower bound of 0.75 on the parameter.

11

Table 2: Calibration Targets and Results for j and υParameters Value Target Data

j 0.057 Value of Residential Housing over Total Output 133%υ 0.041 Value of Commercial Housing over Total Output 88%

ments: the average value of residential real estate over total output and the average valueof commercial real estate over total output from the Flow of Funds to calibrate the twoparameters. We solve for the steady-state of the benchmark model and find (j, υ) suchthat the model’s moments match exactly the empirical moments.20 The empirical valuesof the two moments and the calibrated parameters (j, υ) are presented in Table 2.

In order to use the global nonlinear method described in Appendix B, we need todiscretize the process for At by a finite number of points. For At we use a three pointprocess, At ∈ {A + ∆, A, A− ∆}, which corresponds to expansions, st = G, normal times,st = N, and recessions, st = B.21 We assume the following form of transition matrix

T =

τ1 1− τ1 01−τ0

2 τ01−τ0

2

0 1− τ1 τ1

. (17)

The exogenous stochastic process of productivity is totally symmetric. However, dueto incomplete markets, the resulting dynamics of the economy become asymmetric asshown in Subsection 2.5 below.

The values of τ0 and τ1 are calibrated using historical U.S. data. According to thedefinitions of business cycle expansions and recessions by NBER, there are 12 recessionsin total from post-WWII, January 1945 to December 2013, with an average length of eachrecession of 3.6 quarters. The share of months spent in recessions is 15.7%. Given thetransition matrix T above, the average length of each recession is 1

1−τ1so τ1 = 72.04%. τ0

is chosen such that the probability of a recession in the stationary distribution for At, i.e.,At = A− ∆ matches the share of months spent in recession which implies τ0 = 87.2%.22

As we normalize A to 1, in numerical simulations, we vary ∆ from 1% to 6%, in orderto study the non-linear effects of shocks with different sizes. 23 In the benchmark set of

20We have also verified that the two moments do not change significantly outside of the steady-state.21A two-state process is enough to illustrate the amplification and asymmetric effects, but in order to

match several moments of the productivity process in the U.S. we need at least three states.22From the transition matrix for At, the probability of recession in the stationary distribution is1−τ0

2(2−τ1−τ0).

23Given the exogenous process for productivity, the standard deviation of productivity is given by√1−τ0

2−τ1−τ0∆. From our own estimates using quarterly TFP data from St Louis Fed (see empirical appendix

12

parameters, we choose an intermediate value for ∆, ∆ = 3%.

2.5 Numerical Results

We apply the nonlinear global solution method for the benchmark model in AppendixB to solve for the dynamic equilibrium of this model. We also solve for the equilib-rium using log-linearization around the steady-state and examine the accuracy of thelog-linearized solution. The equations that determine the log-linearized dynamics arepresented in Online Appendix F. In this subsection, we report the numerical results forthe benchmark model with the size of the productivity shock chosen at 3% and other pa-rameters taken from Table 1. The most important properties of the numerical solutionsare the following.

First of all, the global nonlinear solution for Markov equilibrium features an occasion-ally binding collateral constraint. Collateral constraint (6) binds when the entrepreneurs’wealth is sufficiently low. In this binding region, endogenous variables including landprice and output are more sensitive to changes in wealth distribution. Second, the equi-librium is asymmetric with respect to bad shocks versus good shocks despite the factthat the stochastic structure of the shocks is totally symmetric. Third, when the collat-eral constraint binds, the responses of land price and output strongly amplify negativetechnology shocks.24

Figure 1 shows the policy functions (consumption of the entrepreneurs and the house-holds and the total output) and the pricing function for land conditional on the exogenousstate st and the endogenous state ωt , in expansions (st = G, solid blue lines), normaltimes (st = N, purple dash-dotted lines) and recessions (st = B, dashed red lines). Eventhough the global nonlinear method solves for the policy and pricing functions for thewhole range of ωt, Figure 1 is restricted to the values of ωt in the support of the station-ary distribution of ωt. Similarly, Figure 2 shows the current and future policy and landprice functions (at the same ωt) for the current normal times (solid blue lines), futurenormal times (dashed red lines) and recessions (purple dashed-dotted lines).

Because of the collateral constraint, the functions exhibit significantly more nonlinear-

D), ∆ is approximately 1.8%. This low value of ∆ is consistent with the the estimates from Liu, Wang, andZha (2013), which yield ∆ slightly lower than 1%. However, to generate the annual standard deviation ofproductivity in the U.S. economy of about 15% in Khan and Thomas (2013), ∆ is approximately 6%. Giventhe wide range of the possible values of ∆, we choose ∆ = 3% at the middle of the range. In OnlineAppendix I, we examine other values of ∆ in the range.

24We define amplification as the ratio of the average responses of economic variables to exogenousshocks relative to the average responses of economic variables in the model with complete markets withoutany financial friction.

13

0.05 0.1 0.157.5

8

8.5

9

9.5

Land Price

ωt

0.05 0.1 0.150.9

0.95

1

1.05

Total Output

ωt

0.05 0.1 0.150

0.01

0.02

0.03

0.04

Consumption of Entrepreneurs

ωt

st=G

st=N

st=B

0.05 0.1 0.150.84

0.86

0.88

0.9

0.92

0.94

0.96

Consumption of Households

ωt

Figure 1: Policy Functions for Model 1a

ity when ωt is close to zero where the collateral constraint is binding. When the collateralconstraint binds, Figure 2 shows that the standard feed-back effect in Kiyotaki and Moore(1997) kicks in: after a negative productivity shock, even temporary, the entrepreneurshave to cut back their land holding, ht, in order to smooth consumption; they cannotborrow more due to the collateral constraint. This reduction in land demand depressesland price, qt, which through the collateral constraint, forces the entrepreneurs to reducetheir debt, partly by reducing consumption, and further cut back their land holding. Thisvicious cycle results in a significant decline in land price, as well as in the entrepreneurs’land holding and the total output. There is also an inter-temporal feedback: lower currentwealth of the entrepreneurs leads to lower future wealth and lower future land prices.25

Given that the current land price is a weighted average of current per unit profit and fu-ture land prices, as shown in the asset pricing equations, lower future land prices in turnleads to lower current land price. The entrepreneurs are more efficient in using land thanthe households (see the discussion in footnote 3); so a significant decrease in their landholding leads to a significant decrease in output.

The advantage of the nonlinear solution method used in this paper is that, we can seeclearly two regions of the state space in which the economy follows different dynamics.

25The entrepreneurs borrow less due to shrinking borrow ability but they reduce land holding by muchmore, leading to lower future wealth than their current wealth as shown in Figure 4.

14

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

0

0.5Land

ht,s

t=N

ht+1

,st+1

=B

ht+1

,st+1

=N

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

-3

-2

-1

0Bond

bt,s

t=N

bt+1

,st+1

=B

bt+1

,st+1

=N

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

7.5

8

8.5

9Land Price

qt,s

t=N

qt+1

,st+1

=B

qt+1

,st+1

=N

Figure 2: Portfolio Choice and Land Price for Model 1a in Normal Times

In one region, when the collateral constraint is binding (or nearly binding), the feed-backeffect is important. In the other region when the collateral constraint is far from binding,asset price and output are less sensitive to changes in the wealth distribution.26,27 Figure3 helps us determine the neighborhood of wealth distribution in which the collateral con-straint starts to bind. The upper panel shows the portfolio choice of the entrepreneurs asa function of ωt and st = N. On the left of the vertical green line (in both panels), aroundωt = 0.05, the collateral constraint is close to be binding and will bind if st+1 = B. Thelower panel shows the ergodic distribution of ωt conditional on st = N. The probabil-ity that the collateral constraint is binding in the ergodic distribution is significant and isapproximately 45.1% in recessions.28

One important ingredient of our solution method is that for each st and ωt - the en-trepreneurs’ wealth share (wealth distribution), we have to solve for the future wealthdistribution ωt+1 for each realization of st+1. The left panel of Figure 4 shows ωt+1 as

26We can also see different dynamics between the two regions by looking at dxtdωt

, x = q, Y, as functions ofωt, i.e. the marginal effect of redistributing wealth from the households to the entrepreneurs on land priceand output. Figure 1 suggests that these functions are decreasing in ωt and are much higher at lower ωtwhen the collateral constraint is binding.

27Mendoza (2010) also emphasizes the two different dynamics, Sudden Stops versus normal cycles, inhis model (with exogenous processes for interest rate). But his solution method does not allow for a visualpresentation of the two dynamics like ours.

28The binding probability is 0% in good states and 0.2% in normal states respectively. The overall bindingprobability is 7.2%.

15

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16-4

-2

0

2

4Entrepreneurs Portfolio

q*h

b

collateral constraint

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

0

2

4

6

8

10

PD

F

Stationary Density

Figure 3: The Entrepreneurs’ Portfolio Choice and the Stationary (Ergodic) Distributionof Wealth Share in Model 1a in Normal Times

functions of ωt and st+1 given that st is in the normal state, i.e. st = N. Given thatthe entrepreneurs are more exposed to the productivity shock because they are the onlyagents who can produce, their wealth increases (relative to the households’) as the goodshock hits next period (solid blue line), and decreases as the bad shock hits next period(dashed red line). If st+1 stays at the normal state, then the wealth distribution remainsalmost unchanged as the future wealth function (dash-dotted purple line) stays close tothe 45o line (dashed black line). The transition functions for the entrepreneurs’ wealthshare combined with the transition matrix of the exogenous states determine the ergodicdistribution of the wealth share in the right panel (we plot the density of the ergodic dis-tribution). Given the small share of land in the aggregate production function, the wealthshare of the entrepreneurs always stays below 15% in the ergodic distribution. As shownin Figure 4, when ωt is close to 0, ωt+1 > ωt, thus the lowest level of wealth share ωt = 0is never reached in the ergodic distribution. This result is in contrast to the one in the in-complete markets model presented below (or in Brunnermeier and Sannikov (2014) andHe and Krishnamurthy (2013)). The density becomes zero before ωt reaches 0.29

To understand the asymmetric and amplification effects, Figure 5 plots the impulse-response functions (IRFs) of land price and output, starting from st = N and ωt at 10% or

29The numerical solution also shows that ωt+1 < ωt when ωt is sufficiently high, even when st = G.This is not visible in Figure 4 since the range of ωt is restricted to a relevant interval.

16

0 0.05 0.1 0.150

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Transition Functions

ωt

ωt+

1

st+1

=G

st+1

=N

st+1

=B

45o

0.05 0.1 0.150

1

2

3

4

5

6

7

Stationary Density

ωt

PD

F

st=G

st=N

st=B

Figure 4: Transition and Distribution for Entrepreneurs’ Wealth Share

50% of the ergodic distribution of wealth shares, conditional on st = N. To compute theseIRFs, we assume that st+1 changes to G (or B) and simulate forward the realizations ofst+r, r > 1, using the transition matrix (17), the Markov equilibrium transition functionsfor ωt, the policy and pricing functions. We then take the averages of land price andoutput across simulation paths. The IRFs are reported as the differences between theseaverages and the averages in simulations assuming that st+1 stays at N. We also comparethese IRFs to the linear IRFs in the log-linearized solution. In the case of bad shocks, weplot the minus (flip) of the relative changes in land price and output.

Figure 5 shows the asymmetric effects: land price and output decrease after a badshock, st+1 = B, (red thick solid lines for ω at the 10th percentile of the ergodic distributionand blue dash-crossed lines for ω at the 50th percentile of the ergodic distribution) bymore than they increase after a good shock, st+1 = G, (red thin solid lines for ω at 10th

percentile of the ergodic distribution and blue dash lines for ω at 50th percentile of theergodic distribution). The asymmetric effect is much stronger for lower ω than for higherω. Figure 5 shows the amplification effects, i.e., land price and output decrease (increase)by more than 3% after productivity decreases (or increases) by 3%. The amplificationeffect is strongest when ω is low and st+1 = B, in which land price decreases by almost6% and output decreases by almost 5% at impact of 3% decrease in productivity.

The black dotted line shows the IRFs under log-linearization. Under log-linearization,the responses to shocks are perfectly symmetric so we only plot the IRFs under positiveshocks. As shown in the figure, by assuming that the collateral constraint is always bind-

17

0 5 10 15 20 250

2

4

6

8

t

%

Percentage Deviation of Land Price after 3% shocks

dynamics (flipped) after B shock, ω at 10%

dynamics after G shock, ω at 10%



loglinearized result

0 5 10 15 20 250

1

2

3

4

5

6

t

%

Percentage Deviation of Output after 3% shocks





loglinearized result

Figure 5: IRFs for Non-Linear versus Log-Linear Models

ing, log-linearization overstates the effects of shocks. Land price changes by close to 8%and output close to 6% at impact and the changes are also more persistent.

Another way to capture the asymmetric effect of the collateral constraint is to calculatethe average (weighted by the ergodic distribution of ωt) percentage changes in land priceand output in normal times, i.e., xt+1−xt

xt, where x = q or Y, when the aggregate shock in

the next period switches to expansion or recession respectively.30 The first row of Table3 reports the percentage changes in the long run under complete markets Model 0, de-scribed in Appendix A. In Model 0, there is no asymmetric nor amplification effects: a 3%increase or decrease in productivity leads to exactly 3% increase or decrease in output andland price.31 The second row of Table 3 shows the percentage changes for the benchmarkmodel (Model 1a). Compared to the complete markets outcomes (the first row, Model 0),the benchmark model exhibits asymmetric effects, i.e., positive shocks increase land priceand output by less than negative shocks decrease land price and output. For example, a3% fall in productivity generates a larger response in output (−3.65%) than the responseto an increase of the same size in productivity (+3.25%). Amplification effects are alsopresent, for example, the responses in output (3.25% to a positive shock and −3.65% to a

30Land price and output remain almost unchanged if the productivity shock stays in the normal state.31For the exact parameters used in this numerical exercise, the sizes of the responses of land price and

output to shocks in Model 0 are exactly the size of the shocks.

18

Table 3: Average Land Price and Output Changes After a 3% Increase or 3% Decrease inAggregate Productivity

Land price OutputType of Friction Expansion Recession Expansion RecessionComplete Markets (Model 0) 3.00% -3.00% 3.00% −3.00%Collateral Constraint (Model 1a) 3.60% -4.22% 3.25% −3.65%Collateral Constraint (alt, Model 1b) 3.62% -4.30% 3.26% −3.70%Incomplete Markets (Model 2) 3.46% -3.81% 3.21% −3.43%Complete Hedging (Model 3) 3.00% -3.00% 3.00% −3.00%Partial Hedging (Model 4) 3.71% -2.92% 3.20% -2.95%

Note: This table shows the long run average responses of land price and output after productivity changes from normal to either high (Expansion)or low (Recession). Productivity can take on one of three realizations, (normal, high, and low), where low and high realizations are 3% away fromnormal.

negative shock) are larger than the size of the shocks (3%).32

2.6 Alternative Collateral Constraints

In the collateral constraint (6), we use the expected future land price. This constraint canbe micro-founded under limited commitment and has been used in a large number ofpapers including Kiyotaki and Moore (1997), Iacoviello (2005), Fostel and Geanakoplos(2008), Geanakoplos (2010), and Cao (2011).33 However, in practice collateralized con-tracts are often written using current asset prices and this is also assumed in a large num-ber of papers, for example Mendoza (2010) and Bianchi (2011). Figure 4 shows that thewealth distribution, ωt, moves very slowly over the business cycles, and as a result landprice also moves slowly. Therefore using current or expected future land price should notimply quantitatively significant differences between the two models. We can show thisresult rigorously by solving an alternative model in which the collateral constraint (6) is

32The average amplification effects in Models 1a, as well as in Model 1b and Model 2 are slightly largerthan the average amplification effects of technology shocks found in the existing empirical and quantitativestudies of business cycles such as King, Plosser, and Rebelo (1988), Gali (1999), and Smets and Wouters(2007). We also show in the Online Appendix K that the effect will be significantly larger if we increase theshare of land, ν, in the production function as in Kocherlakota (2000). Moreover, land price also declinesmuch more after bad shocks when the collateral constraint is binding, i.e. conditional on low values of ωas shown by the conditional IRFs in Figure 5.

33Cao (2011) shows that when the lender can seize a fraction of the asset upon default, the collateralconstraint arises endogenously and has the form

bt(st) + mht minst+1|st

qt+1

(st+1

)≥ 0.

If we use this collateral constraint the average responses of land price and output after 3% negative shocksare −3.96% and −3.41% respectively.

19

replaced by the following alternative collateral constraint:

bt + mqtht ≥ 0.

Fortunately, this model is a special case of the general Markov equilibrium defini-tion and the solution method in Cao (2010). We solve for the Markov equilibrium underthis alternative collateral constraint for the same parameters used in Subsection 2.5. Thesolution is quantitatively similar to the solution of the collateral constraint model in Sub-section 2.5. For example, Row 3 in Table 3 shows that the amplification and asymmetriceffects are only slightly higher than the ones in the benchmark model (Row 2).

3 Incomplete Markets Model

The benchmark model, Model 1a presented in the previous section, with incomplete mar-kets and collateral constraint delivers amplified and asymmetric responses of the econ-omy to exogenous shocks. To understand the importance of market incompleteness ver-sus collateral constraint for the amplification and asymmetric effects, in this section, westudy a model in which markets are incomplete, i.e. the entrepreneurs can only borrow byissuing state non-contingent bonds, but they do not face a collateral constraint. Instead,we impose only the no-Ponzi condition on the entrepreneurs, i.e., for all t and st,34

lim T→∞Et

[(T

∏t′=t

pt′

)(bT + qT+1hT)

]≥ 0.

We call this model Model 2. The definitions of the competitive equilibrium and Markovequilibrium are similar to those in the benchmark model.

The first-order conditions (F.O.C.s) in ht and bt in the maximization problem of theentrepreneurs imply

(πt − qt)c−σ1t + γEt[qt+1c−σ1

t+1 ] = 0 (18)

and− ptc

−σ1t + γEt

[c−σ1

t+1

]= 0. (19)

We rewrite (18) as

34In the proof of Lemma 1, we show that, in some cases, this no-Ponzi condition is equivalent to a naturalborrowing limit.

20

qt = πt + γEt

[qt+1

(ct+1

ct

)−σ1]

= Et

[∞

∑s=0

γs(

ct+s

ct

)−σ1

πt+s

].

The right hand side of this equation shows that, from the entrepreneurs’ point of view,land price is the present discounted value of present and future profits from productionusing land and the discount factor depends on the marginal utility of the entrepreneurs.Notice that, compared to the pricing equation (16), land no longer has collateral values tothe entrepreneurs.

Despite the fact that we do not impose any constraint on the entrepreneurs’ borrowingexcept for the no-Ponzi condition, the following lemma shows that, in equilibrium, thefinancial wealth of the entrepreneurs is endogenously bounded from below by 0.

Lemma 1. In any competitive equilibrium, thus any Markov equilibrium, we must have ωt ≥ 0for all t and st.

Proof. Online Appendix E.

One consequence of this lemma is that the competitive, or Markov equilibrium willnot change if we replace the no-Ponzi schemes condition on the entrepreneurs by a morestringent borrowing constraint

ωt+1(st+1) = bt(st) + qt+1(st+1)ht(st) ≥ 0

for all t, st.When 0 < σ1 ≤ 1, we also show that this is the natural borrowing limit, i.e. in equi-

librium, any feasible plan - plan that implies positive consumption at all times and in allstates - must satisfy the inequality ωt ≥ 0 for all t and st.

3.1 Numerical Results

In this subsection, we present the numerical results for the incomplete markets model -Model 2 - with the parameters in Subsection 2.4.

Quantitatively, Row 4 in Table 3 shows the average (over the ergodic distribution ofωt) of the changes in land price and total output given the current normal state, st = N.We observe significant amplification and asymmetric effects under incomplete markets.

21

By comparing the amplification effects in this model (Row 4 in Table 3) to the amplifi-cation effects in the benchmark model (Row 2 in Table 3) - Model 1a with both marketincompleteness and collateral constraints - we can say that market incompleteness ac-counts for about two thirds of the amplification effects.35

Under market incompleteness only, the amplification and asymmetric responses comefrom the net worth effect as follows. An initial negative shock decreases land productiv-ity and consequently land price. Because of market incompleteness, the value of debtholding of the entrepreneurs, which is determined from the previous period, remainsunchanged. Therefore the decrease in land price leads to a decrease in the net worth ofthe entrepreneurs, which in turn reduces their demand for land for production. Sincethe entrepreneurs are more efficient users of land than the households (as explained infootnote 3), the reduction in their land demand further depresses the price of land, andfurther lowers their net worth, setting off a vicious cycle of falling land price and fallingnet worth. Thus the effects of the negative productivity shock on land price and produc-tion are amplified. Moreover, because the utility and production functions are concave, anegative productivity shock has larger effects than a positive productivity shock throughthe net worth channel. In Appendix C, we study a two-period version of our model withlog utilities to derive analytically the net worth effect. We also show that, under com-plete markets, this effect disappears because the entrepreneurs can perfectly insure their(relative) net worth against productivity shocks.

Figure 6 shows the policy functions (consumptions of the entrepreneurs and the house-holds and total output) and the pricing function for land conditional on the exogenousstate st = B and the endogenous state ωt (blue solid lines - for comparison, the dashedred lines are the functions in Model 1a). Figure 6 is restricted to the values of ωt in thesupport of the ergodic distribution. We observe that, despite the absence of a collateralconstraint, land price and output functions are nonlinear in the wealth distribution. Inparticular they are more sensitive to changes in the wealth distribution when the wealthof the entrepreneurs is low.36

Figure 7 is the counterpart of Figure 4 for Model 1a, showing the transition function(left panel) and the stationary distribution (right panel) when st = N compared to the

35For example, after a 3% negative TFP shock, on average, output decreases by −3.43% with marketincompleteness alone compared to −3.65% in the model with market incompleteness and collateral con-straints. As a result, about 66% ( 3.43%−3%

3.65%−3% ) of the amplification in output is accounted for by market in-completeness. Similarly, about 66% ( 3.81%−3%

4.22%−3% ) of the amplification in land price is accounted for by marketincompleteness.

36Similar to the observation in footnote 26, another way to see the nonlinearity is to look at dxtdωt

, x = qor Y, as functions of ωt. Figure 6 suggests that these functions are decreasing in ωt and are much higher atlower ωt.

22

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

7

7.5

8

8.5

9Land Price

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

0.86

0.88

0.9

0.92

0.94

0.96

0.98Total Output

Incomplete Markets (Model 2)

Collateral Constraint (Model 1a)

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035Consumption of Entrepreneurs

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

0.8

0.82

0.84

0.86

0.88

0.9Consumption of Households

Figure 6: Policy and Pricing Functions under Incomplete Markets (Compared to theBenchmark), st = B.

ones in Model 1a. Nonlinearity implies asymmetric responses of the equilibrium landprice and output with respect to productivity shocks. Starting from st = N, the goodshock, i.e., st+1 = G increases the entrepreneurs’ wealth and the bad shock st+1 = Bdecreases the entrepreneurs’ wealth, as shown in Figure 7. But conditional on the samechange in entrepreneurs’ wealth, land price and output increase after a good shock byless than they decrease after a bad shock due to nonlinearity. This leads to the asym-metric responses of equilibrium land price and output to symmetric shocks, as shownquantitatively in Table 3.

Despite the amplification and asymmetric effects in Model 2 due to the net worth effectcoming from market incompleteness, a collateral constraint, in addition to market incom-pleteness, does amplify exogenous shocks more, especially when the collateral constraintbinds. This can be seen from the more nonlinearity of the policy and pricing functionsfor Model 1a compared to Model 2 in Figure 6 when ω is low. It can also be seen moreclearly from the conditional IRFs (as defined in Subsection 2.5) depicted in Figure 8. Thethick red lines are the IRFs in Model 2 and the crossed blue lines are the IRFs in Model 1a.To simplify the figure, we only plot the IRFs after a negative shock. The left panels showthe (flipped) impulse responses of land price and output after a −3% productivity shock,conditional on st = N and ωt at the 10th percentile of the ergodic distribution in eachmodel. Similarly, the right panel shows the impulse responses conditional on st = N andωt at the 50th percentile. The figure shows that when ωt is high the responses are similar

23

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ωt

0

0.05

0.1

0.15

ωt+

1

Transition Functions, st+1

=B



45o

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ωt

0

1

2

3

4

5

6

7

8

PD

F

Stationary Density, st=N



Figure 7: Transition and Stationary Distribution of Wealth Share (Compared to the Bench-mark), st = N.

in both models. However, when ωt is low the responses are significantly larger in Model1a with collateral constraint. Table 4 shows the conditional responses at the impact of theshocks and at different percentiles of the entrepreneurs’ wealth share (Row 2, Model 1aversus Row 4, Model 2) and tells a similar story.

Then, why is it the case that Model 2 without the collateral constraint accounts forclose to two thirds of the average amplification effect in Model 1a? The answer lies in thetransition functions and ergodic distributions depicted in Figure 7. The left panel showsthat due to precautionary saving, in Model 1a, the entrepreneurs refrain from borrowingtoo much, compared to the amount of borrowing in Model 2, to avoid a binding collat-

Table 4: Land Price and Output Changes After a 3% Decrease in Aggregate ProductivityConditional on Entrepreneurs’ Wealth Shares.

Land price OutputPercentile of ω 50 percent 10 percent 50 percent 10 percentComplete Markets (Model 0) -3.00% -3.00% -3.00% -3.00%Collateral Constraint (Model 1a) -3.98% -5.35% -3.44% -4.45%Collateral Constraint (Model 1b) -4.01% -5.59% -3.44% -4.65%Incomplete Markets (Model 2) -3.81% -4.09% -3.43% -3.56%Complete Hedging (Model 3) -3.00% -3.00% -3.00% -3.00%Partial Hedging (Model 4) -2.96% -2.90% -2.97% -2.93%

Note: This table shows the responses of land price and output, in different models, after productivity changes from normal to low (Recession),i.e. productivity decreases by 3%. The responses are reported at the 50th and 10th percentiles of the entrepreneurs’ wealth share in the ergodicdistribution of each model.

24

0 10 20 30

t

0

1

2

3

4

5

6

%

Change in Land Price after -3% shocks, ω at 10%

dynamics (flipped), Model 2

dynamics (flipped), Model 1a

0 10 20 30

t

0

1

2

3

4

5

%

Change in Output after -3% shocks, ω at 10%

0 10 20 30

t

0

1

2

3

4

5

6

%

ω at 50%

0 10 20 30

t

0

1

2

3

4

5

%

ω at 50%

Figure 8: Impulse Response Functions in Model 2 versus Model 1a.

eral constraint. Consequently, as depicted in the right panel, the ergodic distribution ofwealth shares in Model 1a has less mass at lower levels of ω compared to the ergodic dis-tribution in Model 2. This difference attenuates the significant amplification effect at lowω’s (when the collateral constraint binds) in Model 1a. This intuition also suggests that inmodels with larger amplification effects due to a collateral constraint, the precautionarysaving channel is stronger and the collateral constraint binds less often. Therefore marketincompleteness will account for more of the amplification in these models. We confirmthis intuition in Online Appendix K.

4 Hedging with Collateral Constraints

The comparison between the benchmark model - Model 1a - in Section 2 and the incom-plete markets model - Model 2 - in Section 3 suggests that one of the main ingredients forthe amplification and asymmetric effects is market incompleteness beside the collateralconstraint. To further demonstrate this point, we study two variations of the benchmarkmodel, which we call the complete hedging model (Model 3, Subsection 4.1) and partialhedging model (Model 4, Subsection 4.2) and in which the entrepreneurs have access tosome Arrow securities. In Model 3 - model with complete hedging - the entrepreneurscan trade a complete set of state-contingent securities with the households subject to a

25

collateral constraint on each state-contingent security.37 In Model 4 - model with partialhedging - the entrepreneurs can borrow in state non-contingent bonds but can also buyinsurance against the worst realization of aggregate shocks. However, the total payofffrom bond and insurance contracts is subject to a collateral constraint.

4.1 Complete Hedging

In this model, we maintain the collateral constraint, however we allow the agents to tradea complete set of Arrow securities, subject to a collateral constraint for each security. Inhistory st, let pt (st+1) denote the price of the Arrow security that pays off one unit of con-sumption good if st+1 happens and nothing otherwise. Let φt (st+1) and φ′t (st+1) denotethe holdings of the entrepreneurs and the households, respectively, of the Arrow security.The definition of competitive equilibrium as well as Markov equilibrium are exactly thesame as in the benchmark model, except now we need to impose the condition that themarkets for the Arrow securities clear, i.e. φt (st+1) + φ′t (st+1) = 0.

In this model, the budget constraint of the entrepreneurs becomes

ct + qt(ht − ht−1) + ∑st+1|st

pt (st+1) φt (st+1) ≤ φt−1 (st) + Yt − wtLt.

and the collateral constraint (6) is now

φt (st+1) + mqt+1(st+1)ht ≥ 0 ∀st+1|st. (20)

We can solve numerically for the Markov equilibrium in this model similarly for pre-vious models presented above. However, for the parameters in Table 2, we can solve forthe Markov equilibrium analytically. The details of the solution are presented in OnlineAppendix G. In this subsection, we analyze the properties of the solution and comparethem to the benchmark model.

Figure 9 shows the differences between the price and policy functions in this currentmodel with complete hedging and the benchmark model with a collateral constraint andincomplete markets in Section 2 (when st = B). The pricing functions and policy func-tions in the two models are close to each other. In particular, even with complete hedging,Model 3 exhibits strongly nonlinear dependence of land price and output on wealth dis-tribution at low levels of the entrepreneurs’ wealth. This nonlinearity is due to binding

37We would like to thank referees for insightful comments that substantially improve the exposition ofthis subsection.

26

ωt

0.05 0.1 0.157.6

7.8

8

8.2

8.4

8.6

8.8Land Price

ωt

0.05 0.1 0.150.9

0.92

0.94

0.96

0.98

1Total Output

Complete Hedging (Model 3)Collateral Constraint (Model 1a)

ωt

0.05 0.1 0.150

0.005

0.01

0.015

0.02

0.025

0.03


ωt

0.05 0.1 0.150.85

0.86

0.87

0.88

0.89

0.9


Figure 9: Price and Policy Functions under Complete Hedging - Model 3 - versus theBenchmark Model when st = B.

collateral constraints and might potentially lead to asymmetric and amplification effectsas in the benchmark model.

However, from the analytical solution, we find that there is no asymmetric effect andand the amplification effect is negligible. Indeed, the analytical solution shows that

ωt+1 =

γωt

β−υ(β−γ)ωtif ωt ≥ ω

(1−m) h(ωt)H otherwise,

(21)

where ω is a threshold above which the collateral constraints (20) do not bind and belowwhich they bind. h(ωt) is determined by a quadratic equation independent of st, st+1:

(j + υ)

Hωt=

υ(1− β) +[υj(1− γ) + υ2(β− γ) + γ(j + υ)

]ωt

ωt(1−m)h (ωt)− (1− υωt)mj(1− β)

ωt(1−m)[H − h (ωt)].

In addition, prices and allocations depend only on ωt and At. For example, when ωt < ω:

q(ωt, st) = A(st)(L(ωt)− (1− ν))

(1− γ)Hωt

(h(ωt)

L(ωt)

)ν

.

Therefore there is no asymmetric effects, i.e. starting from st = N, bad shock st+1 = N increases

land price and output as much as good shock st+1 = G increases land price and output. The

amplification effect is also negligible since ωt+1 is very close to ωt, as shown in Figure 10.

In addition, the dynamics for ωt described by (21) show that, unlike the cases withincomplete markets (with or without collateral constraint), the long run ergodic distribu-

27

tion of wealth share ωt is degenerated and concentrates on a value ω∗, regardless of theexogenous state. At ωt = ω∗, ωt+1 = ωt, therefore the amplification is entirely absent.38

In Online Appendix G, we present the other equations that determine the equilibriumat ω∗ and ω and we show that

ω∗ =υ(1− β) (1−m)

j(1− υ)(1− γe) + υ(1− β)(1 + jm)< ω,

where γe = mβ + (1−m)γ. Therefore, at ω∗, the collateral constraint is binding for all fu-ture states, and land demand of the entrepreneurs is given by h∗ and ω∗ = qt+1h∗−mqt+1h∗

qt+1H =

(1−m) h∗H . For the parameters in Subsection 2.4, ω∗ = 0.0435 < ω = 0.0467.

In order to understand how the complete set of Arrow securities help the entrepreneursto insure against shocks, Figure 10, lower panel, shows the entrepreneurs’ optimal choicesof φt (st+1) as functions of the wealth distribution and the future exogenous state st+1,given the current state st = N. For clarity we only plot the choices of φt for st+1 = G andst+1 = B. Below ω, the collateral constraints are binding for all future states, therefore

ωt+1 =qt+1ht −mqt+1ht

qt+1H= (1−m)

ht

H

regardless of the realization of st+1.39 However, above ω, the collateral constraints arenot binding, and the entrepreneurs borrow relatively more from the future good statecompared to the future bad state, so that ωt+1 is also independent of st+1. All in all, theupper panel in Figure 10 shows that the transition functions for wealth shares ωt+1 asfunctions of ωt is independent of st+1 (solid lines, relative to the dashed 450 line), andst. In addition, ωt+1 > ωt for ωt < ω∗ and ωt+1 < ωt for ωt > ω∗. Therefore, ω∗ isthe only stationary and stable level of wealth shares in the long run. The portfolio choiceas a function of ωt in Figure 10 echoes the recent findings in Rampini and Viswanathan(2010) on risk management and the distribution of debt capacity. In particular, whennet worth is low, the entrepreneurs forgo insurance, i.e. the collateral constraint bindsin all future states, in the interest of investing in land and production. While at higher

38In light of this result, one might wonder why there is amplification in Kiyotaki and Moore (1997) sinceit is a deterministic model, and so markets are effectively complete, i.e., the number of bonds is equal to thenumber of future states (one). The difference is that in the paper, the initial shock is unexpected (but thesubsequent responses of land prices and outputs are rationally anticipated) while shocks in our Model 3are fully expected and hedged against. In addition, in Model 3, when the shocks are somehow unexpected,for example, when a bad shock reduces productivity by more than rationally expected by the agents, i.e.At+1 = A′ < A(st+1 = B), we recover the amplification effect. The details of this exercise are presented inOnline Appendix G.

39In this case, the form of collateral constraint (20) forces the entrepreneurs to perfectly insure theirwealth against shocks.

28

ωt

0.02 0.03 0.04 0.05 0.06 0.07 0.08

ωt+

1

0.02

0.03

0.04

0.05

0.06

0.07

0.08Wealth Dynamics, All States

ωt+1

45o

ωt

0.02 0.03 0.04 0.05 0.06 0.07 0.08-3.5

-3

-2.5

-2

-1.5

-1

Arrow Securities Holdings, st = N

φt(st+1 = G)constraint,st+1 = G

φt(st+1 = B)constraint,st+1 = B

Figure 10: Land and Arrow Securities Holdings.

levels of net worth, the entrepreneurs actively manage risk by leaving the collateral con-straint slack in some future states (conserving debt capacity, in the language of Rampiniand Viswanathan (2010)). In addition, the entrepreneurs buy some insurance against theworse future state, i.e. φt+1(st+1 = B) > φt+1(st+1 = G). At ωt = ω∗, the entrepreneursdo not engage in risk-management.

To summarize, the solution of this current model with collateral constraint but withcomplete markets shows that when the agents have access to a complete set of securi-ties, amplification and asymmetric effects disappear.40 This point is also made by Krish-namurthy (2003) in a simple 3-period economy and DiTella (2014) in a continuous-timemodel.41

4.2 Partial Hedging

In this subsection, we present a model with partial hedging - Model 4 - in which theagents can trade in a state non-contingent bond and an Arrow security that pays off oneunit of consumption if the worst state is realized next period. The entrepreneurs are also

40In Online Appendix G, we show this result analytically for the parameters given in Subsection 2.4.For other commonly used parameter values, quantitatively similar results apply. For example, we find thatthe ergodic distribution of wealth shares has very small variance if not degenerated: in the model withσ1 = σ2 = σh = 2 and η = 1.5, the mean of wealth shares ω’s in the ergodic distribution is 0.0206, and thestandard deviation is 8.7× 10−5.

41However, in DiTella (2014), the experts face an equity constraint instead of collateral constraints usedin our paper. The equity constraint arises endogenously because of moral hazard problems as derived inHe and Krishnamurthy (2011).

29

subject to the collateral constraint. Formally, in history st, let pinst denote the price of the

Arrow security that pays off one unit of consumption good if the worst state happens att + 1, i.e. st+1 = B and nothing otherwise. Let φt (st+1 = B) and φ′t (st+1 = B) denotethe holdings of the entrepreneurs and the households, respectively, of the Arrow security.The definition of competitive equilibrium as well as Markov equilibrium are exactly thesame as in the benchmark model, except for the extra condition that the markets for theArrow security clear, i.e. φt (st+1 = B) + φ′t (st+1 = B) = 0. The entrepreneurs and thehouseholds can also trade in non-state contingent bonds as in the benchmark model. Thetotal promises by the entrepreneurs from the Arrow securities and the bonds have to becollateralized by land.

In this model, the budget constraint of the entrepreneurs becomes:

ct + qt(ht − ht−1) + ptbt + pinst φt ≤ φt−11{st=B} + bt−1 + Yt − wtLt.

and the collateral constraint (6) is now

bt(st) + φt(st+1 = B)1{st+1=B} + mqt+1(st+1)ht(st) ≥ 0 ∀st+1|st.

The algorithm to compute the Markov equilibrium in this model with partial hedging isalso similar to the ones for the benchmark model and the complete hedging model in theprevious subsection. Below, we report the numerical findings.

Figure 11 shows the differences between the price and policy functions in this currentmodel - Model 4 with partial hedging and in the benchmark model in Section 2 (whenst = B). Figure 12 shows the differences between the transition functions and the er-godic distributions in the two models. We observe that, even though both models featurehighly nonlinear price and policy functions for lower values of wealth, in Model 4 withpartial hedging, the entrepreneurs are able to insure themselves against the worse shocks.Therefore, the stationary distribution has much lower masses at lower values of wealthcompared to the benchmark model. Consequently, Table 3 and Table 4 show that the av-erage and conditional amplification effects in Model 4 are much smaller, virtually absentfor st = B, compared to the benchmark model.

5 Some Empirical Evidence

The results from our model show that productivity (TFP) shocks have asymmetric effectson the output and land prices. First, negative TFP shocks have larger effects than positive

30

0.05 0.1 0.15

ωt

7.6

7.8

8

8.2

8.4

8.6

8.8Land Price

0.05 0.1 0.15

ωt

0.9

0.92

0.94

0.96

0.98

1Total Output

Partial Hedging (Model 4)


0.05 0.1 0.15

ωt

0

0.01

0.02

0.03


0.05 0.1 0.15

ωt

0.85

0.86

0.87

0.88

0.89


Figure 11: Price and Policy Functions under Partial Hedging - Model 4 - Compared to theBenchmark Model, st = B.

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ωt

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

ωt+

1

Transition Functions, st+1

=B



45o

0.02 0.04 0.06 0.08 0.1 0.12 0.14

ωt

0

5

10

15

20

25

30

PD

F

Stationary Density, st=N



Figure 12: Transition and Stationary Distribution of Wealth Share under Partial Hedging- Compared to the Benchmark Model, st = N.

31

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

-1

-0.8

-0.6

-0.4

b

qh


s=G

s=N

s=B

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

ωt

-1

-0.8

-0.6

-0.4

b

qh


Figure 13: Real Estate Leverage Ratio and Wealth DistributionsNote: The y-axis is the the ratio of the entrepreneurs’ bond holding to the value of their land holding, which is the negative of the real estate leverage

ratio.

TFP shocks as indicated in Table 3. Second, the effects also depend on the wealth distribu-tion: they are larger when the wealth share of the entrepreneurs is low. Lastly, combiningthe previous two points, the responses should be the strongest when we have both lowTFP shocks and low wealth share of the entrepreneurs. We offer some empirical evidenceof these implications in this section.42 In particular, we use four quarterly U.S. time series:the TFP shock, the real price of land, the real output and the real estate leverage ratio of thenon-financial business (corporate and non-corporate) sector. The real estate leverage ratiois defined and constructed as the ratio of the credit market debt to the value of real estatewhich is used as a proxy for the wealth share of the entrepreneurs. As can be seen fromFigure 13, the real estate leverage ratio of the entrepreneurs is strictly decreasing in theirwealth share in incomplete markets model with or without collateral constraints. Thesample covers the period from 1951:Q4 to 2010:Q4. Appendix D provides more detaileddescriptions of the data.

Our specifications take the following three forms:

∆ log xt = α1 + βt f p highIt∆ log TFPt + βt f p low(1− It)∆ log TFPt + γ1Zt + ε1t (22)

∆ log xt = α2 + βlev highIlt−1∆ log TFPt + βlev low(1− I l

t−1)∆ log TFPt + γ2Zt + ε2t (23)

42Following Liu, Wang, and Zha (2013), we interpret the entrepreneurs in the model as the non-financialbusiness sector in the data.

32

∆ log xt = α3 + βlev high, t f p highIlt−1It∆ log TFPt + βlev high,t f p lowI

lt−1(1− It)∆ log TFPt (24)

+ βlev low, t f p high(1− Ilt−1)It∆ log TFPt

+ βlev low,t f p low(1− I lt−1)(1− It)∆ log TFPt + γ3Zt−1 + ε2t

where xt is either real output or real land price, It is an indicator for TFP shock whichequals value 1 when TFP shocks are high, and value 0 when they are low43. Similarly, I l

t

is an indicator for the real estate leverage ratio which equals value 1 when the leverageratio is high, and value 0 when the leverage ratio is low. We classify the leverage ratio ashigh when they are above the linear trend over the sample period44. As a state variable,we use the lagged value for I l

t in the regressions. Zt is a vector of control variables.In equation (22), due to the asymmetric effect on TFP shocks, we expect βt f p low

larger than βt f p high. In equation (23), the responses to TFP shocks depend on the realestate leverage ratio in the business sector. Our theoretical results predict that TFP shocksshould have larger effects when the wealth share of the entrepreneurs are low, and thusβlev high should be larger than βlev low. In equation (24), the responses to TFP shocksdepend both on whether the real estate leverage ratio is high or low and whether the TFPshock is high or low, so we have four different values of β for four different situations.We expect βlev high,t f p low to be the largest among the four.

Table 5 presents the regression results about the effect of TFP shocks on changes inland price. Specification (1) does not differentiate whether the TFP shocks or the realestate leverage ratio is high or low. We find that on average 1 percent of TFP shock in-creases land price by 0.59 percent. This result is significant at the 10% level. Specification(2), which corresponds to equation (22), depends on whether the TFP shock is high or low,and shows that the asymmetric effect on this dimension is strong. The effect of a high TFPshock is small and not significant, while a low TFP shock has large and significant effectsuch that a 1% TFP shock has strong amplified effect of 1.83% on housing price. Specifica-tion (3), which corresponds to equation (23) shows that when the real estate leverage ratiois high, the responses are large and significant at 5% level. On the contrary, when the realestate leverage ratio is low, the responses to TFP shocks are insignificant. Specification (4),which corresponds to equation (24), combines the previous two cases of high or low TFPshock and high or low real estate leverage ratio, resulting in four possible combinations.We find that TFP shocks have significant effects on land price only when TFP shocks arelow, and the effect is the largest when we have both a high real estate leverage ratio and

43We use the first quintile of the TFP shocks as the threshold value. As indicated by our model, a negativeTFP shock generates the amplified effects only when it is low enough.

44As robustness checks, we also use other two definitions for I lt including comparing the leverage ratio

with its quadratic time trend and with the sample mean. The results obtained are similar.

33

a low TFP shock. All the results shown here are consistent with the predictions of ourmodel.

Table 6 presents the regression results about the effect of TFP shocks on changes inreal GDP. Similarly, the coefficients show high asymmetric effects depending on whetherthe TFP shock or the real estate leverage ratio is high or low. Compared to the results ofland price in Table 5, the responses of output are smaller, but the asymmetric effects arevery robust.

The empirical findings in Tables 5 and 6 complement the recent results in Guerrieri andIacoviello (2015a), which emphasizes the asymmetric responses of economic activities,including consumption and employment, to changes in house prices. Using state andMSA-level data, they find that changes in economic activities correlate more with houseprices when house prices are low. This is consistent with our results because in our modelland price is strictly increasing in the entrepreneurs’ wealth share, and thus is strictlydecreasing in real estate leverage ratio.

6 Conclusion

In this paper, we have shown that market incompleteness, independently of the collat-eral constraint, plays a quantitatively significant role in the amplified and asymmetricresponses of the economy to exogenous shocks. For simplicity, we only consider one typeof shock - productivity shocks. However, it is easy to extend the paper to incorporateother shocks such as housing preference shocks. It would also be interesting to incor-porate money into the model to consider the effects of nominal versus inflation indexeddebts and of monetary shocks and as in Iacoviello (2005) and Cordoba and Ripoll (2004a).

The current model does not have capital. We can consider adding capital into themodel to examine how the amplification and asymmetric effects affect the capital accu-mulation and the aggregate production processes. Cao (2010) offers a way to introducecapital into this kind of model which enables the use of a similar global nonlinear solu-tion method to the one in Appendix B. However, in this case we need to keep track of twoendogenous state variables: the wealth distribution and the aggregate capital.

Another promising venue is to evaluate the effects of different financial markets struc-tures on welfare of different agents in the economy. One can ask whether the exists somefinancial markets reform that improves welfare of every agents in the economy (Paretoimprovement) while reducing aggregate volatility. If not, how much do different agentshave to be compensated to adopt financial market structures that imply more stability,and when are the better times for such reforms? We preliminarily investigate these ques-

34

tions in Online Appendix J.45

References

Becker, R. (1980). On the long-run steady state in a simple dynamic model of equilibriumwith heterogeneous households. Quarterly Journal of Economics 95(2), 375–382.

Bernanke, B. S., M. Gertler, and S. Gilchrist (1999). Chapter 21 the financial accelerator in aquantitative business cycle framework. Volume 1, Part C of Handbook of Macroeconomics,pp. 1341 – 1393. Elsevier.

Bianchi, J. (2011). Overborrowing and systemic externalities in the business cycle. Ameri-can Economic Review 101(7), 3400–3426.

Bianchi, J. and E. G. Mendoza (2013, December). Optimal time-consistent macropruden-tial policy. Working Paper 19704, National Bureau of Economic Research.

Brunnermeier, M. and Y. Sannikov (2014). A macroeconomic model with a financial sector.American Economic Reviews 104(2), 379–421.

Cao, D. (2010). Collateral shortages, asset price and investment volatility with heteroge-neous beliefs. Georgetown University Working Paper.

Cao, D. (2011). Speculation and financial wealth distribution under belief heterogeneity.Georgetown University Working Paper.

Cordoba, J. C. and M. Ripoll (2004a). Collateral constraints in a monetary economy. Jour-nal of the European Economic Association 2(6), 1172–1205.

Cordoba, J.-C. and M. Ripoll (2004b). Credit cycle redux. International Economic Re-view 45(4), 1011–1046.

Davis, M. A. and J. Heathcote (2007). The price and quantity of residential land in theunited states. Journal of Monetary Economics 54, 2595–2620.

DiTella, S. (2014). Uncertainty shocks and balance sheet recessions. Stanford GSB WorkingPaper.

Duffie, D., J. Geanakoplos, A. Mas-Colell, and A. McLennan (1994). Stationary markovequilibria. Econometrica 62(4), 745–781.

45We would like to thank one of the referees for suggesting this interesting exercise.

35

Fernald, J. G. (2014). A quarterly, utilization-adjusted series on total factor productivity.FRBSF Working Paper 2012-19.

Fostel, A. and J. Geanakoplos (2008). Leverage cycles and the anxious economy. AmericanEconomic Reviews 98(4), 1211–1244.

Gali, J. (1999). Technology, employment, and the business cycle: Do technology shocksexplain aggregate fluctuations? American Economic Review 89(1), 249–271.

Geanakoplos, J. (2010). Leverage cycles. In K. R. Daron Acemoglu and M. Woodford(Eds.), NBER Macroeconomics Annual 2009, Volume 24, pp. 1–65. University of ChicagoPress.

Guerrieri, L. and M. Iacoviello (2015a). Collateral constraints and macroeconomic asym-metries.

Guerrieri, L. and M. Iacoviello (2015b). Occbin: A toolkit for solving dynamic modelswith occasionally binding constraints easily. Journal of Monetary Economics 70, 22 – 38.

He, Z. and A. Krishnamurthy (2011). A model of capital and crises. The Review of EconomicStudies 82(2).

He, Z. and A. Krishnamurthy (2013). Intermediary asset pricing. American EconomicReviews 103(2), 732–770.

Iacoviello, M. (2005). House prices, borrowing constraints, and monetary policy in thebusiness cycle. American Economic Reviews 95(3), 739–64.

Jeanne, O. and A. Korinek (2013, January). Macroprudential regulation versus moppingup after the crash. Working Paper 18675, National Bureau of Economic Research.

Khan, A. and J. Thomas (2013). Credit shocks, and aggregate fluctuations in an economywith production heterogeneity. Journal of Political Economy 121(6), 1055–1106.

King, R. G., C. I. Plosser, and S. T. Rebelo (1988). Production, growth and business cycles.Journal of Monetary Economics 21(2), 195 – 232.

Kiyotaki, N. and J. Moore (1997). Credit cycles. Journal of Political Economy 105(2), 211–248.

Kocherlakota (2000). Creating business cycles through credit constraints. Minneapolis FedQuarterly Review 24(3), 2–10.

36

Krishnamurthy, A. (2003). Collateral constraints and the amplification mechanism. Jour-nal of Economic Theory 111, 277–292.

Krishnamurthy, A. (2010). Amplification mechanisms in liquidity crises. American Eco-nomic Journal: Macroeconomics 2(3), 1–30.

Kubler, F. and K. Schmedders (2003). Stationary equilibria in asset-pricing models withincomplete markets and collateral. Econometrica 71(6), 1767–1793.

Liu, Z., P. Wang, and T. Zha (2013). Land-price dynamics and macroeconomic fluctua-tions. Econometrica 81(3), 1147–1184.

Lorenzoni, G. (2008). Inefficient credit booms. The Review of Economic Studies 75(3), 809–833.

Mendoza, E. (2010). Sudden stops, financial crises and leverage. American Economic Re-views 100(5), 1941–1966.

Mian, A. and A. Sufi (2014). House of Debt. University of Chicao Press.

Rampini, A. A. and S. Viswanathan (2010). Collateral, risk management, and the distri-bution of debt capacity. The Journal of Finance 65(6), 2293–2322.

Shleifer, A. and R. Vishny (1997). The limits of arbitrage. Journal of Finance 52(1), 35–55.

Smets, F. and R. Wouters (2007). Shocks and frictions in us business cycles: A bayesiandsge approach. American Economic Review 97(3), 586–606.

Appendix

A Complete Markets

When markets are complete, because the entrepreneurs are less patient than the house-holds, they tend to consume all their net worth in the present. If they are risk-neutral asin Brunnermeier and Sannikov (2014), they will consume their entire net worth at time0, but in our model, they are risk-averse, so they consume their net worth overtime,and their wealth relative to the households’ wealth goes to zero as time goes to infin-ity. The entrepreneurs can do so by issuing equity to households thanks to frictionlessfinancial markets. We consider the long run limit, when the wealth of the entrepreneursapproaches zero. The economy converges to a representative household economy, with

37

the production technology of the entrepreneurs. Due to the Markovian feature of uncer-tainty, the endogenous variables depend only on the aggregate state st: xt = x (st).

Given that the households own the whole production sector, the marginal utility fromthe marginal profit per unit of land should be equal to the marginal utility of one unitof land consumption, i.e., (c′t)

−σ2 πt = j (h′t)−σh . To evaluate the marginal utility of con-

sumption, we observe that the households consume the whole output in the long runso

c′t = Yt = Athυt L1−υ

t .

From the equalization of the marginal utility from the marginal profit and the marginalutility from land consumption, and using the expression for wage and profit in Subsection2.3, we have

j (H − ht)−σh =

(Athυ

t L1−υt

)−σ2υAthυ−1

t L1−υt . (25)

The consumption and labor trade-off equation (15) implies

(1− υ) Athυt L−υ

t

(Athυ

t L1−υt

)−σ2= Lη−1

t . (26)

The two equations (25) and (26) help us solve for the two unknowns (ht, Lt) for each At.Now, given πt as a function of At,ht, and Lt, land price is determined by the pricing

kernel using the marginal utility of the representative household:

q (st)− π (st) = ∑st+1|st

βu′ (ct+1 (st+1))

u′ (ct (st))q (st+1)Pr (st, st+1) .

Given the solution, we can compute the changes of land price and output after the shockst changes from normal to expansion or recession. The result is shown in the first row ofTable 3.

Under log utilities σ1 = σ2 = σh = 1 and linear cost of labor η = 1, i.e., the parametersin Table 1, we can derive the analytical solutions for the policy functions and wealthdynamics of the complete market economy (outside the long run limit described above).In particular, we find that

1. The wealth dynamics ωt+1(ωt) is strictly increasing, and is independent of both st

and st+1.

2. The land holding of the entrepreneurs, h and labor demand L depend on ωt only. Inparticular, They are independent of st.

3. The entrepreneurs consume a constant fraction 1 − γ of their financial wealth in

38

each period:46

c (ωt, st) = (1− γ)q (ωt, st) Hωt.

The analytical solutions are listed below:

ωt+1 (ωt) =γωt

β− υ(β− γ)ωt(27)

h (ωt) =υ

(j + υ)

(1− β)H + [j(1− γ) + υ(β− γ)] Hωt

υ(β− γ)ωt + (1− β)(28)

L (ωt) = (1− υ)(1− β) + [j(1− γ) + υ(β− γ)]ωt

(1− β)(1− υωt)(29)

c (ωt, st) = At(1− γ)

[(1− υ)(j + υ)

(1− β)

]1−υ ωt

(1− υωt)

[υH(1− υωt)

(1− β) + υ(β− γ)ωt

]υ

(30)

c′ (ωt, st) = (1− υ)At

[(1− υ)(j + υ)

(1− β)

]−υ [ υH(1− υωt)

(1− β) + υ(β− γ)ωt

]υ

(31)

qt = Atυ

[(1− υ)(j + υ)

(1− β)

]1−υ [υH(1− υωt)]υ−1

[(1− β) + υ(β− γ)ωt]υ . (32)

From equation (27), we can see that ωt → 0 as t→ ∞. The limiting values of the othervariables are L∞ = 1 − υ, h∞ = υ

(j+υ)H > 0, c∞ = 0, c′∞ (st) = Y∞ (st) = A(st)(1 −

υ)[

υH(1−υ)(j+υ)

]υ, and q∞ (st) = υA(st)

(1−β)

[υH

(1−υ)(j+υ)

]υ−1. It is easy to verify that (h∞, L∞)

solves (25) and (26).

B Global Nonlinear Method

We solve the exact nonlinear equilibrium of the model using the algorithm in Kubler andSchmedders (2003) and Cao (2010). In particular, we solve for the Markov equilibriumin this economy. The original algorithm in Kubler and Schmedders (2003) is for endow-ment economies. Cao (2010) extends this algorithm to a production economy with capitalaccumulation. In the current paper, we show that the original algorithm works similarlywhen we add labor choice as well as housing consumption decision of the households.We present the details of this algorithm for the benchmark model - Model 1a - but thisalgorithm can be adapted to Models 1b, 2, 3, 4 with minor changes.

46As we will show in Online Appendix J, this relation holds for all the market structures with log utilitiesand linear cost of labor.

39

Our algorithm looks for a Markov equilibrium mapping from the financial wealth dis-tribution, ωt - defined in (8), and aggregate shock, st, to land price, qt and bond price, pt,the allocations {ct, ht, bt, c′t, h′t, b′t, L′t} and wage wt, as well as future financial wealth distri-butions, ωt+1(st+1), depending on the realization of future aggregate shocks, st+1. Indeedgiven the mapping from ωt+1 to

{qt+1, ct+1, c′t+1

}, for each ωt and st, we can solve for

{pt, qt, wt} and {ct, ht, bt, µt, c′t, h′t, b′t, L′t} and ωt+1 using equations (9),(10),(11),(12), (13),(14), (15), the budget constraints of the entrepreneurs and the households, the consump-tion good, land, and bond market clearing conditions, as well as the consistency of fu-ture wealth distributions with current portfolio choices, equation (8) for each future state.Here, we follow the procedure in Cao (2010), instead of the one in Kubler and Schmed-ders (2003), in solving for ωt+1 simultaneously with other unknowns. The additionalequations needed to solve for ωt+1 are equation (8) applied to each of the future statest+1: ωt+1 = qt+1(ωt+1,st+1)ht+bt

qt+1(ωt+1,st+1)H , in which the mapping from future wealth distribution andexogenous state to land price, qt+1(ωt+1, st+1), is determined from the previous iterationof the algorithm. It is easy to verify that the number of unknowns are exactly the same asthe number of equations.

We solve for the Markov equilibrium using backward induction. The algorithm startsby solving for the equilibrium mapping for 1-period economy. Then given the map-ping from t = 0 to t = 1 for T-period economy, we can solve for the mapping for(T + 1)-period economy following the procedure described above. The algorithm con-verges when the initial mappings for T−period economy and (T + 1)-period economyare sufficiently close to each other.

An important difference relative to Kubler and Schmedders (2003) and Cao (2010) isthat in Model 2 the entrepreneurs are only subject to the natural borrowing limit, thereforewhen ωt is exactly at the natural borrowing limit ωt = ωt = 0, ct = 0 and the first-order conditions (18) and (19) are not well-defined.47 To deal with this issue, we solveseparately the equilibrium of the economy at ωt. At ωt, we know ct = 0, so we solve for{ht, bt, c′t, h′t, b′t, L′t, ωt+1} given the same set of equations described above.48 Given thatthe behavior of the economy is highly nonlinear around ωt, when we discretize [ωt, 1],we put more points in the range of low values of ωt.49

47When the entrepreneurs have some labor endowment, ωt can actually be slightly negative.48We can show that ωt+1 = 0. In Appendix C, we provide an analytical example that it is possible (even

in finite horizon) to have ωt+1 = 0 and ct+1 = 0 but ht+1, qt+1 > 0.49Given that the households are allowed to borrow from the entrepreneurs, the upper bound for ωt

should exceed 1. However, ωt tends to fall below 1, because of the households’ tendency to lend given theirhigher discount factor.

40

C Two-Period Economy

In this appendix, we present a simple version of the general model with two periods andlog utility in consumption and land, and linear utility in leisure.

Consider a two-period version of the benchmark model in Section 2, with σ1 = σ2 =

σh = η = 1. That corresponds to the utility of the households,

E0

1

∑t=0

βt {log c′t + j log h′t − L′t}

.

and the utility of the entrepreneurs

E0

1

∑t=0

γt log ct.

At time t = 0, the entrepreneurs are endowed with a fraction ω0 of the total land sup-ply, and the households are endowed with the remaining fraction, 1−ω0. The sequentialbudget constraints, production function, and competitive and Markov equilibria underdifferent financial markets structure are defined as in the main body of the paper.

In order to solve for the Markov equilibrium of this economy, we first solve for theequilibrium in the last period t = T = 1 given the financial wealth share of the en-trepreneurs, ωT.50

The entrepreneurs maximize consumption,

maxcT ,hT ,LT

log cT

subject to the budget constraint

cT + qThT = qT HωT + AThυT L1−υ

T − wT LT.

From the F.O.Cs

wT = (1− υ)AT

(hT

LT

)υ

and qT = υAt

(hT

LT

)υ−1

.

50This financial wealth share is determined endogenously from the equilibrium in period t = T− 1, butat period T, the agents take it as given.

41

Substituting these expressions into the entrepreneurs’ budget constraint implies

cT = qT HωT.

The households maximize

maxc′T ,h′T ,L′T

{log c′T + j log h′T − L′T}

subject toc′T + qTh′T = qT H(1−ωT) + wT L′T.

From the F.O.Cs in c′T, L′T, h′T, we obtain

c′T = wT andqT

c′T=

jh′T

.

ThereforeH − hT

j=

wT

qT=

(1− υ)

υ

hT

LT,

where the second equality comes from the F.O.Cs of the entrepreneurs. So

LT =j(1− υ)

υ

hT

H − hTand qT = υAT

[υ

j(1− υ)(H − hT)

]υ−1

.

Combining these results with the market clearing condition in the good markets: cT +

c′T = YT, we obtain

hT =1 + jωT

j/υ + 1H and h′T =

j/υ− jωT

j/υ + 1H,

and

qT = υAT

[υ(1−ωTυ)

(1− υ)(j + υ)H]υ−1

. (33)

We can also solve for cT and c′T explicitly:

cT =qT HωT = υAT

[υ(1− υωT)

(1− υ)(j + υ)H]υ−1

HωT

c′T =υAT

[υ (1− υωT) H(1− υ)(j + υ)

]υ−1 (1− υωT) Hj + υ

.

42

These expressions also imply that

cT

c′T= (j + υ)

ωT

1− υωT, (34)

which is strictly increasing in ωT. We will use this result later when we analyze the twoperiod economy.

The following propositions summarize the key properties of the equilibrium in thelast period:

Proposition 1. qTAT

is strictly increasing in ωT and qT > 0 when ωT = 0.

Proof. From (33),qT

AT= υ

[υ(1−ωTυ)

(1− υ)(j + υ)H]υ−1

,

which is strictly increasing in ωT given that υ < 1. Moreover, when ωT = 0, qT =

υAT

[υ

(1−υ)(j+υ)H]υ−1

> 0.

Proposition 2. YTAT

and YTAT

are strictly increasing in ωT, where YT is the output taking intoaccount the imputed value of housing services to the household defined as in equation (7):

YT = YT + jc′T.

Moreover, YT and YT > 0 when ωT = 0.

Proof. After lengthy algebra manipulations, we arrive at

YT

AT= Hυυυ(1− υ)1−υ 1

(j + υ)υ1 + jωT

(1− υωT)1−υ

,

andYT

AT= Hυυυ(1− υ)1−υ 1

(j + υ)υ1 + j + j(1− υ)ωT

(1− υωT)1−υ

.

It is easy to see that both functions are strictly increasing in ωT and strictly positive evenwhen ωT = 0.

Propositions 1 and 2 suggest that the entrepreneurs are more efficient than the house-holds in using land, so the more wealth (or land share) they own, the higher aggregateoutput is, which in turn leads to higher land price. Indeed from the first order conditions,we have:

43

qT = υAThυ−1T L1−υ

T =jc′Th′T

.

Therefore, the imputed rental value of one additional unit of land consumed by the house-holds is equal to the additional output produced by the entrepreneurs, keeping labordemand as given. At the margin, the entrepreneurs are as efficient in using land to pro-duce as the households are in consuming land. However, this identity does not take into

account the fact that the entrepreneurs optimize on labor demand: LT =((1−υ)AT

wT

) 1υ hT.

When they are given an additional (infinitesimal) dhT amount of land, they demand morelabor, resulting in total increase in output of

AT

((1− υ)AT

wT

) 1−υυ

dhT = AThυ−1T L1−υ

T dhT

> υAThυ−1T L1−υ

T dhT =jc′Th′T

(−dh′T).

Therefore total output, including the imputed value of land consumption by the house-holds, increases. For this reason, we can think the entrepreneurs as the “natural buyers”of land in Shleifer and Vishny’s language.

Given the equilibrium at t = T = 1, we consider the equilibrium of the economy att = T − 1 = 0 under two different financial markets structure. In the first case, financialmarkets are complete, i.e., from the first period t = T − 1 = 0, the entrepreneurs andhouseholds have access to a complete set of securities contingent on the realizations ofthe aggregate productivity AT in the second period. In the second case, financial marketsare incomplete, i.e. from the first period t = T− 1 = 0, the entrepreneurs and householdscan only trade bonds with payoff independent of the realizations of the aggregate pro-ductivity AT in the second period. Under complete markets, we show that qT

ATand YT

AT, YT

AT

are independent of AT. While under incomplete markets, we show that qTAT

and YTAT

, YTAT

arestrictly increasing in AT.

Complete Markets:At time t = 0, the entrepreneurs and households can trade state contingent securities -

φ0(s1) and φ′0(s1) denoting the holdings of the entrepreneurs and the households respec-tively - which pays off one unit of consumption good if state s1 is realized at t = 1 and0 otherwise. The prices of these securities are p0(s1) and are determined such that themarkets for these securities clear, i.e. φ0(s1) + φ′0(s1) = 0, for each s1. The entrepreneursand households can also trade in the land market at price q0, and land market clears, i.e.h0 + h′0 = H. The entrepreneurs are initially endowed with a fraction ω0 of the total stock

44

of land.At time t = 0, the entrepreneurs solve

maxc0,c1(.),h0,φ0(.),L0,h1(.),L1(.)

log c0 + γE [log c1]

subject to:c0 + q0h0 + ∑

s1

p0 (s1) φ0 (s1) = q0Hω0 + A0hυ0 L1−υ

0 − w0L0,

andc1 + q1h1 = q1Hω1 + A1hυ

1 L1−υ1 − w1L1,

where ω1 = q1h1+φ0(s1)q1H . The households maximize their inter-temporal expected utility

subject to similar constraints.Proposition 3 below shows that, with complete markets, land price and output at time

t = 1 are proportional to the realization of productivity A1.

Proposition 3. In any competitive equilibrium, the ratios q1A1

, Y1A1

, Y1A1

are independent of the real-ization of s1.

Proof. From the F.O.Cs of the entrepreneurs:

q0 − υA0hυ−10 L1−υ

0c0

= γE

[q1

ct

]p0(s1)

c0= γπ(s1|s0)

1c1(s1)

w0 = (1− υ) A0hυ0 L−υ

0

Similarly, from the F.O.Cs of the households

q0

c′0=

jh′0

+ βE

[q1

c′1

]p0(s1)

c′0= βπ(s1|s0)

1c′1(s1)

c′0 = w0.

From the FOCs in state-contingent security holding of the entrepreneurs and the house-holds,

c0

c′0=

β

γ

c1(s1)

c1(s1).

45

From the equilibrium in the last period, in particular, the equation (34), c1c′1

is strictly in-creasing in ω1(s1). Thus, the last equation implies that ω1(s1) is independent of s1. Alsofrom the analysis of the equilibrium in the last period, as qT

AT, YT

AT, YT

ATare strictly increasing

functions of ω1(s1), these ratios are independent of s1.This proposition implies that symmetric shocks to productivity lead to symmetric re-

sponses of land price and output.51

Incomplete Markets:In contrast to complete markets, under incomplete markets, at time t = 0, in addition

to trading in the market for land, the entrepreneurs and households can only trade in astate non-contingent bond - b0 and b′0 denoting the bond holdings for the entrepreneursand the households respectively - which pays off one unit of consumption good inde-pendent of the realization of state s1 at t = 1. Bond price p0 is determined such that themarket for the bond clears, i.e. b0 + b′0 = 0. Similar to the model with complete mar-kets, the entrepreneurs and households can also trade in the market for land at price q0,and land market clears, i.e., h0 + h′0 = H. The entrepreneurs are initially endowed with afraction ω0 of total land supply.

At time t = 0, the entrepreneurs solve

maxc0,c1(.),h0,b0,L0,h1(.),L1(.)

log c0 + γE [log c1]

subject to:c0 + q0h0 + p0b0 = q0Hω0 + A0hυ

0 L1−υ0 − w0L0,

andc1 + q1h1 = q1Hω1 + A1hυ

1 L1−υ1 − w1L1,

where ω1 = q1h1+b0q1H .

Proposition 4 below shows that, under some conditions on the initial wealth distribu-tion, because of market incompleteness, the ratio of land price and output to productivityat time 1 to productivity depends on the realization of productivity itself.

Proposition 4. In a competitive equilibrium with b0 < 0, in the second period, q1A1

, Y1A1

, Y1A1

are allstrictly increasing in A1.

51The magnitude of the response of course depends on the initial wealth share of the entrepreneurs, ω0,as it determines the equilibrium at time 0, q0, p0(.) and h0, φ0(.).

46

Figure 14: Two-Period Equilibrium

Proof. From the analysis of the equilibrium in the last period,

q1 =υA11

(1−ω1υ)1−υ

[υ

(1− υ)(j + υ)H]υ−1

, (35)

and from the definition of ωt,

ω1 =h0

H+

b0

q1H. (36)

These two equations help determine q1 and ω1 as functions of h0, b0 and A1.52 In Figure14, the equilibrium values of q1 and ω1 are determined by the intersection of the twocurves that relate entrepreneurs’ wealth share and land price at t = 1 - BS curve (green)which corresponds to the balance sheet equation (36) and Q curve (red) that correspondsto pricing equation (35). When b0 < 0, both curves are upward sloping. As depicted inthe figure, a decrease in A1 shifts the Q curve to the left (red curve to dotted red curve),while the BS curve remains unchanged. Thus this shift decreases the equilibrium ω1, aswell as equilibrium price q1. Because the pricing equation is strictly concave in ω1 and thebalance-sheet function is strictly increasing and concave in ω1, a decrease in A1 leads to alarger decrease in ω1 than an increase in A1 does.

This amplification effect is similar to the balance sheet effect described in Krishna-murthy (2010). However, here we do not impose the additional collateral constraint asin the paper. First observe that land price is increasing in the wealth share (relative networth) of the entrepreneurs (equation (35)). A negative shock to aggregate productiv-

52Given the solutions for q1 and ω1 from equations (35) and (36), h0, b0 are determined in equilibrium attime 0.

47

ity, keeping the wealth share as given, decreases land price. This decreases wealth share(equation (36)); further decreases land price (equation (35)), setting off the vicious circleof falling land price and wealth share.

D Data Description and Empirical Results

Our empirical results are based on four U.S. aggregate series: the TFP shock, the real priceof land, the real output and the real estate leverage ratio in the non-financial business sec-tor. The data of real price of land covers the period 1975:Q1 to 2010:Q4. For the other threeseries, the sample period is from 1951:Q4 to 2014:Q4. All series are seasonally adjustedusing the X-13ARIMA-SEATS program provided by the U.S. Census Bureau.

The TFP shock is retrieved from Federal Reserve Bank of St. Louis. We use utilization-adjusted quarterly-TFP series for the U.S. Business Sector, produced by John Fernald (Fer-nald, 2014). We divide the original series by four to cancel his annualization adjustment.

The real price of land is from Liu, Wang, and Zha (2013). As claimed in their paper,fluctuations in real estate values are mostly driven by changes in land prices. They useliquidity-adjusted price index for residential land, which they construct from the FHFAHome Price Index using the method provided in Davis and Heathcote (2007) . The seriesis adjusted using the price index of nondurables consumption and services from Bureauof Economic Analysis. See Appendix A of their paper for more details on this series.

The real output data is the real GDP from Federal Reserve Bank of St. Louis usingchained dollars method (2009=100). We divide the original series by four to cancel theirannualization adjustment.

The real estate leverage ratio is the ratio of the net value of credit market debt (creditmarket debt - credit market instruments) to the market value of real estate in the non-financial business sector. All the data used to construct this ratio is from the FinancialAccounts of United States provided by the Federal Reserve Board.

48

Table 5: Regression of Housing Prices on TFP Shocks(1) (2) (3) (4)

∆TFPt 0.59*(0.34)

∆TFPt high 0.02(0.46)

∆TFPt low 1.83**(0.75)

∆TFPt, lev high 1.04**(0.47)

∆TFPt, lev low 0.16(0.46)

∆TFPt high, lev high 0.60(0.57)

∆TFPt low, lev high 2.08*(1.14)

∆TFPt high, lev low -0.67(0.61)

∆TFPt low, lev low 1.74*(0.91)

∆hpt−1 0.13 0.12 0.13 0.11(0.09) (0.09) (0.09) (0.09)

Observations 142 142 142 142Note: Regressions of housing price changes on TFP shocks using quarterly U.S. aggregate data from 1975Q1 to 2010Q4. Standard errors in

parenthesis. ***, **, * mean coefficients significant at 1%, 5% and 10% levels respectively. The regressand is the percentage change in land prices.∆TFPt represents TFP shocks. ∆TFPt high means the current TFP shock is high, and ∆TFPt low means the current TFP shock is low. ∆TFPt,lev high is the TFP shock when the leverage ratio in the previous quarter is high, and ∆TFPt, lev low is the TFP shock when the leverage ratio inthe previous quarter is low. ∆TFPt high, lev high means both the current TFP shock and the leverage ratio from the previous period are high. Theother three cases are defined similarly. ∆hpt−1 is the percentage change in housing price in the previous quarter. See Appendix D for more detailsfor the data description.

49

Table 6: Regression of Real Output on TFP Shocks(1) (2) (3) (4)

∆TFPt 0.11***(0.03)

∆TFPt high 0.065*(0.04)

∆TFPt low 0.21***(0.07)

∆TFPt, lev high 0.16***(0.04)

∆TFPt, lev low 0.05(0.04)

∆TFPt high, lev high 0.13***(0.046)

∆TFPt low, lev high 0.20**(0.09)

∆TFPt high, lev low -0.02(0.05)

∆TFPt low, lev low 0.23**(0.09)

∆outputt−1 0.38*** 0.37*** 0.37*** 0.36***(0.06) (0.06) (0.06) (0.06)

Observations 252 252 252 252Note: Regressions of real GDP changes on TFP shocks using quarterly U.S. aggregate data from 1951Q4 to 2014Q4. Standard errors in parenthe-

sis. ***, **, * mean coefficients significant at 1%, 5% and 10% levels respectively. The regressand is the percentage change in land prices. ∆TFPtrepresents TFP shocks. ∆TFPt high means the current TFP shock is high, and ∆TFPt low means the current TFP shock is low. ∆TFPt, lev high isthe TFP shock when the leverage ratio in the previous quarter is high, and ∆TFPt, lev low is the TFP shock when the leverage ratio in the previousquarter is low. ∆TFPt high, lev high means both the current TFP shock and the leverage ratio from the previous period are high. The other threecases are defined similarly. ∆outputt−1 is the percentage change in real GDP in the previous quarter. See Appendix D for more details for the datadescription.

50

E Proof of Lemma 1

We prove this result by contradiction. Suppose that in a competitive equilibrium withthe optimal plan of the entrepreneurs {ct, ht, bt}t,st , there exists t∗ and history st∗

∗ suchthat ωt∗(st∗

∗ ) = qt∗(st∗∗ )ht∗−1(st∗−1

∗ ) + bt∗−1(st∗−1∗ ) < 0. Given the formula for the profit

maximization of the entrepreneurs and the definition of financial wealth ωt, the budgetconstraint (5) can be re-written as

ct∗(st∗∗ ) +

(qt∗(st∗

∗ )− πt∗(st∗∗ ))

ht∗(st∗∗ ) + pt∗(st∗

∗ )bt∗(st∗∗ ) ≤ qt∗(st∗

∗ )ωt∗(st∗∗ )H.

Pick a λ > 1, and consider an alternative trading and consumption plan{

ct, ht, bt}

t,st forthe entrepreneurs which is the same as the initial plan for t < t∗ but for t ≥ t∗:

{ct, ht, bt

}t>t∗,st|st∗∗

= {λct, λht, λbt}t>t∗,st|st∗∗

and

ct∗(st∗∗ ) = λct∗(st∗

∗ )− (λ− 1)qt∗(st∗∗ )ωt∗(st∗

∗ ) > ct∗(st∗∗ )

ht∗(st∗∗ ) = λht∗(st∗

∗ )

bt∗(st∗∗ ) = λbt∗(st∗

∗ ).

It is easy to verify that{

ct, ht, bt}

t,st satisfy all the constraints, including the no-Ponzischemes condition. Since ct ≥ 0 for all t′, st′ , ct ≥ ct for all t, st and ct∗(st∗

∗ ) > ct∗(st∗). Thiscontradicts the property that {ct, ht, bt}t,st is the optimal plan of the entrepreneurs. So bycontradiction ωt ≥ 0 for all t, st.

When 0 < σ1 ≤ 1, we prove by contradiction that any feasible plans of the en-trepreneurs must have positive financial wealth for all t, st. Suppose that in a competi-tive equilibrium, there is a feasible plan

{ct, ht, bt

}t,st of the entrepreneurs with ωt∗(st∗

∗ ) =

qt∗(st∗∗ )ht∗−1(st∗−1

∗ ) + bt∗−1(st∗−1∗ ) < 0 for some t∗ and history st∗

∗ . As argued above, thebudget constraint (5) can be re-written as

ct∗(st∗∗ ) +

(qt∗(st∗

∗ )− πt∗(st∗∗ ))

ht∗(st∗∗ ) + pt∗(st∗

∗ )bt∗(st∗∗ ) ≤ qt∗(st∗

∗ )ωt∗(st∗∗ )H.

Pick a λ > 1, and consider an alternative trading and consumption plan{

˜ct, ˜ht, ˜bt

}t,st

for

the entrepreneurs which is the same as the initial plan for t < t∗ but for t ≥ t∗:{˜ct, ˜ht, ˜bt

}t>t∗,st|st∗∗

={

λct, λht, λbt}

t>t∗,st|st∗∗

51

and

˜ct∗(st∗∗ ) = λct∗(st∗

∗ )− (λ− 1)qt∗(st∗∗ )ωt∗(st∗

∗ ) > (λ− 1)qt∗(st∗∗ )(−ωt∗(st∗

∗ ))

˜ht∗(st∗∗ ) = λht∗(st∗

∗ )˜bt∗(st∗

∗ ) = λbt∗(st∗∗ ).

Since ct ≥ 0 for all t′, st′ ,

E0

[∞

∑t=0

γt ( ˜ct)1−σ1 − 11− σ1

]≥ E0

[∞

∑t=0

γt (ct)1−σ1 − 11− σ1

]

+ γt∗ Pr(st∗∗ )

((λ− 1)qt∗(st∗

∗ )(−ωt∗(st∗

∗ )))1−σ1 − 1

1− σ1− (ct∗(st∗

∗ ))1−σ1 − 1

1− σ1

.

When σ1 < 1, since (qt∗ (st∗∗ )(−ωt∗ (st∗

∗ )))1−σ1

1−σ1> 0, by choosing λ sufficiently large, this al-

ternative plan{

˜ct′ ,˜ht′ ,

˜bt′}

t′,st′clearly delivers a higher value to the entrepreneurs com-

pared to their equilibrium value while satisfying all the constraints, including the no-Ponzi schemes condition. This contradicts the fact that the competitive equilibrium planis optimal. Therefore for all feasible plans, ωt ≥ 0 for all t and st.

Similarly when σ1 = 1, the last term becomes

γt∗ Pr(st∗∗ )(

log(λ− 1) + log(

qt∗(st∗∗ ))+ log

(−ωt∗(st∗

∗ ))− log ct∗(st∗

∗ ))

which also delivers a higher value to the entrepreneurs compared to their equilibriumvalue for sufficiently high λ; this leads to a contradiction.

F Steady State and Log-Linearization for Collateral Con-

straint Model

Becker (1980) shows that in a neoclassical growth model with heterogeneous discountfactors, long run wealth concentrates on the most patient agents, in this case the house-holds. However, in our model, due to the collateral constraint, the entrepreneurs can onlypledge a fraction of their future wealth to borrow. Therefore, despite their lower discountfactor, their wealth does not disappear in the long run. In particular, the model admitsa long run steady state in the absence of uncertainty. In this subsection, we solve for the

52

steady state in our model.Suppose that there is no uncertainty, i.e., At(st) ≡ A. In steady state, all variables are

constant, so we can omit the subscript t. For the ease of notation, denote

γe = mβ + (1−m)γ, (37)

as the average discount factor for the entrepreneurs’ investment in land as in Iacoviello(2005). The first order condition in b′t ,

b′t : −ptc

′−σ2t + βEt

[c′−σ2

t+1

]= 0. (38)

In the steady state, c′t = c′t+1 = c′, implies that p = β. Because γ < β, the entrepreneurwants to borrow as much as possible up to the collateral constraint. Indeed, the first ordercondition for b implies that the collateral constraint is strictly binding and the Lagrangemultiplier µ on the constraint is strictly positive:

µ = (β− γ) c−σ1 > 0. (39)

Given that the collateral constraint is binding, we have b = −mqh.From the first-order condition in h,

(πt − qt)c−σ1t + µtmEt [qt+1] + γEt

[qt+1c−σ1

t+1

]= 0, (40)

in which the marginal profit

πt = υAt

(ht

Lt

)υ−1

.

In the steady state, we have

q =1

1− γeυAhυ−1L1−υ. (41)

The steady state version of the first order condition in Lt is

w = (1− υ)AhυL−υ. (42)

From the budget constraint of the entrepreneurs, we obtain

c =(1− γ)(1−m)υ

1− γeAhυL1−υ.

Combining with the market clearing condition in the market for consumption good, we

53

have c′ = AhυL1−υ − c. The market clearing conditions in the housing market and labormarket imply, h′ = H − h and L′ = L.

So in the steady-state all the variables can be expressed as functions of two unknowns,h and L. The first-order conditions on h′ and L′ of the households provide two equationsthat help determine the two unknowns:

−q(c′)−σ2 + j

(h′)−σh + βq

(c′)−σ2 = 0

andw(c′)−σ2 =

(L′)η−1 .

For example, when σ2 = 1 and σh = 1 as in Iacoviello (2005), the second equation, com-bined with the labor choice equation at the steady state (42) implies

L =

1− υ

1− (1−γ)(1−m)1−γe

υ

1η

From the first equation, h is determined as

hH

=υ (1− β)

υ (1− β) + j[(1− γe)− (1− γ)(1−m)υ].

Given the steady state levels of h and L, the steady state level of wealth distribution isω = (1−m)h

H .Following Iacoviello (2005), we assume that the collateral constraint always binds

around the steady state. Relative to the standard log-linearization technique, we needto solve for the shadow value of the collateral constraint, i.e., the multiplier µt, in addi-tion to prices and allocations. Given a variable xt, let xt denote the percentage deviationof xt from its steady state value, i.e., xt = xt−x

x . Given the exogenous processes for theproductivity shock At, we solve for the endogenous variables ct, c′t, ht, h′t, b′t, qt, wt, pt, Lt,µt using the method of undetermined coefficients.53 The following linear system charac-terizes the dynamics of the economy around the steady state:

(qt − σ2c′t) = −σh(1− β)h′t + βEt[(qt+1 − σ2c′t+1)]

pt = σ2(c′t −Et c′t+1)

53Given the special 3-state structure of the stochastic shocks assumed in the main paper, we cannotdirectly use Dynare to solve for the log-linearized version of the model.

54

wt − σ2c′t = (η − 1)Lt

(1− γe)[At + (υ− 1)ht + (1− υ)Lt − σ1ct]− (qt − σ1ct)

+ m(β− γ)(µt + Etqt+1) + γEt(qt+1 − σ1ct+1)

= 0

β( pt − σ1ct) = (β− γ)µt − γσ1Et(ct+1)

wt = At + υht − υLt

b′t = ht + Etqt+1

c∗ ct + c∗′ c′t = Y∗[At + υht + (1− υ)Lt]

h∗ht + h′∗h′t = 0

c′ c′t + qh′(h′t − h′t−1) + βb′∗( pt + b′t)

= b′∗b′t−1 + wL(wt + Lt).

G Complete Characterization for Model 3

With the parameters chosen in our calibration, in particular, σ1 = σ2 = σh = 1 and η = 1,we can characterize analytically the policy functions and wealth dynamics. The long-runwealth distribution is degenerated to some positive value ω∗54 (stationary-state) insteadof 0 as in the complete market model (Model 0). The solution involves guess and verifyfunctional forms for the policy functions and wealth dynamics. We summarize the mainresults below:

1. The wealth dynamics ωt+1(ωt) is strictly increasing, and is independent of both st

and st+1. In the long run, ωt converges to a positive value ω∗.

2. The land holding of the entrepreneurs, h and labor demand L depend on ωt only. Inparticular, they are independent of st.

3. The entrepreneurs consume a constant fraction 1 − γ of their financial wealth in

54With other commonly used parameter values, we find that the ergodic distribution has very smallvariance if not degenerated. For example, in an exercise with σ1 = σ2 = σh = 2 and η = 1.5, the meanentrepreneurs’ wealth share ω is 0.0206, and the standard deviation is 8.7× 10−5.

55

each period:c (ωt, st) = (1− γ)q (ωt, st) Hωt

4. There exists a threshold value ω such that the collateral constraints are binding (al-together) when ωt ≤ ω and not binding when ωt > ω. In addition, ω∗ < ω and thecollateral constraints are therefore always binding in the stationary state.

Based on result 1 above and the formula of wealth distribution (8), the Arrow securityholdings are

φ(ωt, st+1) = −m(ωt)q(ωt+1, st+1)h(ωt)

with m(ωt) ≤ m. The equality holds if the collateral constraint binds. We can show thatm(ωt) is decreasing in ωt, and the threshold wealth level ω is determined such that theimplied leverage ratio (as if there were no collateral constraint) m(ω) equals m. Specifi-cally, ω can be solved by the following quadratic equation (pick the root in [0, 1]):

γ(j + υ)ω[1− β + υ(β− γ)ω]

υ[β− ω(β− γ)υ] [(1− β) + [j(1− γ) + υ(β− γ)]ω]= 1−m.

The policy functions for the other variables depend on in which region ωt belongs:

1. If ωt > ω, results from equations (27) to (32) still hold.

2. If ωt ≤ ω, the collateral constraints are binding. The policy function for labor supplyL (ωt), (29) still holds. The policy function for land holding of the entrepreneurs,h (ωt) can be solved by the following quadratic equation (pick the root in [0, H]):

(j + υ)

Hωt=

υ(1− β) +[υj(1− γ) + υ2(β− γ) + γ(j + υ)

]ωt

ωt(1−m)h (ωt)− (1− υωt)mj(1− β)

ωt(1−m)[H − h (ωt)](43)

To obtain (43), we can normalize the Lagrangian parameter by55

µt(st+1) =µt(st+1)ct+1(st+1)

Pr(st+1|st),

and then the F.O.Cs (10) and (12) can be written as:

υAthυ−1t L1−υ

t − qt

ct+ ∑

st+1|st

[(mµt(st+1) + γ)π(st+1|st)

qt+1(st+1)

ct+1(st+1)

]= 0 (44)

55We drop ωt from the equations below to save some notations, but it should be clear that all the variablesare functions of ωt (or ωt+1).

56

pt (st+1)

ct= [µt(st+1) + γ]

π(st+1|st)

ct+1(st+1). (45)

By (14) and (45) we have

ct

c′t=

β

µt(st+1) + γ

ct+1(st+1)

c′t+1(st+1)(46)

which implies that µt(st+1) ≡ µt ∀st+1. From (44) with (9), (15), market clearingcondition of consumption good, and ωt+1 = (1−m) ht

H , we have

1(1− γ)Hωt

=υLt

[Lt − (1− υ)]ht+ (mµt + γ)

1(1− γ)(1−m)ht

(47)

With (46) and (13), we have[1

(1− γ)Hωt− µt + γ

(1− γ)(1−m)ht

]Lt − (1− υ)

1− υ=

jH − ht

(48)

Using (47), (48) and (29), we get (43).

3. The other variables can be derived by the following equations:

ωt+1 (ωt) = (1−m)h (ωt)

H(49)

c′ (ωt, st) = (1− υ)A(st)

(h (ωt)

L (ωt)

)υ

c (ωt, st) = Y (ωt, st)− c′ (ωt, st)

= A(st)[L (ωt)− (1− υ)]

(h (ωt)

L (ωt)

)υ

q (ωt, st) =c (ωt, st)

(1− γ)Hωt

By replacing h (ωt) in (43) by ωt+1 using (49) and setting ωt+1 = ωt, we get the degener-ated wealth level as:

ω∗ =ν(1− β) (1−m)

j(1− ν)(1− γe) + ν(1− β)(1 + jm).

In particular, at ωt ≡ ω∗, all the endogenous variables x(ωt, st) can be written as x(st),and the holdings of Arrow securities φ (ωt, st, st+1) as φ (st, st+1) and their prices as p (st, st+1).

57

These variables take the following value:

h∗ =υ(1− β)H

j(1− υ)(1− γe) + υ(1− β)(1 + jm)

L∗ = 1− jmh∗

H − h∗q(s)A (s)

≡ 11− γe

υ (h∗)υ−1 (L∗)1−υ , ∀s = 1, 2, . . . S. (50)

The price of Arrow securities is

p (st, st+1) =βπ(st+1|st)c′ (st)

c′ (st+1)=

βπ(st+1|st)A (st)

A (st+1). (51)

Denote the Lagrangian multipliers as µ (st, st+1), the first-order-condition of φt (st, st+1)

is− p (st, st+1)

c (st)+ µ (st, st+1) +

γπ(st+1|st)

c (st+1)= 0

Thus we have

µ (st, st+1) = π(st+1|st)

[βA(st)

c (st) A(st+1)− γ

c (st+1)

]=

π(st+1|st)(β− γ)

A (st+1) (h∗)υ (L∗)−υ [L∗ − (1− υ)]

> 0

which suggests imperfect risk sharing due to the financial frictions.At ωt = ω∗, the allocations and prices share the same features with those under com-

plete markets in the long run, Model 0, that allocations are constant over time and acrossstates and land price is proportional to aggregate productivity. However the two equilib-ria are not the same. For example, using the expression above, we can show that h∗ < h∞,the limit value of the entrepreneurs’ land holding in complete market economy (Model 0)in Appendix A.

Amplification with Unexpected Shocks We can think of Kiyotaki and Moore (1997) asa counter part of our complete hedging model in their deterministic setting. Because thetransition in Kiyotaki and Moore is deterministic, the number of state-contingent assets -the bond - other than land is equal to the number future states, i.e., one in Kiyotaki andMoore. So why is it that there is amplification in Kiyotaki and Moore but not in ours at thestationary state? The difference comes from whether shocks are expected or unexpected.In Kiyotaki and Moore, the first negative shock is not expected, i.e., an MIT shock (but the

58

subsequent reactions of the economy to the shock are rationally anticipated by the agentsin the economy). However, in the complete hedging version of our economy, all shocksare expected and hedged against by the agents in the economy using state-contingentsecurities, therefore there is no amplification. Following this reasoning, if we introduce anunexpected shock to the complete hedging model, we should also observe amplificationas in Kiyotaki and Moore.The unexpected shocks that we consider are the following. Assume that at time t = 0, theeconomy is at the stationary state ω0 = ω∗. The households and the entrepreneurs tradecontingent claims for t = 1, i.e. the entrepreneurs hold

{φ0(s1)}s1∈{G,N,B} .

We also know that the collateral constraint is binding at ω0 = ω∗, therefore

φ0(s1) + mq1(s1)h0 = 0

for s1 ∈ {G, N, B}. Now at t = 1 and s1 = B, we assume that productivity is lower thanwhat is expected by the agents, i.e. A′ = 0.96 < A1(B) = 0.97. In this sense the shockis unexpected. We assume that the dynamics of At goes back to the rational expectationversion after this unexpected shock.Figure 15 shows the amplification in land price and output:

q′ − q1(B)q1(B)(A1(B)− A′)

andY′ −Y1(B)

Y1(B)(A1(B)− A′)

as functions of the unexpected decrease in productivity: A1(B) − A′ in percentage. Asin Kiyotaki and Moore, the response of land price and output to the unexpected declinein productivity is amplified. For example when productivity declines 1% more than ex-pected, i.e. A′ = 0.96, land price declines by 3 · 1% = 3% more than when the decreaseis expected, i.e., land price decreases by 6% as opposed to 3%. Similarly output declinesby 2 · 1% = 2% more than when the decrease is expected, i.e., output decreases by 5%instead of 3%.

59

Figure 15: Amplification after Unanticipated Shocks

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

q′−q∗(B

)q∗(B

)∗(A

′−A(B

))-4

-3.5

-3

-2.5Amplification of Land Price

Unexpected Decrease in A (Percentage): (A(B)−A′) ∗ 1000 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

Y′−Y

∗(B

)Y

∗(B

)∗(A

′−A(B

))

-3

-2.8

-2.6

-2.4

-2.2

-2

-1.8Amplification of Output

H Models with Incomplete Markets and Exogenous Bor-

rowing Constraint

In this appendix, we examine an alternative model with exogenous borrowing constraint.The model has the same ingredients as the ones in the benchmark model except for thefollowing exogenous borrowing constraint instead of the collateral constraint:

bt ≥ −B. (52)

The borrowing constraint B is chosen exogenously and we study the behavior of themodel when we vary B. Let µt denote the Lagrangian multiplier associated to this bor-rowing constraint. The first-order conditions with respect to ht and bt in the maximizationproblem of the entrepreneurs are

(πt − qt)c−σ1t + γEt[qt+1c−σ1

t+1 ] = 0 (53)

and− ptc

−σ1t + µt + γEt

[c−σ1

t+1

]= 0, (54)

and the complementary-slackness condition is satisfied:

µt(bt + B

)= 0. (55)

60

Other conditions are the same as in the benchmark model.We first solve for the steady state of this model. From equations (38), (54), and (53),

we have

p = β,

µ = (β− γ) c−σ1

q =1

1− γυAhυ−1L1−υ

This expression of land price is different from equation (41) in the discount factor γ in-stead of γe. Since γ < γe, given the same steady state level of h and L, the land price islower under the exogenous borrowing constraint than under the endogenous collateralconstraint since land loses its value as collateral in the former. Consequently, these twomodels do not share the same steady state.

We can use the global nonlinear solution method presented in Appendix B to solvefor the Markov equilibrium in this economy. Table 7 is the counterpart of Table 3 in thepaper for this model with the exogenous borrowing constraint. In particular, Row 3 ofTable 7 corresponds to Row 4 (Model 2) in Table 3 in the paper, in which there is no up-per bound on the borrowing of the entrepreneurs. To make the experiments comparable,In Row 4 of Table 7, the exogenous limit is equal to the unconditional expectation of theentrepreneur’s borrowing in the economy with the endogenous collateral constraint ofthe benchmark model. When we tighten the exogenous constraint, the amplification andasymmetric effects are actually reduced. At first sight, this result seems counter-intuitive.However, this result is in line with the discussions in Mendoza (2010) and Kocherlakota(2000). The exogenous borrowing constraint reduces the borrowing of the entrepreneurs,and thus reduces the net worth effect in the benchmark incomplete markets model. Animportant difference here compared to Kocherlakota (2000) is that under uncertainty, itis possible to have infinite exogenous borrowing constraint in the incomplete marketsmodel (the entrepreneurs limit themselves from borrowing too much because of the pre-cautionary saving motive). Infinite exogenous borrowing constraint leads to the maximalnet worth effect, thus significant amplification and asymmetric effects.

I Volatility Paradox

In Brunnermeier and Sannikov (2014), the authors discover an interesting feature calledthe volatility paradox, i.e., with incomplete financial market, lower exogenous risk can lead

61

Table 7: Average land price and output changes in normal state, 3% shockLand price Output

Type of Friction Expansion Recession Expansion RecessionComplete Markets (Model 0) 3.00% 3.00% 3.00% 3.00%Collateral Constraint (Model 1a) 3.60% -4.22% 3.25% -3.65%incomplete markets (B = ∞, Model 2) 3.46% -3.81% 3.21% -3.43%incomplete markets (B = 2.73) 3.47% -3.52% 3.24% -3.29%

Table 8: Probabilities of Binding Collateral Constraint∆ Expansion Normal Recession Overall

0.1% 100% 100% 100% 100%0.5% 5.3% 90.7% 99.9% 78.7%1% 0.2% 42.9% 95.5% 44.4%2% 0% 4.9% 68.9% 14.2%3% 0% 0.2% 45.1% 7.2%4% 0% 0% 29.1% 4.6%5% 0% 0% 19.2% 3.0%

to higher endogenous risk. Their intuition is that lower exogenous risk encourages the en-trepreneurs to borrow more and use higher leverage which result in larger output/assetprice declines during crisis. We first show that in Model 1a with collateral constraint, ourfully nonlinear solution does not exhibit the volatility paradox because the entrepreneurs’borrowing capacity is constrained. As we decrease the size of the exogenous shocks, thebinding probability goes to 1 and the nonlinear solution becomes closer to the log-linearsolution, and both converge to the steady state with no endogenous risk. See Table 8 forthe details. The stationary distribution is thus degenerated, as shown in Figure 16 withdifferent sizes of exogenous shocks.

On the contrary, in Model 2 with incomplete market but no collateral constraint, werecover the volatility paradox. The stationary distribution is not degenerated when thesize of the exogenous shock goes to 0, as shown in Figure 17.

J Welfare Analysis

We compare the welfares of the entrepreneurs and the households in different economiesby computing the consumption equivalence, i.e., the percentage increase (decrease) of theconsumptions required to make the agents indifferent between the benchmark economy(Model 1a) and the targeted economy. The way we do this is to assume an unexpectedswitch from the benchmark (B) economy to the targeted (T) economy by keeping agents’

62

Figure 16: Stationary Distributions with Collateral Constraint, st = N (Model 1a)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

10

20

30

40

50

60

70

80

90

∆ =0.1%

∆ =0.5%

∆ =1%

∆ =2%

∆ =3%

∆ =4%

∆ =5%

Figure 17: Stationary Distributions without Collateral Constraint, st = N (Model 2)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20

2

4

6

8

10

12

14

16

∆ =1%

∆ =2%

∆ =3%

∆ =4%

∆ =5%

63

portfolios unchanged, in this case housing h and bond b. We compute the consumptionequivalence for each possible value of wealth, and then present the average welfare effectsusing the ergodic distribution in the benchmark economy. Below are the steps to derivethe results:

1. Pick a wealth level ωBt and aggregate shock st, find the portfolio hB(ωB

t , st) andbB(ωB

t , st) in the benchmark economy using its policy functions. Then fix (h, b),and use the fixed-point iteration to locate the corresponding wealth level ωT

t in the

targeted economy: ωTt =

qT(ωTt ,st)h+b

qT(ωTt ,st)H

, where qT(ωTt , st) is the land pricing function

in the targeted economy.

2. Using ωTt from step 1, we can get the value functions for the entrepreneurs and the

households in the targeted economy: VTe (ωT

t , st) and VTh (ωT

t , st). We would like tocompute the percentage increase of consumption for the entrepreneurs, ρe(ωB

t , st)

and the households ρh(ωBt , st) from the benchmark economy to the targeted econ-

omy to make them indifferent. With log utilities as in the paper, we can get

ρe(ωBτ , sτ) = exp

[(1− γ)

(Ve(ωT

t , st)−Ve(ωBt , st)

)]− 1

ρh(ωBτ , sτ) = exp

[(1− β)

(Vh(ωT

t , st)−Vh(ωBt , st)

)]− 1

3. Computing average welfare effects ρe and ρh across the ergodic distribution in thebenchmark economy.

Table 9: Average Consumption Increase from the Benchmark EconomyType of Friction Entrepreneur HouseholdsIncomplete Markets (Model 2) -3.97% 0.09%Partial Hedging ( Model 4) -3.49% 0.16%Complete Hedging ( Model 3) -6.18% 0.25%Complete Market (Model 0) -11.93% 0.42%

The results for welfare analysis are listed in Table 9. In particular, Model 1a generatesthe highest welfare for the entrepreneurs, followed by the partial hedging model (Model4), incomplete markets model (Model 2) and the complete hedging model (Model 3), andtheir welfare in the complete market economy (Model 0) comes as the lowest. The en-trepreneurs have lower discount factor so they always want to consume early. In thecomplete market case (Model 0), their consumptions are zero in the long run. Model 1ayields the highest welfare to the entrepreneurs because the collateral constraint prevents

64

them from pledging all their future incomes and thus they are able to maintain the highestaverage wealth and consumption among all models.

As an example, we look into the 6.18% welfare loss of the entrepreneurs when mov-ing from Model 1a to Model 3. First, we claim that with the parameters in Table 1, thefollowing relationship holds in all the market structures:

ct = (1− γ)qtHωt (56)

which means the entrepreneurs consume a constant fraction 1− γ of their wealth. We canderive this result using equations (3), (5), (8), (9), (10) and (15).

Figure 18: Compare Model 1a and Model 3, st = N

0 0.05 0.1 0.15 0.20

0.5

1

1.5

2

2.5

3

3.5Stationary Wealth Distribution

0 0.05 0.1 0.15 0.20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2Mapping of Wealth to Model 3 in Normal

Mapping of Wealth

45o

Based on (56), welfare reduction for the entrepreneurs could come from two sources:a reduction in their wealth share ωt, or/and a reduction in land price. We first comparethe entrepreneur’s wealth in these two economies. In the left figure in Figure 18, we plotthe stationary distribution in Normal state for Model 1a and Model 3. The vertical lineis the degenerated wealth level in Model 3. We can see that the wealth share in Model 3is significantly lower than the wealth in Model 1a. Actually the average wealth share ofModel 1a is 0.0645, which is about 50% more than the degenerated wealth level in Model3. Entrepreneurs in the complete hedging economy are simply poorer. On the other hand,average land price in Model 1a is 2% lower than in Model 3. Thus if we directly applyequation (56) the welfare loss should be around 30%. Why is it only 6.18% as in Table9? The reason is by keeping the entrepreneurs’ portfolio (h, b) and switching from Model

65

1a to Model 3, the entrepreneurs get higher wealth distribution ωt immediately. This isbecause their debt holding b is negative in the support of the ergodic distribution, andland price is higher in Model 3 than in Model 1a as shown by Figure 9. Thus after thistransition the entrepreneurs enjoy higher consumption in the first several periods but astime passes by their wealth and consumption get lower gradually. Lastly, ωt varies lessin Model 3 than in Model 1a and together with the concavity of the entrepreneurs’ utilityfunction, this lower variance further reduces the loss from lower ωt.

The ranking of the welfare comparison for the households is in the opposite order.Welfare of the households in complete markets economy (Model 0) is the highest fol-lowed by the complete hedging economy (Model 3). Since the collateral constraints arebinding in Model 3, the allocation of resources is less efficient than in the complete mar-kets economy. Besides, the entrepreneurs survive in Model 3 and share part of the output.The model with incomplete markets and collateral constraints (Model 1a) generates thelowest welfare for the households because of the financial frictions.

We also observe that, moving away from Model 1a, the relative decrease in the en-trepreneurs’ welfare is significantly larger than the relative increase in the households’welfare. This finding suggests that financial markets reforms aiming at reducing aggre-gate volatility might not be popular among some agents in the economy.

Now assume that one still wants to reform the financial markets from Model 1a tothe other models, when is the best time to implement the reform to get support from thehouseholds and face the least resistance from the entrepreneurs? To answer this question,we look at the degree of welfare changes upon reform depending on the state in whichthey are implemented. Figure 19 shows the change in the entrepreneurs’ welfare if wemove from Model 1a to Model 3. The entrepreneurs’ welfare decreases the least when ω

is high and in expansions. To the extent that the entrepreneurs need to be compensatedto support the reform from Model 1a to Model 3, it might be the cheapest to enact such areform in such states.

K Alternative Models with Larger Amplification

In the paper, we show that market incompleteness plays a dominant role in amplifyingshocks instead of collateral constraints. We would like to see whether this result wouldlikely carry over to settings with larger amplification. Kocherlakota (2000) suggests twoways to generate larger amplification: (1) by increasing the weight of collateralizable as-sets (in this case, land) in the production function and (2) by using specifications of theproduction function in which the elasticity of substitution between land and labor is less

66

Figure 19: Changes in Entrepreneurs’ Welfare from Model 1a to Model 3

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

ω

-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

Co

nsu

mp

tio

n E

qu

iva

len

ce

Change in Entrepreneurs Welfare from Model 1a to Model 3

Expansion

Normal

Recession

Table 10: Average land price and output changes in normal state, 3% shockLand price Output

Type of Friction Expansion Recession Expansion RecessionIncomplete Markets (υ = 0.03) 3.34% -3.61% 3.11% -3.28%Collateral Constraint (υ = 0.03) 3.45% -3.95% 3.14% -3.47%Incomplete Markets (υ = 0.041) 3.46% -3.81% 3.21% -3.43%Collateral Constraint (υ = 0.041) 3.60% -4.22% 3.25% -3.65%Incomplete Markets (υ = 0.05) 3.56% -3.97% 3.24% -3.49%Collateral Constraint (υ = 0.05) 3.72% -4.40% 3.29% -3.71%Incomplete Markets (υ = 0.06) 3.66% -4.13% 3.30% -3.60%Collateral Constraint (υ = 0.06) 3.85% -4.59% 3.37% -3.82%Incomplete Markets (υ = 0.07) 3.76% -4.29% 3.37% -3.70%Collateral Constraint (υ = 0.07) 3.96% -4.76% 3.45% -3.93%

than 1. We explore both cases and show that our main result still holds.

K.1 Models with Larger Shares of Land in the Production Function

In the benchmark model, the calibrated share of the land in the production function isυ = 0.041. We vary the value of υ to see the relative importance of market incompletenessand collateral constraint in amplifying shocks. Other parameter values are not changed.The results are listed in Table 10 with two types of financial market structures: incompletemarkets structure without borrowing constraint, and incomplete markets with collateralconstraint.

First, we can see that as υ gets larger, the amplification effect is stronger. For example,with 3% negative TFP shock and collateral constraint, land price will decrease by -3.95%

67

Table 11: Probabilities of Binding Collateral Constraintυ Expansion Normal Recession Overall

0.03 0% 3.64% 51.11% 10.52%0.041 0% 0.2% 45.08% 7.22%0.05 0% 0% 40.07% 6.29%0.06 0% 0% 35.72% 5.61%0.07 0% 0% 32.13% 5.04%

when υ = 0.03, but when υ = 0.07, land price decreases by -4.76%. Second, as υ getslarger, market incompleteness by itself explains a larger part in the total amplification ef-fect. With υ = 0.03 and a 3% negative TFP shock, output decreases by -3.28% with marketincompleteness alone compared to -3.47% in the economy with collateral constraint. Asa result, about 60% (3.28%−3%

3.47%−3% ) of the amplification effect is generated by market incom-pleteness. When υ = 0.07, this ratio increases to 75% (3.70%−3%

3.93%−3% ). The comparison suggeststhat in situations with larger amplification, market incompleteness quantitatively plays alarger role in amplifying shocks, and collateral constraint plays a smaller role. This resultshould not be surprising because with larger amplification, the entrepreneurs’ precau-tionary saving motive is stronger and they will borrow less to avoid regions where thecollateral constraint is binding. Although in regions with binding collateral constraint,higher υ generates stronger amplification, the probability of the economy to enter thoseregions are lower. The binding probabilities with different υ are listed in Table 11. Onaverage collateral constraint has less effect in amplifying shocks as υ gets larger.

K.2 Models with CES Production Function

In this subsection, instead of using a Cobb-Douglas production, we consider a constant-elasticity-of-substitution (CES) production function with the elasticity between land andlabor is set as φ = 0.8, as in equation (57). Reducing land input would have a larger effecton output than the Cobb-Douglas case since more labor would be required to neutralizethe effect due to lower substitutability.

Yt = At

(υh

φ−1φ

t + (1− υ)Lφ−1

φ

t

) φφ−1

, (57)

With CES production function, j - household’s weight of housing in utility, and ν - shareof housing in the production function, are re-calibrated using the steady-state equilib-rium. Again, we use two moments, the average value of residential real estate over total

68

Table 12: Average land price and output changes with CES production, 3% shockLand price Output

Type of Friction Expansion Recession Expansion RecessionCollateral Constraint, (Cobb-Douglas, Model 1a) 3.60% -4.22% 3.25% -3.65%Incomplete Markets, (Cobb-Douglas, Model 2) 3.46% -3.81% 3.21% -3.43%Collateral Constraint, CES 3.58% -4.26% 3.30% -3.80%Incomplete Markets, CES 3.43% -3.83% 3.24% -3.53%

output and the average value of commercial real estate over total output from the Flowof Funds, to calibrate the two parameters. We solve for the steady-state with CES produc-tion function and find (j, υ) such that the model’s moments match exactly the empiricalmoments. We find j = 0.071, and υ = 0.042. The amplification effects are listed in Table12. For convenience, we replicate the results with Cobb-Douglas production function forcomparison.

We find that, in general, using a CES production function with the elasticity of sub-stitution less than 1 increases the amplification effects. For example, with collateral con-straint, output decreases by −3.80% with a negative 3% TFP shock compared to −3.65%with Cobb-Douglas production function. We find market incompleteness still explains alarge part in amplifying shocks. In particular, 66% of the response in land price and also66% of the response in output to negative shocks are explained by market incompleteness.

L Continuous Time Model

In this section, we present a continuous time version of the incomplete markets model,Model 2, in the main paper. The equilibrium in this model can be solved using the meth-ods developed in Brunnermeier and Sannikov (2014) and He and Krishnamurthy (2013).

L.1 Economic Environment

The environment is exactly the same as in the main paper, except that time is continuous.The aggregate productivity At follows a diffusion process

dAt = µA(At)dt + σA(At)dZt,

where Zt is a standard Brownian motion.

69

The dynamics of land price qt is determined by

dqt = µqt dt + σ

qt dZt, (58)

where µqt and σ

qt are endogenously determined.

Given the process of land price qt, interest rate rt and wage rate wt, households maxi-mize a lifetime utility function given by

E0

∫ ∞

0e−ρt

{(c′t)

1−σ2 − 11− σ2

+ j(h′t)

1−σh − 11− σh

− 1η(L′t)

η

}, (59)

where E0 [.] is the expectation operator, ρ > 0 is the discount rate, c′t is consumption attime t, h′t is the holding of land. L′t denotes the hours of work. Households can trade in themarket for land as well as a non state-contingent bond market that yield instantaneousrate of return rt. Let b′t denote the holding of non state-contingent bond of the households.The households are subject to the following constraint on the dynamics of their net worthn′t:

dn′t =h′t(µ

qt dt + σ

qt dZt

)+ rtb′tdt− c′tdt + wtL′tdt

n′t =qth′t + b′t (60)

Given land in the utility function of households, implicitly h′t ≥ 0. The households arealso subject to the No-Ponzi scheme conditions

lim T→∞Et

[e−∫ T

t rsds (qThT + bT)]≥ 0.

Entrepreneurs use a Cobb-Douglas constant-returns-to-scale technology that uses landand labor as inputs. They produce consumption good Yt according to

Yt = Athυt L1−υ

t , (61)

where At is the aggregate productivity which depends on the aggregate state st, ht is realestate input, and Lt is labor input.

The entrepreneurs discount the future at the discount rate γ > ρ. The entrepreneursmaximize

E0

∫ ∞

0e−γt (ct)1−σ1 − 1

1− σ1(62)

70

subject to the following constraint on the dynamic of their net worth, nt:

dnt =ht(µ

qt dt + σ

qt dZt

)+ πtht + rtbtdt− ctdt

nt =qtht + bt. (63)

Output Yt is produced by combining land and labor using the production function (61).Given the production function of the entrepreneurs, we have implicitly ht ≥ 0. The en-trepreneurs are also subject to the no-Ponzi schemes condition:

lim T→∞Et

[e−∫ T

t rsds (qTh′T + b′T)]≥ 0.

L.2 Equilibrium

The definition of the sequential competitive equilibrium for this economy is standard andis a continuous version of the competitive equilibrium in the main paper.

Definition 3. A competitive equilibrium is sequences of prices {qt, rt, wt}∞t=0 and alloca-

tions {ct, ht, bt, Lt, c′t, h′t, b′t, L′t} such that (i) qt follows the dynamics (58), (ii) the {c′t, h′t, b′t, L′t}maximize (59) subject to the dynamic net worth constraint (60) and the no-Ponzi condi-tion and {ct, ht, bt, Lt} maximize (62) subject to dynamic net worth constraint (63) andthe no-Ponzi condition, and production technology (61) given {qt, rt, wt} and initial as-set holdings {h0, b0, h′0, b′0}; (iii) land, bond, labor, and good markets clear: ht + h′t = H,bt + b′t = 0, Lt = L′t, ct + c′t = Yt.

Let ωt denote the normalized financial wealth of the entrepreneurs

ωt =nt

qtH

and ω′t denote the normalized financial wealth of the households:

ω′t =n′t

qtH.

By the land and bond market clearing conditions, we have ω′t = 1−ωt in any competitiveequilibrium. Therefore in order to keep track of the normalized financial wealth distri-bution between the entrepreneurs and the households, (ωt, ω′t), in equilibrium, we onlyneed to keep track of ωt. To simplify the language, we use the term wealth distribution fornormalized financial wealth distribution.

71

Markov equilibrium is also a continuous version of Markov equilibrium in the mainpaper.

Definition 4. A Markov equilibrium is a competitive equilibrium in which prices and al-locations at time t, depend only on the wealth distribution at time t, ωt and the exogenousstate At.

This Markov equilibrium is the same as the Markov equilibria in Brunnermeier andSannikov (2014) and He and Krishnamurthy (2013). We can use the algorithms in their pa-pers to solve for the Markov equilibrium in our paper. However, there are two importantdifferences. First, because of persistent TFP shocks (instead of I.I.D. depreciation shocksto capital stock as in Brunnermeier and Sannikov (2014) or I.I.D. return shocks on divi-dend as in He and Krishnamurthy (2013)), in our Markov equilibrium, we need to keeptrack of both wealth distribution and the current exogenous shock. Second, we allow forgeneral utility functions, instead of linear or log utility functions as in Brunnermeier andSannikov (2014) and He and Krishnamurthy (2013).

M Multiple Production Technologies

In this section, we simplify our model in the main paper in the spirit of Brunnermeier andSannikov (2014) as well as Cordoba and Ripoll (2004b) and Kiyotaki and Moore (1997).We assume that the households do not have a preference for housing but have access toan inefficient production function

Y′ = A(h′)υ′ (L′

)1−υ′ (64)

with min(At) < A < max (At). Households maximize a lifetime utility function givenby

E0

∞

∑t=0

βt{(c′t)

1−σ2 − 11− σ2

− 1η(Lt)

η

}, (65)

where Lt is the hours of work (instead of L′t in the benchmark model). The budget con-straint of the households is

c′t + qt(h′t − h′t−1) + ptb′t ≤ b′t−1 + wt Lt + Y′t − wtL′t. (66)

Housing is no longer in the utility function of households, so we have to impose explicitly

h′t ≥ 0.

72

Given their land holding at time t, ht, the households choose labor demand L′t to max-imize profit

maxL′t

{Y′t − wtL′t

}subject to their production technology (64) if they produce. The first order condition withrespect to L′t implies

wt = (1− υ′)A(h′t)

υ′(L′t)−υ′ ,

i.e. L′t =((1−υ′)At

wt

) 1υ′ h′t and profit

Y′t − wtL′t = π′th′t

where π′t = υ′A((1−υ′)A

wt

) 1−υ′υ′ is profit per unit of land for the households.

Definition 5. A competitive equilibrium is sequences of prices {pt, qt, wt}t,st and alloca-

tions{

ct, ht, bt, Lt, c′t, h′t, b′t, L′t, Lt

}t,st

such that (i){

c′t, h′t, b′t, L′t, Lt

}t,st

maximize (65) sub-

ject to the budget constraint (66), the production technology (64), h′t ≥ 0 and the No-Ponzi condition, and {ct, ht, bt, Lt}t,st maximize (62) subject to the entrepreneur’s budgetconstraint, production technology (61), and a collateral constraint, given {pt, qt, wt} andsome initial asset holdings

{h−1, b−1, h′−1, b′−1

}; (ii) land, bond, labor, and good markets

clear: ht + h′t = H, bt + b′t = 0, Lt + L′t = Lt, ct + c′t = Yt.

In the steady state, the entrepreneurs own the whole supply of land. Outside thesteady state, we can use the definition of Markov equilibrium and the associated solutionmethod as in the benchmark model in the main paper. The main difference between thesolution of this model and the benchmark model is that at the natural borrowing limitfor the entrepreneurs, i.e. ωt = 0, the households start producing using their inefficientproduction function. This puts a higher lower bound on the total output as well as landprice compared to when the households cannot produce as in the main paper.

73

ampliﬁcation and asymmetric effects without collateral...

Documents