Balance Sheet Recessions∗
Zhen Huo
José-Víctor Ríos-Rull
University of Minnesota
University of Minnesota
Federal Reserve Bank of Minneapolis
Federal Reserve Bank of Minneapolis
CAERP, CEPR, NBER
February 2013
Abstract
We explore the effects of financial shocks in heterogenous agent economies (a la Aiyagari/KrusellSmith), with frictions in labor markets, where demand contributes to productivity, and where there are
houses that provide services and are in fixed supply. In this economy households of different wealth
and earnings buy different number (varieties) of goods and different quantities of each good. Unlike in
standard neoclassical models, a financial shock that increases the desire to save triggers (we expect)
a recession with asset market depreciatoin that may be quite severe.
Keywords: Balance Sheet Recession, Endogenous productivity
JEL classifications: E20, E32, F44
∗
Ríos-Rull thanks the National Science Foundation for Grant SES-1156228. We are thankful for discussions with Yan
Bai, Kjetil Storesletten, and Nir Jaimovich. The views expressed herein are those of the authors and not necessarily those of
the Federal Reserve Bank of Minneapolis or the Federal Reserve System.
1
Introduction
The current recession in the U.S. has been accompanied by several stylized features: (1) There is an
unprecedented credit cycle in the household side. The debt-to-income ratio increased quickly to its
historical highest level (about 2.3) until 2008, and declined sharply since the crisis begun. (2) Similar to
the debt-to-income ratio, the housing price almost doubled from 2000 to 2008, followed by a dramatic
drop since the beginning of the current recession. (3) The unemployment rate climbed to 10% in 2009
and remained at a high level for a fairly long period, labelled a ‘slow recovery’. (4) Private spending
(especially durables) dropped dramatically (Berger and Vavra [2012]) . (5) Net exports increased by more
than 400 billions of dollars from mid 2008 to 2010. From a micro perspective, Mian and Sufi [2010,
2012] provide convincing county-level evidence to show that a sudden reduction of households credit and
consumers’ demand is linked of the contraction of economic activities.
Motivated by these facts, a growing literature studies recessions due to a lack of household demand. To
avoid the obvious implication that households want to work harder when facing tougher financial times
this literature has typically assumed that price rigidity overturns individual desires. Moreover many of
these papers also frame the liquidity shock in the context of a representative agent environment where
the notion of financial shock is hard to grasp given the large amounts of wealth in the data at its disposal.
An exception to the first shortcoming is Huo and Ríos-Rull [2012] where the reduction in household
expenditures trigger reduced productivity and drastically lower earnings for firms while exceptions to the
second shortcoming are Guerrieri and Lorenzoni [2011] and Midrigan and Philippon [2011] (who, by the
way, find that the quantitative impact of financial shocks is small).
In this paper we pose a model that addresses these two shortcomings in a context where a large drop in
housing prices can also occur. In our environment, households are subject to aggregate and idiosyncratic
uninsurable risk and there is a financial system that allows them to smooth consumption. There are
aggregate shocks to the ability of households to borrow that work like a (temporary) negative wealth
effect. In the model economy, as in Huo and Ríos-Rull [2012], there are various real frictions that difficult
the expansion that results from negative wealth effects in most models. First, there are real rigidities that
make it quite costly to have a fast expansion of the tradable sector and hence a rapid increase of exports.
Second, there are labor market frictions: firms have to incur in hiring costs and wages are determined
via a bargaining process that gives workers some hold-up power and prevents wages from dropping as
much as they would in a competitive environment. Third, and most importantly, there are goods market
frictions as in Bai et al. [2011] and Huo and Ríos-Rull [2012] that imply that consumption expenditures
increase productivity as households bear some of the efforts to extract output of the economy. Fernald
[2012] shows that total factor productivity dropped since 2008 and started to recover after 2010.
An important feature of the current recession is the large swing of the housing price and private debt.
Also, mortgages are the majority (more than 70%) of households’ liabilities in the U.S. In our environment,
households are allowed to trade both financial assets and housing, where the value of housing can be
served as the collateral for financial loans. We consider shocks to the collateral requirement. A more strict
collateral requirement translates in an attempt to save more by households which triggers a recession.
Moreover, the tightened borrowing constraint also reduces the demand for housing, and in turn the housing
price, resulting in an even more severe reduction of the value of housing that can be used as collateral.
Similar vicious circles have been discussed in Mendoza [2010] and Eggertsson and Krugman [2012]. This
effect is stronger for more disadvantaged households which results in asymmetric loses as documented by
in Mian et al. [2011] and Guerrieri and Iacoviello [2013]. The effects of the tightening can be long in the
sense that the economy takes its time to adjust resulting in a prolonged recession even if there would be
no such recession under a stationary environment with a tight financial constraint.
On the technical side, we extend Huo and Ríos-Rull [2012] and Bai et al. [2011] to a heterogeneous-agent
environment where households of different wealth and income levels purchase different number of varieties
and different amount of goods of each variety. We adopt a structure that distinguishes two kinds of inputs
in production: the first is a pre-installed input, which has to be produced no matter whether there will be
shoppers or not (think of the distribution infrastructure such as shops, buildings, administration). The
second is a variable input, which is only used when shoppers actually show up and that can be transferred
across distribution centers. This allows firms to adjust its output at each location to meet asymmetric
demand. Preferences are such that in the cross-section, poorer households consume a smaller number of
varieties and also smaller quantities. Also, when a given household reduces consumption, it does so by
reducing both the number of varieties and the quantity of each variety so an economy-wide reduction of
consumption results in lower productivity.
This paper is related to a rapidly growing literature on goods market frictions. In Alessandria [2009],
when consumers can exert effort to search for cheaper goods, countries with a lower income level may
pay lower prices for the same goods, departing from the law of one price. Faig and Jerez [2005] pose a
theory of commerce where competitive search for goods plays a prominent role, while Lagos [2006] poses
a theory of TFP based on labor distortions. None of these papers focus on business cycles. Bai et al.
[2011] and Huo and Ríos-Rull [2012], the goods market frictions have effects on TFP, both papers are of
the representative agent variety making it a stretch to talk about financial frictions.
We focus on how changes in the savings of households result in aggregate fluctuations, which is also
related to recent literature on the liquidity trap. Eggertsson [2011], Christiano et al. [2011] and SchmittGrohe and Uribe [2012] trigger a recession by shocking households’ preferences. In these papers, an
increased discount factor is considered as a reduced form approach to modelling an increased desire to
save. Rendahl [2012] engineers liquidity traps with search frictions in the labor market, where a transitory
shock can have relatively persistent effects due to a slow adjustment of the unemployment rate.
Another line of literature closely related to this paper is the boom and bust in the housing market.
3
Kiyotaki et al. [2011] explore the causes the boom of housing market in the 2000s and its redistribution
effects. Favilukis et al. [2012] focus on the effects of the endogenous risk premium on the housing price
and its interaction with exogenous changes of financial conditions. Garriga et al. [2012] explain the
boom and bust of the housing market in a small open economy. They carefully construct a segmented
credit market to make the decline in the world interest rate loose the domestic credit condition but not
increase the capital investment. All of the papers above assume that housing serves as collateral, but they
also assume fixed labor supply, which prevents them from linking the change in housing market to the
aggregate production. Justiniano et al. [2012] add the housing market into an otherwise standard DSGE
model. After an increase of the collateral requirement, the model economy experiences a recession, with a
drop in housing price, aggregate output and household liability. However, the magnitude of the recession
is much smaller than what is observed in the data, and labor supply increases for most households, which
is counterfactual. Ríos-Rull and Sánchez-Marcos [2008] explores the implications for housing prices and
transactions of business cycles.
2
The Model
We consider a Bewley-Imrohoroglu-Huggett-Aiyagari type model in a small open economy (i.e., the interest
rate is set by the rest of the world). There are two production sectors in the economy, tradable and
nontradable goods, subject to adjustment costs when the capital and employment level in these sectors,
that make it hard to reallocate resources fast. Nontradable goods are subject to search frictions as in
Bai et al. [2011] and Huo and Ríos-Rull [2012], firms and consumers have to search for each other before
transactions can take place. We want to think of this as consumers being engaged in shopping which
is costly for them and that we model as a standard search friction. The labor market is also subject
to frictions with firms needing to engage in costly vacancy posting. Wage determination occurs via
bargaining in a centralized manner.
Households can own both financial assets and housing. We assume there is a representative mutual
fund which operates in different financial markets and abstract households from the portfolio problem of
choosing how much of their wealth to hold in the form of shares of the nationally producing firms and
how much to own in fixed interest bonds. The households then can only save and borrow in a single
type of financial asset that consolidates all the relevant assets of the economy (except privately owned
housing). Crucially, borrowing has to be collateralized by housing.
The economy is hit by shocks that we denote θ, with transition matrix Πθθ,θ0 and that we will describe in
more detail later. We now describe each part of the model in detail.
4
2.1
Goods Market
There are two types of goods: tradables, which can be traded across countries and can be used for
consumption and investment, and nontradables, which can be used only for local consumption and are
subject to additional frictions that we now describe in detail.
There is a measure one of varieties of nontradables i ∈ [0, 1], and each variety is produced by a firm which
owns a measure one continuum of locations. Every period, firms posts a price for all the locations and
heterogeneous consumers may demand different amounts of goods at the same price.1 Firms can only
sell their products when shoppers visit their stores. Part of their inputs are used to maintain those stores
open and they are wasted when shoppers do not show up. The other inputs can be transferred across
shops. The firm then delivers the amount of goods demanded at each of the locations that are filled. We
will specify the production function later.
Consumers have to search and find varieties, and they value both the number of varieties and the quantity
consumed of each variety. To obtain varieties, consumers need to search for them, incurring a shopping
disutility while doing so.
Denote the aggregate measure of searching effort or shoppers looking for varieties as D, and recall that
the measure of firms as 1. The total number of matches is determined by a CRS matching function
M g [D(S), 1], where S is the vector of state variables to be specified later. Denote market tightness by
Q g (S) =
1
D(S) ,
which makes the probability that a shopper (a unit of shopping effort and not an agent)
finds a variety is
Ψd [Q g (S)] =
M g [D(S), 1]
,
D(S)
(1)
while the measure of shoppers that firms have access to is
Ψf [Q g (S)] = M g [D(S), 1].
(2)
Given this conventions Ψf [Q g (S)] is also the fraction of shops or locations of each variety that are
occupied.
Firms in the tradable goods sector operate in a standard competitive market, and the price for tradable
goods is used as the numeraire. The production function for tradables is F T (k, `), where ` is the labor
input. Throughout, we use the subindex N to refer to nontradables and T to tradables.
1
In a related paper, Huo and Ríos-Rull [2013], we explore environments where searching and shopping are different and
where firms post different prices to discriminate between costumers. This agrees with the evidence posted by in lieu of the
evidence posted by Aguiar et al. [2011] and Kaplan and Menzio [2013].
5
2.2
Labor Market
Work is indivisible, and all workers are either employed or unemployed. The labor market has a search
friction that we model simply as a requiring firms to pay a hiring cost κ per worker that they want to
hire.2 Denote by Φw (S) the measure of new jobs created in a given period. An employed worker faces a
probability δn of job loss that may depend on its type . An unemployed worker faces a probability
Φw (S)
1−u
of finding a job, where u is the unemployment rate. The transition matrix for an individual worker is then
given by:


1 − δn



 δ
n
Πw
e,e 0 (S) =
Φw (S)


1−u


 1 − Φw (S)
1−u
if e = 1, e 0 = 1
if e = 1, e 0 = 0
if e = 0, e 0 = 1
.
(3)
if e = 0, e 0 = 0
Given these assumptions the average quality of the labor force depends only on the level of employment.
The higher the level of employment the lower the average quality of the labor force as documented by
Kydland and Prescott [1988].3
2.3
Households
There is a continuum of infinitely lived households with total measure normalized to 1.
Endowment and labor market attachment A household has a skill level, or amount of efficient units
of labor that evolves according to a Markov process with transition matrix given by Π,0 .4 Let π ()
be the ergodic measure of type worker in the population. Not all skill types have the same probabilities
of finding or losing a job, so we denote by Πw
e 0 |e, (S) the transition probability of the evolution of the job
given the skill level. If the household is employed, it gets paid w (S), if unemployed it gets w units of
the good, the consumption equivalent of home production.
Housing Households like a good that we call houses and that it has specific properties. First, it is an
asset in limited supply. Second, it has to be owned in order to be enjoyed, and it is subject to capital gains.
Third, households can live with zero housing (cheap renting or moving in with others), and achieve a
saturation point, so the rich need not herd large quantities of it. Fourth, we abstract from any transaction
costs in the resizing of housing, so every period the household can buy and sell as much as it wants.
Clearly, capital gains and loses will be very much a problem for the household.
2
We follow Christiano et al. [2013] into avoiding matching functions in the labor market.
See also Castañeda et al. [1998].
4
In this version unemployment does not deteriorate skills, which would require that we had Π,0 |e .
3
6
Assets markets A household can own housing and financial assets. At the beginning of the period we
denote the household wealth by a, making the portfolio irrelevant given that there are no transaction
costs. The rate of return of the financial asset depends on whether the household is a borrower or a
saver. We write the rate of return of the financial asset, denoted b, as as consisting of two parts. One
that depends on the state of the economy the period of saving and is given by
(
q(θ, b) =
if b ≥ 0
1,
1
1+r ∗
− ς(θ),
if b < 0
(4)
,
where r ∗ is the world interest rate and ς(θ) is the transaction cost when borrowing that may depend on
the aggregate state and it includes whatever expectation lenders may have of household default. Note
that the interest rate of the loan is uncontingent on events happening the period after. The other part
of the return is relevant for to those that hold positive assets and we write it as
(
R(S, S 0 , b) =
1 + r (S, S 0 ),
if b ≥ 0
1,
if b < 0
(5)
,
where r (S, S 0 ) is the net return to saving in the mutual fund, which will be specified later but it is
the return of a mutual fund that owns all the firms of the economy in addition to some foreign net
asset position at fixed interest rate r ∗ . Clearly, positive assets are risky. The amount of negative assets
that a household can hold depends is limited by having house as collateral. In other words, households
cannot have a negative asset position. The amount of collateral that is needed depends on aggregate
circumstances and we denote by, 1 − λ(θ) the down payment requirement.
Preferences Households also like a consumption aggregate cA , housing h, and shopping effort d. The
aggregate consumption basket is valued via an Armington aggregator of tradables and nontradables, while
nontradables themselves aggregate via a Dixit-Stiglitz formulation with a variable upper bound, yielding

Z
cA = ω
0
IN
1
ρ
ρ(η−1)
η
η−1
η
+ (1 − ω)cT
cN,i di


η
η−1
,
(6)
where cN,i is the amount of nontradable good of variety i, IN ∈ [0, 1] is the measure of varieties of nontradable goods that the household has acquired, ρ > 1 determines the substitutability among nontradable
goods, and η controls the substitutability between nontradables and tradables. The period utility function
is given by u(cA , h, d). In addition, the household discounts the future at rate β.
7
Household’s Problem
A household is indexed by its skills, , its employment status e, and its assets a
which are its individual state. The recursive problem of the household is:
max
V (S, , e, a) =
cN,i ,cT ,IN ,h,d
u(cA , h, d) + β
X
0
0 0 0 0 0
Πθθ,θ0 Πw
e 0 |e, (S ) Π,0 V [S , , e , a (S , b, h)],
(7)
0 ,e 0 ,θ0
subject to the definition of the consumption aggregate (15) and
Z
IN
pi (S)cN,i + cT + ph (S)h + q(θ, b)b = a + 1e=1 w (S)s + 1e=0 w
(8)
0
a0 (S 0 , b, h) = ph (S 0 )h + R(S, S 0 , b)b,
(9)
q(θ, b)b ≥ −λ(θ)ph (S)h,
(10)
IN = d Ψd [Q g (S)],
(11)
S 0 = G (S, θ0 ).
(12)
Equation (8) is the household’s budget constraint. Equation 9 describes the evolution of total wealth.
Contingent on the realization of aggregate state S 0 , the total wealth in the next period is the sum of
the value of housing and the value of financial assets. Equation (10) is the down payment requirement.
Equation (11) is the requirement that varieties have to be found, which requires effort d and depends on
the tightness in the goods market.
If we define
cN =
1
IN
Z
0
IN
1
ρ
cN,i
ρ
,
p(S) =
1
IN
Z
IN
[pi (S)]
1
1−ρ
1−ρ
,
(13)
0
we can obtain the standard demand function for each variety
cN,i
pi (S)
=
p(S)
ρ
1−ρ
cN .
(14)
Guessing that all varieties will receive a symmetrical treatment (equal consumption among those are
consumed because all firms choose the same price, pi (S) = p(S)), we can rewrite expenditures on
RI
nontradables as p(S)IN cN instead of 0 N pi (S)cN,i . Also, we can rewrite the aggregate consumption
bundle as
η
η−1
η−1
(η−1)
ρ
η
η
+ (1 − ω)cT
c A = ω IN c N
.
8
(15)
The first order conditions of the household problem are are
(16)
ucN = p(S)IN ucT ,
ud
− d g
,
Ψ [Q (S)]
(17)
uIN = p(S)cN ucT
X
0
0
q(θ, b)ucT = β
Πθθ,θ0 Πw
e 0 |e, (S)Π,0 R(S, S , b)ucT + ζq(θ, b),
(18)
e 0 ,s 0 ,θ0
ph (S)ucT = uh + β
X
0 0
Πθθ,θ0 Πw
e 0 |e, (S) Π,0 ph (S )ucT + ζλ(θ)ph (S).
(19)
e 0 ,s 0 ,θ0
Equation (16) shows the optimality condition between nontradable and tradable goods. Note that increasing the number of varieties results in additional shopping disutility and Equation (17) determines the
optimal number of varieties. Equation (18) is the Euler equation with respect to the holding of financial
asset. ζ is the multiplier associated with the collateral constraint. When the collateral constraint is not
binding, ζ = 0. Equation (19) is the Euler equation with respect to housing. When ζ > 0, housing serves
its additional role as collateral for borrowing.
b for workers with arbitrary wage w
b today,:
For later use, we also define an auxiliary function V
b (S, , e, a; w
b) =
V
max
cN,i ,cT ,IN ,h,d
u(cA , h, d) + β
X
0 0 0
0
Πθθ,θ0 Πw
e 0 |e, (S)Π,0 V [S , e , a(S , b, h)],
(20)
e 0 ,s 0 ,θ0
subject to constraints (9-12) and to
(21)
b + 1e=0 w .
p(S)IN cN + cT + ph (S)h + q(θ, b)b = a + 1{e=1} w
Representation of households We describe the state of all households by means of a probability
measure X defined over an appropriate family of subsets of the wealth position of households and over
the shocks. Given a law of motion of the aggregate state of the economy S 0 = G (S, θ0 ), and individual
decision rules and shocks processes we can obtain the law of motion of X 0 = Φ(S 0 , X ).
2.4
Firms in the Nontradable Goods Sector
A firm’s individual state variable is (k, n), where k is the capital stock and n is the measures of workers
with different types. Let be the average labor efficient units per employment
P
n()
= P
n()
The total labor input for a firm with employment n =
9
P
n()
(22)
is simply ` = n =
P
n().
Each firm owns a continuum of locations with equal probability Ψf [Q g (S)] of being visited. At each
location, there are three kinds of inputs, pre-installed or fixed capital, k and labor `1 , and also variable
labor `2 . The production function is
F N (k, `1 , `2 ) = zN k α0 `α1 1 `α2 2
(23)
When a shopper wants to buy c units of nontradables at a location, the amount of variable labor `2
needed to produce c is
α
1
g (c, k, `1 ) = c α2 k
1
α
− α0 − α2
2`
.
1
(24)
Note that g (c, k, `1 ) is simply the inverse function of F N (k, `1 , `2 ) given pre-installed k and `1 . At the
posted price pi , the demand schedule of a shopper sent by household {, e, a} is
pi
c(pi , S, , e, a) =
p(S)
ρ
1−ρ
cN (S, , e, a)
(25)
where cN (S, , e, a) is the policy function of derived from households’ problem. Applying the law of large
numbers, the total variable labor needed is
f
Z
g
Ψ [Q (S)]
g [c(pi , S, , e, a), k, `1 ] dX
(26)
Firms start the period with total labor input ` = n, and they need to allocate it into pre-installed or fixed
and variable labor. The fact that firms have to pre-install capital and labor at a location no matter whether
shoppers actually visit them later is why demand can affect productivity: with a larger demand or higher
probability of being visited, more capital and labor can be utilized. The relative marginal productivity of
preinstalled versus variable labor is the key margin that determines how important is the role of household
demand in determining output.
Recall that there are frictions in the labor market, firms have to choose labor one period in advance
and engage in hiring costs to do so. Let κ be the hiring cost of getting an additional worker for the
following period (this cost yields a worker of particular skills given the size of the workforce). Both capital
investment and hiring costs are in the form of tradable goods. When hiring a worker, the probability they
meet a worker of type is
f
Φ (, S) =
R
dX (, 0, da)
u
10
(27)
where
R
dX (, 0, da) is the measure of type unemployed worker. The problem of the firms is then
f
N
g
Z
c(pi , S, , e, a)dX − w (S)` − i − κv +
Ω (S, k, n) = max Ψ [Q (S)]pi
i,v ,pi
`1 ,`2
X
Πθθ,θ0
θ0
ΩN (S 0 , k 0 , n0 )
,
1 + r∗
(28)
f
g
Z
`2 ≥ Ψ [Q (S)]
X
n()
`1 + `2 =
subject to
(29)
g [c(pi , S, , e, a), k, `1 ]dX
(30)
k 0 = (1 − δk )k + i − φN (k, i),
h
i
X
Π,0 (1 − δn )n() + v Φf (, S)
n0 (0 ) =
(31)
(32)
S 0 = G (S, θ0 ),
(33)
where φN (k, i) is a capital adjustment cost, which slows down the adaptation of firms to new conditions.
The first order conditions are
`1
α1
=
,
`2
α2
)
(
Z
0
X
1 − δk − (φN,k
1 + r∗
α2 0 f g 0
θ
k )
,
=
Πθ,θ0 − pi Ψ [Q (S )] gk [c(pi , S, , e, a), k, `1 ]dX +
N,k 0
ρ
1 − φN
1
−
(φ
)
0
i
i
θ
(
)
X
X
f
0 α2 0
∗
θ
0
0
0
κ(1 + r ) =
Πθ,θ0
Π,0 Φ (, S)
p − w (S ) + π ( ) 1 − δn κ .
ρ i
0 0
(34)
(35)
(36)
θ ,
Equation 34 is the optimality condition for the allocation of labor. It turns out that the firm always
want to fix the ratio of pre-installed labor to variable labor and the fraction of a certain kind of labor
input equals to their contribution to the production. Equation 35 and 36 are the optimality conditions
for investment and hiring.
Here, we also define the value function with an arbitrary wage w , which will be used in Nash bargaining
b N (S, k, n; w
b ) = max Ψf [Q g (S)]pi
Ω
i,v ,pi
`1 ,`2
Z
b
c(pi , S, , e, a)dX − w
X
n() − i−
κv +
X
θ0
11
Πθθ,θ0
ΩN (S 0 , k 0 , n0 )
. (37)
1 + r∗
2.5
Firms in the Tradable Goods Sector
Unlike the nontradable goods sector, firms in the tradable goods sector operate in a frictionless, perfectly
competitive environment. To accommodate the possibility of decreasing returns to scale, we pose that in
addition to capital and labor, firms also need to use another factor, land, available in fixed supply, as an
input of production. Without loss of generality, we assume that there is a firm that operates each unit of
land. There are also adjustment costs to expand capital and employment, given by functions φT ,k (k, i)
and φT ,n (n0 , n), which makes it difficult for this sector to expand quickly. The problem of the firms in
the tradable goods sector is
ΩT (S, k, n) = max F T (k, `) − w (S)` − i − κv − φT ,n (n0 , n) +
X
i,v
Πθθ,θ0
θ0
ΩT (S 0 , k 0 , n0 )
, (38)
1 + r∗
k 0 = (1 − δk )k + i − φT (k, i),
X
`=
n(),
subject to
(39)
(40)
n0 (0 ) =
X
h
i
Π,0 (1 − δn )n() + v Φf (, S)
(41)
S 0 = G (S).
(42)
The first order conditions are
1 + r∗
,k
1 − φT
i
,n
κ + φT
(1 + r ∗ ) =
n0
X
Πθθ,θ0
θ0 ,0
=
X
Πθθ,θ0
(
FkT
θ0
(
X
0
+
,k 0
1 − δk − (φT
k )
,k 0
1 − (φT
)
i
h
)
(43)
,
i
0
,n 0
0
Π,0 Φf (, S) (F`T )0 − w (S 0 ) − (φT
n ) + π ( ) 1 − δn
)
κ .
(44)
Equation (43) and Equation (44) are similar to the optimality condition for nontradable firms.
The value function with arbitrary wage is
b T (S, k, n; w
b ) = max F T (k, `) − w
b ` − i − κv − φT ,n (n0 , n) +
Ω
i,v
2.6
X
θ0
Πθθ,θ0
ΩT (S 0 , k 0 , n0 )
1 + r∗
(45)
Mutual Fund
To abstract from a portfolio problem faced by individual households, we have assumed that households
can save and borrow a single kind of financial asset, which is operated by a representative and risk neutral
12
mutual fund. The mutual fund owns all the local tradable and nontradable firms, also they hold a position
in international bonds, and the hold the loans to homeowners
The expected rate of return of all these assets has to be the same. Both the nontradable and tradable
firms discount future value at rate
1
1+r ∗ ,
and the expected rate of return equals to 1 + r ∗ . The rate of
return of the international bond, r ∗ has no uncertainty. Loans to households have transaction costs to
implement and have a default risk that arises in case the capital loss of the homeowner is larger than the
sum of its equity in the house and its labor income. We assume that a markup in the interest charged to
the loan, ς(θ) per unit of borrowing, covers both costs exactly.5
Total amount of financial assets in the economy is
Z
(46)
b(S, x)dX
b>0
Total amount of mortgages in the economy are
Z
−b(S, x)dX
L=
(47)
b<0
The total saving in international bond is
0
B+
Z
=
b(S, x)dX − ΩN (S, KN , NN ) − π N (S) + ΩT (S, KT , NT ) − π T (S) +
b>0
1
L
1 + r∗
(48)
The realized rate of return is
1 + r (S, S 0 ) =
2.7
0 +L
ΩN (S 0 , KN0 , NN0 ) + ΩT (S 0 , KT0 , NT0 ) + (1 + r ∗ )B+
R
b>0 b(S, x)dX
(49)
Wage Determination
With labor market frictions, there is a wide range of wages which can be accepted by firms and households,
as discussed in Hall [2005]. Here, we follow the convention in the labor search literature that the wage
rate is determined via (generalized) Nash bargaining. In Krusell et al. [2010] and Nakajima [2012], agents
internalize the effect of additional saving on their bargaining position, but except for those who are close
to the borrowing constraint, the wealth effect on the wage bargaining is tiny. Also, there is no convincing
empirical evidence showing that a higher wealth level increases the worker’s bargaining power. Since our
focus is not to obtain a realistic wage distribution, we assume that individual workers and firms take the
wage as given and act as though a worker-firm pair like themselves bargain over the wage rate.
5
We proceed by guess and verify of the frequency of aggregate loses of the banking sector to determine ς(θ).
13
The objective of a representative worker or a labor union is given by the aggregate surplus for employed
workers
Z
bn (S; w
b) =
V
e=1
h
i
b (S, a, 1, s; w
b (S, a, 0, s; w
b) − V
b ) dX .
V
(50)
b
The value of an additional worker for a representative firm in the nontradable goods sector with wage w
is
α3
b N (S, KN , NN ; w
b N (S)
b) =
Ω
p(S)N (S) − w
n
ρ
0
0
0
X NN ()
X
ΩN
n(0 ) (S , KN , NN )
θ
+
Π,0 (1 − δN )
Πθ,θ0
. (51)
NN
1 + r∗
0
0
,
θ
For a firm in the tradable goods sector, the marginal value of an additional worker is
,n
b T (S, KT , NT ; w
b T (S) − φT
b ) = F`T (KT , `T )T (S) − w
Ω
n
n
+
X NT ()
,0
NT
Π,0 (1
−
)
δN
X
Πθθ,θ0
0
0
0
ΩT
n(0 ) (S , KT , NT )
θ0
1 + r∗
!
−
,n
φT
n0
. (52)
b N may not be the
Firms in nontradable and tradable goods sectors may not value workers equally, i.e., Ω
n
T
b
same as Ω . We assume that the wage that a weighted value of the evaluation of the worker by firms,
n
with weights given by the employment share of each sector, is used in the bargaining process.
With these elements, the Nash bargaining problem becomes
h
iϕ h
i1−ϕ
bn (S; w
b N (S, KN , NN ; w
b T (S, KT , NT ; w
b)
b ) + (1 − χ)Ω
b)
w = max V
χΩ
,
n
n
(53)
b
w
N
where ϕ is the bargaining power of households and χ = NNN+N
is the employment share of the nontradable
T
b yields the first order condition:
goods sector. Taking the derivative with respect to w
h
i
b N (S, KN , NN ; w ) + (1 − χ)Ω
b T (S, KT , NT ; w ) = (1 − ϕ)V
bn (S; w ),
bn,w (S; w ) χΩ
ϕV
n
n
(54)
bn,w (S; w ) is given by
where V
Z
ucT (S, a, e, s; w )dX .
bn,w (S; w ) =
V
e=1
14
(55)
2.8
State Variables and equilibrium
Given the description of the environment and the problem faced by agents, we now turn to specify the
aggregate state of the economy S = {θ, B, KN , KT , NN , NT , X }. Here θ is the set of exogenous shocks
that have hit the economy, B is the net foreign asset position of the mutual fund, KN and KT are the
capital in the nontradable and tradable sector, and NN and NT are the employment vectors in the two
sectors. Recall that a household in the model economy is characterized by x = {, e, a}.
Equilibrium is then a law of motion of the economy S 0 = G (θ0 , S), and pricing functions, p(S), ph (S),
and w (S) together with decision rules such that households and firms maximize and crucially, the housing
price clears.
3
Mapping the model to data: Steady state
To be written
4
Solution Methods
We apply the methods pioneered by Krusell and Smith [1997, 1998] and also used by Castañeda et al.
b instead of the true state of the economy to
[1998] and many others. We use a simple vector of states S
forecast prices. These are the aggregate shocks, and total capacity of tradables and nontradables as well
as the net foreign asset position for a total of 3 continuous variables. Current prices are not forecasted,
instead market clearing prices are found each period.
5
Exploring the effects of credit crunches
To be written
15
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