Adverse Selection, Uncertainty Shocks
and Business Cycles∗
Daisuke Ikeda†
Bank of JapanFirst draft: February, 2011
This draft: November, 2011

Abstract
I study two dynamic economies in which imperfect ﬁnancial markets materialize
uncertainty shocks. In ﬁnancial markets borrowers not only can divert a fraction of
returns but also have better information about the riskiness of their projects than dolenders, resulting in adverse selection as well as moral hazard. Uncertainty shocks
change the degree of uncertainty about borrowers riskiness. I show analytically thatthe uncertainty shocks emerge either as (i) ﬁnancial shocks which change a wedge between a return to capital and a risk-free rate, or nearly as (ii) shocks to the marginaleﬃciency of investment. In both cases quantitative analysis suggests that the uncertainty shocks and mechanisms studied here may constitute important elements ofbusiness cycles.
(JEL ) E30, E44, E52.
Key words: Adverse selection, uncertainty shocks.

∗

I am deeply indebted to my Ph.D. advisor at Northwestern University, Larry Christiano, for his continuous help and invaluable advice. I thank Martin Eichenbaum, Giorgio Primiceri, Matthias Doepke, Mirko
Wiederholt and Matthias Kehrig for their helpful comments. The views expressed in this paper are solely
the responsibility of the author and should not be interpreted as reﬂecting views of the Bank of Japan.
†
E-mail: daisuke.ikeda@boj.or.jp. Address: 2-1-1 Nihonbashi-Hongokucho,Chuo-ku,Tokyo 103-8660
Japan.

1

1. Introduction
The great recession starting in late 2007, accompanied by a sharp rise in interest rate
spreads, has shifted researchers’ focus to ﬁnancial factors. In an attempt to understandthe causes of the great recession in a dynamic general equilibrium framework, two groupsof researchers explore the role of two diﬀerent shocks. The ﬁrst group focuses on ﬁnancial
shocks which change a wedge between a return to capital and a risk free rate (an interest
rate spread) exogenously. It shows that the ﬁnancial shocks can explain the key featuresof the great recession (Hall, 2010, Gilchrist, Yankov and Zakrajsek, 2009, and Gilchristand Zakrajsek, 2010 among others).1 The second group focuses on shocks to the marginaleﬃciency of investment (MEI shocks). It shows that the MEI shocks constitute the main
driving force of business cycles in the U.S. including the great recession (Justiniano, Primiceri and Tambalotti, 2009a,b). Justiniano, et al (2009b) make a conjecture that the MEI
shocks appear as reduced-form shocks related with ﬁnancial intermediation.
While both the ﬁnancial shocks and the MEI shocks deepen the understanding of thegreat recession, both the two shocks share a common drawback. They appear in a reducedform in a dynamic general equilibrium model. The ﬁnancial shocks change an interest rate
spread exogenously. The MEI shocks may have a connection with ﬁnancial factors, but thedetail remains unspeciﬁed in a model. Consequently economic mechanisms behind the two
shocks remain unknown.
Apart from those two shocks, shocks to the degree of uncertainty have got attention asa source of business cycles. Uncertainty, measured by various second moments, appears toincrease after major economic and political incidents (Bloom, 2009) and during recessions
in the U.S. including the great recession (Bloom, Floetotto and Jaimovich, 2010).2 In
an informal level, the great recession saw an economy face a lot of uncertainty about thequality and the riskiness of assets, which might be reﬂected to a sharp rise in interest rate
spreads.
Motivated by the above observations, I build and study two dynamic general equilibrium economies in which imperfect ﬁnancial markets materialize uncertainty shocks. In thetwo economies both an agency problem and asymmetric information make ﬁnancial markets
incomplete. Speciﬁcally, entrepreneurs (borrowers) not only can divert a fraction of returns
1

Gilchrist, et al (2009) and Gilchrist and Zakrajsek (2010) call the ﬁnancial shocks as credit supply
shocks. They use the stripped-down model of Bernanke, Gertler and Gilchrist (1999, BGG hereafter)
in which they abstract loan rates from the original BGG model. According to Gilchrist and Zakrajsek(2010) one interpretation of the credit supply shocks is the risk shocks considered by Christiano, Mottoand Rostagno (2010, CMR hereafter). Though the two shocks become similar after log-linearization, therisk shocks do not exactly coincide with the credit supply shocks.
2
Also, using ﬁrm-level data or macro data, Gilchrist, Sim and Zakrajsek (2010) and Kiley and Sim(2011) provide empirical evidence on uncertainty shocks as a source of business cycles. Kehrig (2011),
using plant-level data, shows that the dispersion of total factor productivity is procyclical.

2but also have better information about the riskiness of their projects than do intermediaries (lenders), resulting in adverse selection in ﬁnancial markets as well as moral hazard.
Uncertainty shocks I consider change the degree of uncertainty about an entrepreneur’s
riskiness. I show analytically that the uncertainty shocks emerge as the ﬁnancial shocks in
one economy and nearly as the MEI shocks in another economy after log-linearization.
Thus, the uncertainty shocks, combined with imperfect ﬁnancial markets, go beyond
reduced-form shocks, providing micro foundations for both the ﬁnancial shocks and the
MEI shocks. Also, imperfect ﬁnancial markets serve as another mechanisms through whicha change in uncertainty aﬀects real economic activities, in addition to real factors consideredby Bloom (2009) and Bloom, et al (2010).
In modeling imperfect ﬁnancial markets I focus on one speciﬁc source of asymmetric
information in ﬁnancial markets: the riskiness of projects run by entrepreneurs, as in Stiglitz
and Weiss (1981).3 Each entrepreneur has a project with diﬀerent degree of riskiness, which
is private information to the entrepreneur. Intermediaries provide funds to entrepreneurs
taking into account the distribution of entrepreneurs riskiness while the intermediaries do
not know the riskiness of speciﬁc entrepreneur’s project. This asymmetric information
results in adverse selection in which some safe entrepreneurs do not get funded.
In contrast with Stiglitz and Weiss (1981) and the others who ﬁx the scale of investment,
I extend their model allowing for the variable scale of investment. In doing so I introduce anagency problem, similar to Gertler and Karadi (2010), in which entrepreneurs can divert afraction of returns. This agency problem puts a balance sheet constraint, limits the amountof loans provided by intermediaries and help endogenize the scale of investment.4
I solve a non-trivial problem featuring both adverse selection and moral hazard. To
ensure the existence of equilibrium, I consider the following timing of events in making
a loan arrangement. First, competitive intermediaries provide the schedule of contracts.
Second, observing the other intermediaries’ schedules an intermediary decides to exit orstay in the market. Third, entrepreneurs choose an intermediary among those staying in
the market and choose a contract from the schedule provided by the intermediary. Unlike
the standard timing of events, my model features the second step in which intermediaries
have an option to withdraw their loan arrangements. Without this option an intermediary
may take advantage of the other intermediaries schedules which screen entrepreneurs by
their private information. This feature reminds of the non-existence of equilibrium incompetitive insurance markets analyzed by Rothchild and Stiglitz (1976). The second stepin a loan making process in the model is inspired by Wilson (1977) and Hellwig (1987)
3

Mankiw (1986) and Bernanke and Gertler (1990) extend Stiglitz and Weiss (1981) model and analyze
the eﬀect of adverse selection from a macroeconomic perspective in a static partial equilibrium framework.
4
If one allowed the variable scale of investment while keeping the framework of Stiglitz and Weiss (1981),
only the riskiest entrepreneur would invest and the resulting equilibrium would be eﬃcient, as mentioned
by Christiano and Ikeda (2011).

3

who propose a similar idea to resolve the non-existence problem of Rothchild and Stiglitz(1976).
I embed credit frictions, characterized by both adverse selection and moral hazard, intoa real business cycle model. I consider two real business cycle models which diﬀer in a
ﬁnancial market subject to credit frictions. In the ﬁrst model (Model-I), I embed credit
frictions into the demand side of capital. Entrepreneurs own, trade and rent out capital,
whose activities constitute the aggregate demand for capital. The uncertainty shocks playa role as a shifter of the demand curve. In Model-I, the uncertainty shocks emerge as theﬁnancial shocks.
In the second model (Model-II), I embed credit frictions into the supply side of investment. Entrepreneurs produce investment goods, whose activities constitute the aggregate
supply of investment. In Model-II, the uncertainty shocks emerge nearly as the MEI shocks.
I show quantitatively that the uncertainty shocks generate signiﬁcant ﬂuctuations consistent with business cycles in both the two models, as long as the models have ampliﬁcation
mechanisms. The ampliﬁcation mechanisms consist of a counter-cyclical markup in wages
and variable capital utilization rates. I embed the markup exogenously to focus on the roleof ampliﬁcation mechanisms. With the markup the marginal product of labor is equatedto the markup times the marginal rate of substitution between consumption and hours.
The model overcomes a co-movement problem, ﬁrst pointed by Barro and King (1984), if
the counter-cyclicality of markup is great enough.5 .
The negative uncertainty shocks increase the degree of overall uncertainty about theriskiness of projects. With imperfect ﬁnancial markets, an increase in uncertainty implies anincrease in the degree of asymmetric information, aggravating adverse selection in ﬁnancial
markets. Intermediaries respond by raising loan rates oﬀered to entrepreneurs, resulting ina spike of external ﬁnance premium and a drop in aggregate loans. In Model-I, the demandfor capital and the price of capital decrease initially and decrease further through a balance
sheet channel. In Model-II, the supply of investment decreases and also decreases further
through a balance sheet channel. Enhanced by the ampliﬁcation mechanisms, investment,
output, consumption and hours all fall, while the external ﬁnance premium rises, consistent
with the U.S. business cycles. A slight diﬀerence appears in the price of capital and the net
worth. While the price of capital and the net worth decrease in Model-I consistent withbusiness cycles, they increase in Model-II.
The uncertainty shocks I consider in my model share the same spirits with Williamson(1987) who considers shocks to the riskiness of the entrepreneur’s return in the framework
5

In standard neoclassical environments where the marginal rate of substitution between consumption
and hours is equated to the marginal product of labor, consumption and hours move to an opposite direction
without a change in aggregate technologies or preferences. The uncertainty shocks change neither aggregate
technologies nor preferences, so that the uncertainty shocks cannot overcome the co-movement problem bythemselves4

of Townsend (1983) costly state veriﬁcation model. CMR (2010) consider similar shocks
named risk shocks in a dynamic stochastic general equilibrium model with ﬁnancial frictions
a la BGG (1999). I show that the uncertainty shocks in Model-I share similar quantitative
implications to the risk shocks while the two shocks are notionally diﬀerent.
The two dynamic general equilibrium models I present in this paper have a close connection with a growing literature on adverse selection in macroeconomic settings (Eisfeldt
2004, Kurlat 2010, Bigio 2010, 2011). The two models in this paper diﬀer from thosepapers with respect to the source of asymmetric information. While I focus on asymmetric information on the riskiness of projects as in Stiglitz and Weiss (1981), they focus onasymmetric information on the quality of assets (projects). House (2005) studies Stiglitz
and Weiss (1981) model in an overlapping generations framework, keeping the ﬁxed scaleof investment. I allow the variable scale of investment and embed adverse selection into a
dynamic general equilibrium model in a reasonable manner.
I organize the rest of the paper as follows. In Section 2, I present a partial equilibrium
model with asymmetric information and an agency problem. I start from a symmetric
information model with an agency problem as a baseline and proceed to an asymmetric
information model. In Section 3, I embed the partial equilibrium model into dynamic general equilibrium models and present two models. In each model I make clear the role ofuncertainty shocks analytically. In Section 4, I conduct simulations to explore the quantitative eﬀects of uncertainty shocks. In Section 5, I discuss two issues. First, I compare
the uncertainty shocks with the risk shocks analyzed by CMR (2010). Second, I consider
another shocks to make clear the role of uncertainty shocks. In Section 6, I conclude thepaper.

2. Adverse Selection: A Partial Equilibrium ModelI study a ﬁnancing problem between entrepreneurs (borrowers) and intermediaries (lenders),
taking as given returns. I solve for the optimal contract between entrepreneurs and intermediaries. In this section I omit time subscript t for notational simplicity. In the subsequent
section I embed this partial equilibrium ﬁnancing problem into a dynamic general equilibrium model and explore the impact of uncertainty shocks.
2.1. Environment
Overview: There exist many entrepreneurs and intermediaries. A ﬁnancing problem isstatic in that it involves only one time borrowing and lending. The ﬁnancing problem
evolves in the following ﬁve steps. Initially, nature draws and assigns entrepreneur’s type(private information) which characterizes the riskiness of investment project. In the ﬁrst
step, an intermediary provides a schedule of contracts to entrepreneurs without knowing

5the entrepreneurs’ private information. In the second step, after observing the other intermediaries’ schedules of contracts an intermediary decides whether to stay in the market
or to leave the market. In the third step, an entrepreneur chooses an intermediary and its
speciﬁc contract, and invests in project. In the fourth step, after realizing the outcome of
project an entrepreneur decides whether to divert its return or not. Finally, an entrepreneur
and an intermediary receive returns depending on the previous actions.
The ﬁnancing problem features both adverse selection and moral hazard. Adverse
selection occurs due to entrepreneur’s private information about the riskiness of project.
Moral hazard occurs due to entrepreneur’s possible action of diverting its return.
Entrepreneurs: There exist many risk neutral entrepreneurs whose objective is to maxi-

mize their return. Initially an entrepreneur starts its business with net worth Nn in unitof goods, indexed by n. An entrepreneur has a project with its success probability p,
which is private information to the entrepreneur and is drawn from distribution function
F : [p, p̄] → [0, 1] with 0 < p < p̄ ≤ 1, independently and identically across entrepreneurs.
I assume that distribution F (p) has full support.
A set of index, (n, p), characterizes an entrepreneur. The type-(n, p) entrepreneur has
net worth Nn and has a project with probability of success p. I assume that for given n
there exist many entrepreneurs so that distribution F (·) coincides with the distribution of
the type-n entrepreneurs.
An entrepreneur chooses an intermediary and a contract among those oﬀered by theintermediary. The type-(n, p) entrepreneur receives loan Bn (p) from the intermediary. The
sum of loan Bn (p) and net worth Nn constitutes the entrepreneur’s asset. The entrepreneur
invests its asset in its project.
After the entrepreneur invests, the type-p project results in a success or a failure. Incase of success the project yields the gross return θ(p)Re per unit of goods invested. In caseof failure the project yields zero return. I assume θ(p) = 1/p for analytical tractability sothat the expected gross return, just after the realization of the project’s outcome (successor failure), becomes the same for all projects. This simpliﬁcation allows me to focus only
on the riskiness of project as private information.
I assume that the limited liability law protects entrepreneurs. Entrepreneurs do nothave any liability after paying to intermediaries. The limited liability assumption implies
that intermediaries can not force entrepreneurs to pay when the entrepreneurs fail in theirprojects because the entrepreneurs have nothing to pay at hand.
Under the limited liability assumption, an intermediary oﬀers a schedule of contracts
specifying the amount of loan and payment {Bn (p), Xn (p)}p , where Bn (p) denotes theamount of loan and Xn (p) denotes the amount of payment conditional on the success ofproject. Without loss of generality I consider a truth telling schedule of contracts such that
type-(n, p) entrepreneur chooses {Bn (p), Xn (p)} voluntarily and such that the entrepreneur

6

does not divert its return. The payment conditional on the failure of project must be zero
because of the limited liability assumption.
In addition to asymmetric information, an entrepreneur has a moral hazard problem
such that an entrepreneur can steal a fraction of the gross return made from a project.
Given a pair of loan and payment, {Bn (p), Xn (p)}, chosen by the type-(n, p) entrepreneur,and given the success of its project, the return when the entrepreneur abides by the contract
is given by
Return|abide = θ(p)Re (Bn (p) + Nn ) − Xn (p),= (1/p) {Re Nn + [Re − pXn (p)/Bn (p)] Bn (p)} ,(1)

where θ(p) ≡ 1/p denotes the productivity of the type-p project conditional on the successof the project, and I have used θ(p) ≡ 1/p in the second equality. The ﬁrst line of equation(1) implies that the entrepreneur’s return is given by the gross return from investment
minus the payment to an intermediary. The term in bracket in the second line of equation(1) refers to an expected excess return per unit of loan. If the excess return were negative,the entrepreneur would not borrow. If the excess return were positive, the entrepreneur
would borrow as much as possible.
The entrepreneur can break the contract and steal a fraction, 0 < ϕ < 1, of the gross
return from investment, θ(p)Re (Bn (p) + Nn ). Then, the entrepreneur’s return when theentrepreneur steals is given by
Return|steal = ϕ(1/p)Re (Bn (p) + Nn ).

(2)

An intermediary collects the rest of the return when the entrepreneur steals.6
Intermediaries: There exist a small number of risk neutral intermediaries relative to en-

trepreneurs. An intermediary takes in deposits from households with risk-free rate Rf and6

Another agency problem generates the same incentive problem. Instead of stealing the return supposethat an entrepreneur can declare the failure of its project even when the entrepreneur actually succeeds inthe project. When the entrepreneur declares the failure intermediaries monitor the entrepreneur and detect
the false claim with probability 1 − ϕ without no costs. An entrepreneur realizes the project’s outcome(success or failure) and reports the outcome to an intermediary.
In this case, given the project’s success, the return when the entrepreneur abides by the contract is givenbyReturn|abide = θ(p)Re (Bn (p) + Nn ) − Xn (p),where θ(p) ≡ 1/p. When the entrepreneur misbehaves the return is given by
Return|steal = ϕθ(p)Re (Bn (p) + Nn ).
The incentive constraint requiring that entrepreneur does not misbehave turns out to be the same as theone when the entrepreneur can steal a fraction of return.

7

makes loans to entrepreneurs. In the process of making a loan contract, ﬁrst, the intermediary provides a schedule of contracts{Bn (p), Xn (p)}p as a function of entrepreneur’s typep. Since the intermediary can observe the amount of net worth owned by entrepreneurs
the menu also depends on the net worth indexed by n. In the second step, the intermediary observes the other intermediaries’ schedules and decides whether to leave the market
or not. The intermediary’s proﬁts become zero when the intermediary leaves the market.
In the third step, entrepreneurs choose an intermediary. Because I restrict my attention
to a truth telling schedule without loss of generality, the type-(n, p) entrepreneur chooses
contract {Bn (p), Xn (p)} oﬀered by its chosen intermediary
I assume that intermediaries compete in lending. That is, intermediaries compete inproviding a schedule of contracts to attract entrepreneurs. I also assume that entrepreneurs
choose intermediaries randomly if the schedule of contracts oﬀered by the intermediaries
are indiﬀerent.
2.2. Symmetric Information Benchmark
Before solving the model with asymmetric information I consider a model with symmetric
information as a benchmark to make clear the role of asymmetric information. In an
economy with symmetric information intermediaries observe the riskiness of project, p,
while entrepreneurs still can misbehave and steal a fraction of return.
I consider, without loss of generality, an intermediary’s loan arrangement in which an
entrepreneur does not steal its return voluntarily. Given a unit of loan, an intermediarysets the borrowing interest rate, Rb (p), in a competitive manner, resulting in the following
equation:pRb (p) = Rf .
(3)
The left-hand-side of equation (3) denotes the expected payment from one unit of loan,
while the right-hand-side of equation (3) denotes the cost of one unit of loan. The perfectcompetition among intermediaries results in the intermediary’s zero proﬁts condition, (3).
The intermediary faces an agency problem: an entrepreneur has an incentive to steal afraction of return if the intermediary provides too much loans. The intermediary limits theamount of loan to ensure that the entrepreneur does not misbehave. Speciﬁcally, the loan
provided by the intermediary has to satisfy an incentive constraint such that the return
when the entrepreneur does not misbehave, given by (1), is equal or greater than the
return when the entrepreneur misbehaves, given by (2). Noting that the payment becomesXn (p) = Rb (p)Bn (p) = Rf Bn (p)/p from the intermediary’s zero proﬁt condition, (3), I
express the incentive constraint as
Re Nn + (Re − Rf )Bn (p) ≥ ϕRe (Bn (p) + Nn ),8or
Bn (p) ≤(1 − ϕ)(Re /Rf )Nn .
1 − (1 − ϕ)(Re /Rf )(4)

Constraint (4) implies that the amount of loan is limited by the amount of net worth heldby the entrepreneur.
I assume positive premium Re /Rf > 1, otherwise an entrepreneur has no incentive toborrow from intermediaries.7 I also assume that the degree of moral hazard controlledby ϕ is great enough for the denominator in the right-hand-side of (4) to be positive.
Then, entrepreneurs borrow the maximum amount subject to the incentive constraint, (4).
Consequently, the incentive constraint, (4), binds in equilibrium.
The equilibrium contract oﬀered by an intermediary consists of {Bn (p), Xn (p)}, where
the amount of loan is given by the left-hand-side of (4) and the amount of repayment incase of success is given by Rb Bn (p) with Rb given by (3). Because the intermediary earns
zero proﬁts under the contract, the intermediary has no incentive to exit the market, which
would result in zero proﬁts, after observing the other intermediaries’ loan contracts. An
entrepreneur chooses an intermediary randomly because all intermediaries oﬀer the same
contract. Because there is no proﬁtable deviation for both intermediaries and entrepreneurs,
the contract constitutes an equilibrium in a model with symmetric information.
The equilibrium contract has a convenient feature for aggregation. Because the righthand-side of (4) is linear in net worth Nn and is independent of type p, I can aggregate theindividual loan over n and p and express the aggregate loan as
B=(1 − ϕ)(Re /Rf )N,
1 − (1 − ϕ)(Re /Rf )(5)

where B and N denote the aggregate loan and the aggregate net worth respectively. The
aggregate loan is increasing in the premium, Re /Rf , and is increasing in the aggregate net
worth, N . More importantly the aggregate loan is independent of the distribution of theriskiness of project, F (p). A change in F (p) aﬀects neither the aggregate loan nor any
other aggregate variables in a model with symmetric information.
2.3. Asymmetric Information
Now I consider a model with asymmetric information. I ﬁrst deﬁne an equilibrium in agame-theoretic manner and then solve the model.
Subgame Perfection: A game I consider consists of two types of players: many en-

trepreneurs and a small number of intermediaries relative to entrepreneurs. The gameevolves in three stages. Initially, nature draws and assigns type p, which characterizes
7

In a general equilibrium model studied in the next section I endogenize the returns, Re and Rf , which
satisfy Re /E f < 1 in equilibrium around steady state.

9the riskiness of project, to an entrepreneur. In the ﬁrst stage, an intermediary, indexed
by i, oﬀers a schedule of contracts, σ1,i ≡ {Bni (p), Xni (p)}p , which speciﬁes the amount ofloan, Bni (p), and the amount of payment in case of project’s success, Xni (p), as a functionof p. In the second stage, the intermediary observes the other intermediaries’ schedules
oﬀered in the ﬁrst stage, and decides whether to leave or stay in the market. In the third
stage, entrepreneurs choose an intermediary. If an entrepreneur chooses the i-th intermediary the entrepreneur chooses contract {Bni (p), Xni (p)}, because I limit my attention toa truth telling contract, without loss of generality. Finally, given that the contract does
not make entrepreneurs misbehave, the actions from stage 1 to 3 determine the returns ofentrepreneurs and intermediaries.
The i-th intermediary’s strategy consists of {σ1,i, , σ2,i } where σ1,i ≡ {Bni (p), Xni (p)}pdenotes a schedule of contracts, and σ2,i speciﬁes a decision of whether to stay (σ2,i (σ1 ) = 1)
or leave (σ2,i (σ1 ) = 0) as a function of a set of schedules, σ1 ≡ {σ1,i }i . An entrepreneur’s
strategy speciﬁes a decision of which intermediary to choose as a function of σ1 and σ2 ≡{σ2,i }i . I limit my attention to a deterministic strategy, yet I assume that an entrepreneurchooses an intermediary randomly if the entrepreneur ﬁnds some intermediaries indiﬀerent.
The i-th intermediary competes in lending and chooses schedule σ1,i in the ﬁrst stage
using backward induction. In the third stage, given σ1 and σ2 , entrepreneurs choose anintermediary which oﬀers the most proﬁtable contract. In the second stage, given σ1 andthe other intermediaries’ strategy of whether to stay or leave, σ2,−i (σ1 ) ≡ {σ2,j (σ1 )}j̸=i , thei-th intermediary chooses whether to stay or leave. If the intermediary chooses to stay,set {σ1 , σ2,−i (σ1 )} determines the distribution of entrepreneurs choosing the intermediary,denoted by F i (p; σ1 , σ2,−i ). The intermediary chooses to stay, σ2,i (σ1 ) = 1 if and only if∫ p̄i[pXni (p) − Rf Bni (p)]dF i (p; σ1 , σ2,−i ) ≥ 0,
(6)
Vn (σ1 , σ2,−i ) ≡pwhere Vni (σ1 , σ2,−i ) denotes the intermediary’s proﬁts when choosing to stay, given {σ1 ,σ2,−i (σ1 )}. The bracket in (6) denotes the expected proﬁts made from lending to a single
type-p entrepreneur. With probability p the intermediary receives repayment Xni (p) for
amount of loan Bni (p) with cost Rf per unit of loan. Integrating the proﬁts over p yields
the intermediary’s proﬁts, given by (6).
Let {σ2∗ (σ1 )} denote a set of Nash equilibria in the second stage game, given σ1 . Given
one of the Nash equilibria, set {σ1 , σ2∗ (σ1 )} determines the distribution of entrepreneurs
choosing the i-th intermediary in the third stage, denoted by F i (p; σ1 , σ2∗ (σ1 )). In the ﬁrst
stage, the i-th intermediary chooses its schedule σ1,i to maximize its proﬁts, given by∫ 1∗iVn (σ1 , σ2 (σ1 )) ≡(7)
[pXni (p) − Rf Bni (p)]dF i (p; σ1 , σ2∗ (σ1 )) ,p

where σ1 ≡ {σ1,i , σ1,−i }.
10Now I deﬁne an equilibrium of a model with asymmetric information.
Definition 1: A subgame-perfect equilibrium for a model with asymmetric information∗consists of the schedule of contracts oﬀered by intermediaries in the ﬁrst stage, σ1∗ ≡ {σ1,i}i, the intermediaries’ strategies whether to leave or stay in the market in the second stage,∗σ2∗ ≡ {σ2,i}i , and the entrepreneurs’ strategies choosing an intermediary in the third stage,
satisfying (i) entrepreneurs choose an intermediary providing the most proﬁtable contract,∗∗(ii) given σ1 and σ2,−i, the i-th intermediary chooses to stay, σ2,i(σ1 ) = 1, if and only if∗
the proﬁts from doing to are non-negative, for all i, and (iii) given σ1,−iand σ2∗ , the i-th∗intermediary chooses σ1,ito maximize its proﬁts, for all i. An equilibrium is symmetric if∗∗= σ1,jforand only if the strategy in the ﬁrst stage is the same for all entrepreneurs: σ1,i
all i, j.

Symmetric Equilibrium: In deriving an equilibrium I limit my attention to a symmetricequilibrium in which all intermediaries employ the same strategy. I have already characterized an intermediary’s equilibrium strategy in the second stage as in Deﬁnition 1. In thefollowing I derive an equilibrium strategy in the ﬁrst stage, that is, I solve problem (iii) in∗
Deﬁnition 1, given the assumption of symmetric equilibrium: σ1,i= {Bn (p), Xn (p)} for alli.
My procedure to derive an equilibrium consists of three steps. First, I characterize a
set of strategies which satisfy incentive and participation constraints. Second, I make aconjecture that one of the strategies constitutes an equilibrium strategy. Third, I showthat the candidate for an equilibrium actually constitutes a unique symmetric equilibrium.
Step 1 : I characterize a set of equilibrium schedules of contracts, {Bn (p), Xn (p)}p . An
equilibrium schedule of contracts has to satisfy the following four constraints. First, I
restrict my attention to a truth-telling contract so that an equilibrium schedule has tosatisfy an incentive constraint:
Re Bn (p) − pXn (p) ≥ Re Bn (p̃) − pXn (p̃),∀p, p̃.

(IC1)

Constraint (IC1) ensures that the type-p entrepreneur chooses {Bn (p), Xn (p)} voluntarily.
The left-hand-side of (IC1) denotes the entrepreneur’s return from getting loans whenchoosing pair {Bn (p), Xn (p)} and the right-hand-side of (IC1) denotes the return whenchoosing the other pair.
Second, an equilibrium schedule has to satisfy another incentive constraint:
Re [Nn + Bn (p)] − pXn (p) ≥ ϕRe [Nn + Bn (p)],

∀p,

(IC2)

The second incentive constraint, (IC2), ensures that an entrepreneur does not misbehave.
The left-hand-side of (IC2) denotes the entrepreneur’s return when the entrepreneur does11

not misbehave, while the right-hand-side of (IC2) denotes the return when the entrepreneur
misbehaves and walks way with some returns.
Third, an equilibrium schedule has to satisfy an entrepreneur’s participation constraint:
Wn (p) ≡ Re Bn (p) − pXn (p) ≥ 0,

∀p.

(PCe)

The entrepreneur’s return has to be non-negative, otherwise the entrepreneur would notparticipate in this loan arrangement.
Finally, an equilibrium schedule has to satisfy an intermediary’s zero proﬁt condition:∫ p∗∗
0 = Vn (p ) =[pXn (p) − Rf Bn (p)]dF (p).
(8)
pHere I make a conjecture that there exists p∗ < p̄ such that (PCe) holds with equality forp = p∗ and the amount of payment is strictly positive, Xn (p∗ ) > 0. The conjecture parallels with Stiglitz and Weiss (1981) model in which entrepreneurs with p above a certain
threshold do not get loans because of adverse selection. Given threshold p∗ , the intermediary’s proﬁts consist of a repayment from the type-p entrepreneur, pXn (p), and the costof funds, −Rf Bn (p), integrated over distribution F (p). Because I restrict my attention toa symmetric equilibrium, all intermediaries oﬀer the same schedule of contracts in equilibrium. Entrepreneurs are assumed to choose an intermediary randomly if intermediaries
are indiﬀerent. Then, an intermediary faces distribution F (p) up to a constant scaling
factor in a symmetric equilibrium. The zero proﬁt condition follows from the assumptionof competitive intermediaries.
Step 2 : I narrow down a set of equilibrium schedules and pick up one schedule as a candidate
for an equilibrium schedule. I sketch the procedure in ﬁve steps.
First, I start from a guess that there exists threshold p∗ above which entrepreneurs do
not get loans. This guess, combined with (IC1), implies thatfor p > p∗

Re Bn (p) − pXn (p) = 0,

for p < p∗

Re Bn (p) − pXn (p) > 0.

(9)That is, (PCe) holds with equality for p > p∗ and (PCe) holds with strict inequality forp < p∗ . From (IC1) and (9), for p > p∗ I obtain:
0 ≥ Re Bn (p) − p∗ Xn (p) = (p − p∗ )Xn (p).

(10)

I have derived the inequality in (10) from (IC1) for p = p∗ . I have derived the equality in(10) from (9). Since p − p∗ > 0 the payment has to be zero for p > p∗ , otherwise condition(10) would not hold, violating either (IC1) or (9). Therefore, for p > p∗ both the payment
and the loan become zero: Xn (p) = Bn (p) = 0. For p < p∗ it is straightforward to show
12

that both the payment and the loan has to be positive. Given threshold p∗ , this ﬁrst step
ensures that a schedule satisﬁes (PCe) automatically.
Second, I replace (IC1) by the local incentive compatibility constraint and the monotonicity constraint as in a standard mechanism design problem:
Re [dBn (p)/dp] − p[dXn (p)/dp] = 0,∀pdXn (p)/dp ≤ 0,

(11)
(12)

Intuitively, constraints (11) and (12) correspond to the ﬁrst and second order conditions ofan entrepreneur’s problem respectively. In the problem the type-p entrepreneur maximizes
its proﬁts by choosing pair {Bn (p), Xn (p)}.
Third, I express the proﬁts of entrepreneurs, Wn (p) ≡ Re Bn (p) − pXn (p), as a functionof payment schedule Xn (p) using local incentive compatibility constraint (11) and theenvelope theorem as follows:∫p∗

Xn (x)dx.

Wn (p) =(13)

p

Expression (13) implies that the type-p entrepreneur’s proﬁts are increasing in payment
Xn (p′ ) for p′ ≥ p. The intuition behind (13) has to do with the fact that risky entrepreneurs
with lower p repay less in expected values than safer entrepreneurs with higher p. For
example, if the highest type-p∗ entrepreneur re-payed more, it would have to receive moreloan to satisfy (PCe). This new pair of loan and payment for p = p∗ is also available
to the other entrepreneurs. For p < p∗ , an increase in loan would increase the type-p
entrepreneur’s expected gross return by the same amount as the type-p∗ entrepreneur,while the expected repayment would be less for the type-p entrepreneurs than the type-p∗
entrepreneur. Therefore, risky entrepreneurs with lower p can enjoy rents accrued from alower repayment in expected values.
Using (13) I express loan schedule Bn (p) as a function of Xn (p):][∫ p∗
Bn (p) = pXn (p) +Xn (x)dx /Re .
(14)p

In this procedure I have substituted out local incentive compatibility constraint (11). Constraint (11) is satisﬁed as long as a loan schedule is given by (14).
Fourth, using loan schedule (14) I rewrite (IC2) as(1 − ϕ) e(1 − ϕ)Xn (p) ≤R Nn +
ϕp
ϕp∫ p∗
Xn (x)dx.

(15)pHere I make a conjecture that the inequality in (15) holds with equality for p ≤ p∗ . Under
the conjecture equation (15) becomes an integral equation for unknown function Xn (p). I

13

solve the integral equation and obtain a schedule of contract as a candidate for an equilibrium:[] ( ) ϕ1
(1 − ϕ)Re ∗ 1−ϕ1(p ) ϕXn (p) =
Nn ,
(16)
ϕ
p
{
[( ) 1−ϕ
]}
1
p∗ ϕ
1−ϕ
1+
Bn (p) =
−1
Nn ,
(17)
ϕ
1−ϕ
p
where I have derived equation (17) from equations (14) and (16). Note that Xn (p) is
strictly decreasing in p and satisﬁes monotonicity condition (12). Then, the schedule, (16)and (17), constructed so far satisﬁes (PCe), (IC1) and (IC2).
Fifth, I pin down threshold p∗ using the intermediary’s zero proﬁts condition, (8). Usingloan schedule (14), I express the intermediary’s zero proﬁt condition as follows:∗∫ p∗

0 = Vn (p ) =ω(p)Xn (p)dp,

(18)

p

where ω : (p, p̄) → R is given by
Rf
Rf
ω(p) = pf (p) − e pf (p) − e F (p).
R
R

(19)

At this point I simply assume that a solution to (18) uniquely exists. That is, I assume
that there exists a unique p∗ ∈ (p, p̄) such that V (p∗ ) = 0, given by (18).
The candidate for an equilibrium, given by (16), (17) and (18), has an important feature.
The fact that the second incentive constraint, (IC2) (or (15)), holds with equality implies
that the schedule maximizes payment Xn (p) for p ≤ p∗ , which, in turn, implies thatthe schedule maximizes the entrepreneur’s proﬁts, given by (13), for all p ≤ p∗ . For
entrepreneurs with p > p∗ , proﬁts are zero. I summarize the above argument in Lemma 1.
Lemma 1: Suppose that a solution to (18) uniquely exists. Then, among loan and pay-

ment schedules satisfying (IC1), (IC2), (PCe) and the intermediary’s zero proﬁt condition,(8), the candidate for an equilibrium, given by (16) and (17), maximizes the type-p entrepreneur’s proﬁts for all p.
Step 3 : Finally, I show that the candidate for an equilibrium, given by (16), (17) and(18), constitutes a unique symmetric equilibrium. The statement consists of two parts: anequilibrium part and a uniqueness part. I summarize the result in the following propositionand provide the proof in the appendix.
Proposition 1: Suppose that a solution to (18) uniquely exists. Then, the candidate foran equilibrium, given by (17), (16) and (18), constitutes a unique symmetric equilibrium.
14

Intuitively, the candidate for an equilibrium maximizes the type-p entrepreneur’s proﬁts
for all p, as shown in Lemma 1, and attracts all entrepreneurs, so that an intermediary has
no incentive to deviate and the candidate actually constitutes an equilibrium. The conditionin Proposition 1 is satisﬁed for some distributions and some values for Rf /Re < 1. For
example, as I employ in simulating a model later, the condition in Proposition 1 is satisﬁed
for a uniform distribution with support [p, p̄] with 0 < p < p̄ ≤ 1 for Rf /Re < 1 close tounity.
2.4. Equilibrium Loan Contract
I study the properties and the implications of the equilibrium loan contract. First, I study
its micro properties. Second, I study its macro implications.
Micro properties: I summarize the simple properties of the equilibrium loan contract in

the following Lemma.
Lemma 2: An equilibrium payment and loan schedules, (16) and (17), satisﬁes the following

properties:(i) Both loan schedule (17) and payment schedule (16) are decreasing in riskiness p andincreasing in threshold p∗ .
(ii) The borrowing interest rate, Rb (p) = Xn (p)/Bn (p), is decreasing in riskiness p.
(iii) Threshold p∗ is increasing in the premium, Re /Rf .
Both Lemma 2 (i) and (ii) imply that entrepreneurs with higher risk project get moreloans and pay back more conditional on the success of a project. Intuitively entrepreneurs
face a trade-oﬀ between the amount of loans and the incentive to misbehave. On the one
hand, entrepreneurs with p < p∗ would like to borrow as much as possible because the returnfrom borrowing is positive. On the other hand, more borrowing induces entrepreneurs to
misbehave. Entrepreneurs with lower p have less incentive to misbehave because they havelower expected payment, as is clear from (IC2). Therefore, the risky entrepreneurs canborrow and pay more in the equilibrium contract.
Lemma 2 (iii) implies that an intermediary oﬀers more loans to entrepreneurs as thereturn earned by entrepreneurs gets higher. Given threshold p∗ a rise in the premiumincreases the intermediary’s proﬁts. Consequently the intermediary loosens lending andincreases p∗ until the intermediary’s zero proﬁt condition holds.
Macro implications: The equilibrium loan contract, summarized in Proposition 1, has

two nice properties for aggregation. First, threshold p∗ does not depend on the amount of
net worth. This property has to do with the fact that the net worth does not appear in(18). This property implies that the same threshold applies to all entrepreneurs. Even an
15

entrepreneur with great amount of net worth cannot get loans if its project’s probabilityof success exceeds the threshold: p > p∗ .
Second, both loan schedule Bn (p), given by (17), and payment schedule Xn (p), givenby (16), are linear in net worth. This implies that I do not have to pay attention to the
distribution of net worth in aggregating loan and payment.
As a consequence of the above two properties, the aggregate loan has a simple expression. From loan schedule (17) I can express the aggregate loan, B, as∫ ∫ p∗B=

Bn (p)dF (p)dH(n),
n

p

[
]
∫ p∗ ( ∗ ) 1−ϕ
ϕ
1−ϕ
ϕ
1
p
∗
=
dF (p) −F (p ) N,
ϕ
1−ϕ p
p
1−ϕ

(20)

where F (·) denotes the distribution of p and H(·) denotes the distribution of net worth
indexed by n. Using the intermediary’s zero proﬁt condition, I can also express B as
∫ ∫ p∗
1
B= f
pXn (p)dF (p)dH(n),
R n p
[
]
( ∗ ) 1−ϕe ∫ p∗
ϕp
1−ϕR=
dF (p) N,
(21)
f
ϕ R
p
p
From (20) and (21) I obtain a simple expression for the aggregate loan:B=(1 − ϕ)(Re /Rf )F (p∗ )N.
1 − (1 − ϕ)(Re /Rf )(22)

The aggregate loan is increasing in the premium, Re /Rf , decreasing in the degree of moral
hazard, ϕ, and increasing in threshold p∗ . From (22) the aggregate loan is bounded fromabove as(1 − ϕ)(Re /Rf )B<N,
1 − (1 − ϕ)(Re /Rf )because F (p∗ ) < 1. The upper bound denotes the aggregate loan without asymmetric
information, given by (5).
The expression for the aggregate loan, (22), summarizes the impact of asymmetric
information. With asymmetric information entrepreneurs with p > p∗ do not get loans
because a borrowing interest rate is too high to earn non-negative proﬁts, reﬂecting thehigh default rates of the other entrepreneurs. Asymmetric information causes a lemonsproblem and decreases the aggregate loan by [1 − F (p∗ )] × 100 percent relative to theaggregate loan without asymmetric information.

16

2.5. Equivalence Result
In order to isolate the pure role of asymmetric information I establish an equivalence result
between two economies: one with asymmetric information and another with symmetric
information and with taxes on the entrepreneur’s net worth.
As shown in Section 2.4, a loan-net-worth ratio, Bt /Nt , under asymmetric information
is lower than that under symmetric information. A wedge between the two ratio is given by
1 − F (p∗ ). I add the same wedge in the model with symmetric information by introducing
taxes on the entrepreneur’s net worth.
I consider an economy with symmetric information, analyzed in Section 2.2. A government imposes taxes on the entrepreneur’s net worth. The government injects the tax
revenue to entrepreneurs and makes them invest the tax revenues in a project on behalf ofthe government. The government rebates the return to entrepreneurs in the beginning ofnext period.
Let τ denotes the tax rate. The after-taxed net worth becomes (1 − τ )N in aggregate.
Following the same argument in Section 2.2, I obtain a loan-net-worth ratio, which coincides
with that under asymmetric information, if the tax rate is set toτ = 1 − F (p∗ ).
After receiving the tax revenue from the government, entrepreneurs have an asset in aggregate, amounting to (1 − τ )N + B + τ N = N + B, where B coincides with that underasymmetric information, given by (22). Therefore, both the aggregate loan, B, and the aggregate asset, coincide with those under asymmetric information, showing the equivalence
result.
The equivalence result implies that adverse selection plays a similar role to a shock to thenet worth explored quantitatively by Christiano, Motto and Rostagno (2010) and Gilchrist,
Ortiz and Zakrajsek (2009) among others. The net worth shock transfers resources fromentrepreneurs to households, reducing the amount of loan. While the tax I consider does
not transfer resources from one to another, it eventually reduces the amount of loan andhas a similar eﬀect to the net worth shock.

3. General Equilibrium Models
Now I embed adverse selection in ﬁnancial markets, analyzed in the previous section, intoa dynamic general equilibrium model. In doing so I restrict my attention to one-periodﬁnancing contract so that I can apply the results in the previous section directly. Set aside
the adverse selection, the basic framework is the standard real business cycle model withexogenous countercyclical markups and endogenous capital utilization. As I show later
those additional features serve as critical mechanisms amplifying uncertainty shocks.

17

I study analytically the eﬀect of uncertainty shocks which change the distribution ofentrepreneurs riskiness, in two real business cycle models. In the ﬁrst model, I embed theadverse selection into the demand side of capital as in BGG (1999). In the second model, I
embed the adverse selection into the supply side of investment as in Carlstrom and Fuerst(1997).
In the following I ﬁrst describe technologies, preferences and shocks which are common
between the two models. Then, I describe the two real business cycle models separately.
In doing so I make clear the role of uncertainty shocks in the two models. I show analytically that the uncertainty shocks emerge as ﬁnancial shocks in the ﬁrst model, while theuncertainty shocks emerge as investment shocks in the second model.
3.1. Common Building Blocks
Preferences: There is a continuum of household, with preferences given by

Et

∞
∑

β s U (Ct , Lt ),0 < β < 1,

(23)

s=0

where Ct denotes consumption, Lt denotes labor supply and U (·) satisﬁes U1 > 0, U2 < 0,2
U11 , U22 < 0 and U11 U22 − U12> 0.
A household, with measure unity, consists of the large number of family members whoare either workers or entrepreneurs with their population f and 1−f respectively where 0 <f < 1. Family members switch their job occupation randomly. Speciﬁcally, entrepreneurs
become workers randomly with probability 1 − γ where 0 < γ < 1. I call γ as the surviving
probability of entrepreneurs. The same number of workers become entrepreneurs randomly
so that the proportion of workers or entrepreneurs stays constant over time.
The household, as a representative agent of the family members, consumes and saves.
The household provides the perfect consumption insurance among its family members.
Workers within the household supply labor and earn wage income. Entrepreneurs within
the household specialize in investing in a project and accumulate their net worth.
Entrepreneurs can transfer their net worth to the household to which they belong to at
the beginning of period. The entrepreneur chooses to accumulate the net worth over time
and transfers the net worth only when it becomes to a worker, because the average returnfrom a unit of goods invested in its project is strictly greater than the risk-free returnearned by the household who makes deposits in intermediaries. While the entrepreneur issubject to a risk associated with its project, the household cares only about the average
return because there are many family members of entrepreneurs within the household. Thismodeling device justiﬁes the assumption of risk-neutral entrepreneurs analyzed in Section
2. This setting, adopted from Gertler and Karadi (2010), allows me to embed entrepreneurs
into the standard real business cycle model, keeping the representative agent framework ina reasonable manner.
18

A household can save only through deposits in intermediaries with risk-free rate Rt+1 .
Then, the ﬂow budget constraint is given by
Ct + Bt = Rt Bt−1 + wt Lt + Θt ,(24)

where Bt denotes the amount of deposit at the end of time t, wt denotes wages, and Θt
includes the sum of the net transfer from entrepreneurs who belong to the household and thenet payment to the state-contingent securities on the return to capital. The net payment
matters only in Model-II.
I introduce an exogenous countercyclical markup in wages to make clear its role as theampliﬁcation mechanisms of uncertainty shocks. I model the markup in the simplest but
an ad-hoc manner for the sake of exposition.8 Speciﬁcally, I assume that workers havea monopolistic power over labor supply and set real wages equal to the markup over themarginal rate of substitution between consumption and labor. I assume the markups are
exogenous and countercyclical, given by( )−ω
Yt
λw,t = λw, λw > 1, ω > 0,(25)
Y
where Yt denotes output and Y denotes output in steady state. The time-varying markup,
λw,t , is decreasing in output and countercyclical by construction.
Maximizing utility (23) subject to budget constraint (24) yields the ﬁrst order conditions:[]
U1 (Ct+1 , Lt+1 )1= Et β,(26)
Rt+1
U1 (Ct , Lt )wt = −λw,t U2 (Ct , Lt )/U1 (Ct , Lt ),(27)

where λw,t denotes an exogenous countercyclical markup in wages. Equation (26) implies
that the gross interest rate on deposit, Rt+1 , is risk-free. Equation (27) implies that wage
is equated to the markup over the marginal rate of substitution between consumption andlabor.98

One can introduce a countercyclical markup in wages in a rigorous manner by introducing nominal
rigidities in wages as in Erceg, Henderson and Levin (2000). While the introduction of nominal rigidities inwages provides a solid micro-foundation for the countercyclical markup, it complicates the model. Because
I focus on the counter-cyclical markup as ampliﬁcation mechanisms, but not on the micro-foundation of
the countercyclical markup, I introduce the countercyclical markup in the simplest manner for the sake of
exposition.
9
Here I simply assume that workers set real wages equal to the markup over the marginal rate of
substitution between consumption and labor. One can derive exactly the same condition under a more
rigorous set up considered by Erceg, et al (2000) with ﬂexible wages and an exogenous countercyclicalmarkup in wages.

19Technologies: There are competitive representative consumption goods producers, withproduction technologies given by
Yt = (ut Kt )α L1−α,t

0 < α < 1,(28)

where Kt denotes the aggregate capital, Lt denotes the aggregate labor and ut denotes thecapital utilization rate. Factor prices are equated to marginal products:
rtk = α(ut Kt )α−1 L1−α,twt = (1 − α)(ut Kt )

α

L−α
t ,

(29)
(30)

where rtk denotes the rental rate of capital services, ut Kt .
Capital providers set the capital utilization rate, ut . As I explain later the capitalproviders are entrepreneurs in Model-I while they are entrepreneurs or intermediaries inModel-II. The capital providers earn the rental rate, rtk ut , per unit of capital, with theassociated cost of capital utilization, a(ut ), in consumption goods unit. As in CEE (2005),
the cost satisﬁes a′ (ut ), a′′ (ut ) > 0 and a(1) = 0. Maximizing the rental rate minus the costwith respect to ut results in
rtk = a′ (ut ).
(31)
As is clear from equation (31), the capital utilization rate depends only on the net returnon capital, rtk , and is increasing in rtk .
The goods market clearing requires that
Yt = Ct + It + a(ut )Kt ,(32)

where It denotes the aggregate investment. Term a(ut )Kt denotes the total cost associatedwith capital utilization rate ut . The law of motion for capital satisﬁes:
Kt+1 = (1 − δ)Kt + I¯t ,0 < δ < 1,(33)

where I¯t denotes the newly produced capital goods, which is diﬀerent from investment It
in terms of consumption goods, and δ denotes the capital depreciation rate.
Uncertainty Shocks: As in the partial equilibrium model in Section 2, an entrepreneur

has a project with success probability p, which follows distribution Ft (p). Here I introduce
an exogenous disturbance to the distribution of the riskiness of project and discuss theimplications to aggregate variables. In order to get a simple analytical expression andconduct simulations later I assume that the distribution of the riskiness of project is uniform
over interval [pt , 1]:
p − ptFt (p) =, pt ≡ peυt , 0 < p < 1,(34)
1 − pt20

withυt = ρυ υt−1 + ϵυ,t ,0 < ρυ < 1,

where ϵυ,t denotes a disturbance i.i.d. with mean zero. When the negative uncertainty
shock, ϵυ,t < 0, hits the economy, the distribution becomes more dispersed and the entrepreneur’s project becomes more risky on average. Intermediaries face higher degree ofasymmetric information about the riskiness of project, p, than before.
3.2. Model-I: Adverse Selection in the Demand Side of Capital
I embed adverse selection in ﬁnancial markets, analyzed in Section 2, into the demandside of capital as in BGG (1999). In this model, entrepreneurs own, trade and rent out
capital. In trading capital entrepreneurs purchase capital so that the entrepreneurs’ activities determine the demand for capital. In the supply side of capital there are competitive
capital goods producers subject to investment adjustment costs. If the demand for capitalincreases, the price of capital increases and so does investment. In this model uncertainty
shocks emerge as ﬁnancial shocks which change a wedge between a return to capital and arisk-free rate.
Entrepreneurs and Intermediaries: There exist many entrepreneurs. At time t an en-

trepreneur starts its business with some amount of net worth in unit of consumption goods.
The entrepreneur makes a one-period contract with an intermediary and receives a loan
from the intermediary. Combining the net worth with the loan the entrepreneur purchases
capital goods from capital goods producers with price qt . In aggregate the balance sheetof entrepreneurs satisﬁes:
qt Kt+1 = Nt + Bt .
(35)
The left hand side of equation (35) denotes the value of capital purchased by entrepreneurs
and the right hand side of equation (35) denotes the liability side of balance sheet, consistingof the aggregate net worth, Nt , and the aggregate loan, Bt .
At the end of time t the entrepreneur invests the capital goods in its project with successprobability p and transforms the capital goods into specialized capital goods readily use for
production. The riskiness of project, p, is private information to the entrepreneur. I assume
that on average one unit of capital goods generates one unit of specialized capital goods
for all projects. If the project fails the entrepreneur has nothing at hand. If the projectsucceeds the entrepreneur rents out the specialized capital goods to ﬁrms and earns rentalk
ut+1 per unit of eﬀective capital goods (specialized capital goods times capitalrate rt+1
utilization rates) at the beginning of time t + 1. The entrepreneur incurs the cost of capitalutilization rates, a(ut+1 ), per unit of specialized capital goods. After renting out capital
goods, the entrepreneur sells the depreciated capital goods to capital goods producers withprice qt+1 . Consequently, on average the return from investing one unit of consumption
21

kgoods, Rt+1
, is given by,
k
Rt+1
=

k
rt+1
ut+1 + qt+1 (1 − δ) − a(ut+1 )
.
qt

(36)

k
The return consists of the net return from renting out capital goods, rt+1
ut+1 , plus thecapital gain from the depreciated capital goods, qt+1 (1 − δ), minus the cost associated withcapital utilization rate, a(ut+1 ), per unit of capital purchased at price qt .
After earning returns, the entrepreneur makes a loan payment to the intermediary.
The remaining amount of consumption goods constitutes the entrepreneur’s net worth attime t + 1. At the beginning of time t + 1, an idiosyncratic occupation shock hits theentrepreneur and it switches its job to a worker randomly with probability 1 − γ, and it
stays an entrepreneur with probability γ, where 0 < γ < 1. If the entrepreneur becomes aworker, it brings the net worth to the household to which it belongs. If the entrepreneur
remains to be an entrepreneur, it starts its business with its net worth at time t + 1 again.
Those who just have become entrepreneurs from workers and those who do not have net
worth receive a small amount of goods from the household to which they belong, so thatthey run their projects.
The ﬁnancing problem between entrepreneurs and intermediaries proceeds as in thepartial equilibrium model in Section 2. The returns, Re and Rf in the model in Section 2,kcorresponds to Et Rt+1
and Rt+1 in this general equilibrium model respectively. In applying
the solution in Section 2, I assume that the payment, Xn (p), does not depend on states attime t + 1. This assumption is innocuous because entrepreneurs behave as if they were risk
neutral.
Now I derive equilibrium conditions by applying the solution in Section 2. First, I
derive an equation for the demand for capital. Substituting (20) into (35) and using theexpression for Ft (·), (34), I can express the value of purchased capital as[]∗(1 − ϕ)st pt − pt
Nt ,
(37)
qt Kt+1 = 1 +
1 − (1 − ϕ)st 1 − ptkwhere st ≡ Et Rt+1
/Rt+1 denotes the discounted return to capital and pt = peυt denotes
the lower bound of the support of the distribution of riskiness. Equation (37) deﬁnes thedemand for capital. If the right-hand-side of equation (37) increases, for example, due to achange in the net worth, the demand curve of capital shifts upward. The price of capital,
qt , increases, which, in turn, increases investment and output.
Next, I consider an equation that determines threshold p∗t above which entrepreneurs
do not get loans. According to Proposition 1 the threshold is given by the zero proﬁt
condition, (18). Given uniform distribution Ft (·) I can express zero proﬁt condition (18) as])∫ p∗t [(
p
−
p
1
1
p
1
t
vt+1,n (p∗t ) =p− ϕ dp = 0,
1−
−
(38)
st 1 − pt st 1 − ptpt22Under an assumption that vt+1,n (1) < 0, equation (38) has a unique interior solution.10
Expanding equation (38), I obtain
(
)[
]
[
]
2ϕ−1
pt ϕ
2ϕ−1
1−ϕ
− 1−ϕ
ϕ
2
−
ϕ
ϕ
∗
∗
0=
1−
(pt ) ϕ − pt−(pt ) ϕ − pt,
(39)
2ϕ − 1st
st 1 − ϕwhere I have assumed ϕ ̸= 1/2. Equation (39) deﬁnes p∗t given st and pt .
Next, I derive an equation for an external ﬁnance premium, which constitutes an important ﬁnancial variable in the model. I deﬁne an external ﬁnance premium, EFPt , as a ratioof the interest rate of external ﬁnance to the opportunity cost of internal funds (or risk-freeinterest rates), Rt+1 . In this model the interest rate of external ﬁnance corresponds to aloan interest rate. Because a loan interest rate diﬀers among entrepreneurs with diﬀerent
level of riskiness, as shown in Lemma 2, I use the average loan interest rate as the interest
rate of external ﬁnance.
I derive an expression for the average loan interest rate ﬁrst and derive an expression foran external ﬁnance premium. Conditional on the success of project, an entrepreneur pays
back to an intermediary following the payment schedule, (16). Reminding that Re in thekpartial equilibrium model corresponds to Et Rt+1
in this model and aggregating (16) overriskiness p and net worth Nj , I obtain the aggregate payment conditional on the success ofproject,( ) 1−ϕ∫ ∫ p∗tϕk∗
Et Rt+1  pt
(40)
− 1 Nt
Xn (p)dFt (p)dHt (n) =Xt+1 =
1
−
p
p
n p
t
t
t

I deﬁne the average loan interest rate as a ratio of the aggregate payment to the aggregatebloan: Rt+1
= Xt+1 /Bt , where Bt = qt Kt+1 − Nt is given by (37). Then I can express theexternal ﬁnance premium, EFPt , as follows:( ) 1−ϕϕ∗
1 − (1 − ϕ)st  ptb
− 1 .
(41)
EFPt = Rt+1
/Rt+1 =
(1 − ϕ)(p∗t − pt )ptThe external ﬁnance premium must be positive or EF Pt > 1.11
Finally I derive the law of motion for aggregate net worth. In aggregate entrepreneursk(Nt + Bt ) and pay Rt+1 Bt to intermediaries at the beginning of period
earn return Rt+1
t + 1. The aggregate payment amounts to Rt+1 Bt because the intermediary’s zero proﬁt
10

The assumption, vt+1,n (1) < 0, holds for a reasonable range of parameter values of st and ϕ.
To see this note that the conditional repayment, given by (40), is strictly greater than the av∫ ∫erage repayment, n p pXn dFt (p)dHt (n), and the average repayment over Bt is Rt+1 from the zeroproﬁt condition, (18). Without asymmetric information the external ﬁnance premium would become
EFPt = (1 − pt )−1 log(1/pt ), which purely reﬂects the default risk of entrepreneurs.
1123condition holds.12 A fraction, 1 − γ, of entrepreneurs become workers and bring their
net worth to households. The same number of workers become new entrepreneurs. The
new entrepreneurs and those who do not have any net worth receive the small amount ofstart up funds from households. I assume that the transfer from households at time t is
proportional to output, given by ξYt , where 0 < ξ < 1. Then I can write the law of motion
for aggregate net worth askk
Nt+1 = γ[(Rt+1− Rt+1 )Bt + Rt+1
Nt ] + ξYt+1 ,(42)

where Bt = qt Kt+1 − Nt is given by (37)
In sum, four equations (37), (39), (41) and (42) describe the aggregate consequences of
the ﬁnancing problem between entrepreneurs and intermediaries.
Capital goods Producers: Competitive capital goods producers run two types of business.

First, they produce new capital goods. Second, they purchase depreciated capital goods
from entrepreneurs and sell new capital goods to entrepreneurs.
A capital goods producer purchases consumption goods, It , and transforms it into newcapital goods, I¯t , using following technologies,
[
(
)]
It
¯
It = 1 − S
It ,
(43)
It−1
where function S(·) denotes investment adjustment costs satisfying S(1) = S ′ (1) = 0 and
S ′′ (1) > 0 in steady state. This functional form, introduced by CEE (2005), allows themodel to generate a hump-shaped response of investment and output to various shocks,
consistent with VAR-based evidence.
The capital goods producer purchases depreciate capital goods from entrepreneurs withprice qt , combines them with newly produced capital goods and produce capital goods,
using the linear capital accumulation technology, (33). Then the capital goods producer
sells the capital goods with price qt to entrepreneurs. Given the price of capital, qt , thecapital goods producer chooses the amount of investment to maximize the expected proﬁt:
max Et{It }∞∑β s U1 (Ct+s , Lt+s )s=0

U1 (Ct , Lt ){qt+s Kt+s+1 − [It+s + qt+s (1 − δ)Kt+s ]} ,(44)

subject to production technologies, (33) and (43). Because a household owns capital goods
producers, the capital goods producer discounts future proﬁts using the household’s discount factor. Because of linearity in producing capital goods in (33), perfect competition
results in the same price between new capital goods and old capital goods.
12Intermediaries take in deposits Bt from households with risk-free interest rate Rt+1 and lend to entrepreneurs. The intermediary’s zero proﬁt condition results in the aggregate payment equal to Rt+1 Bt , asshown in the ﬁrst line of equation (21), where the right-hand-side denotes the aggregate payment divided
by Rt+1 .

24

Equilibrium of Model-I: In deﬁning a competitive equilibrium I suppose that the economy

starts from period 0. Given the set of initial condition, {K0 , N0 , I−1 , A−1 } and theprocesses of shocks, a recursive competitive equilibrium consists of the decision rules for the
allocation, {Yt , Ct , It , Kt+1 , Lt , Nt+1 , ut , p∗t }, and the pricing rules for the set of prices {rtk ,wt , qt , Rt+1 }, where the both rules are the functions of the states of the economy, satisfying:• Given the pricing rules, the decision rules satisfy the intermediate goods producers’
ﬁrst order conditions, (29) and (30), the household’s ﬁrst order conditions, (26) and(27), the condition for capital utilization rate, (31), and solve capital goods producer’s problem (44). Also, the decision rules satisfy the optimality conditions ofentrepreneurs and intermediaries, (37) and (39), and the law of motion for capital,(42).
• All markets clear. That is, (32) and (33) hold, where Yt in (32) is given by (28) andI¯t in (33) is given by (43).
Uncertainty Shocks as Financial Shocks: Now I study the role of uncertainty shocks.

In order to make clear the role of uncertainty shocks I log linearize the model around its
steady state. Let x̂t denote the deviation of variable xt from its steady state at time t.
Log-linearizing equations (37) and (39) and substituting out p∗t I obtain:()
ŝt = −χ1 N̂t − q̂t − K̂t+1 − χ2 υt , χ1 , χ2 > 0.
(45)
For the derivation of equation (45), see Appendix.
Equation (45) summarizes both the eﬀect of the uncertainty shocks and the role ofcredit frictions. Without the uncertainty shocks, υt = 0, equation (45) coincides with theequation of the demand for capital in BGG (1999) after log-linearizion. The discounted
return to capital, st , rises when the net worth decreases. The negative relationship between
the discounted return to capital and the net worth serves as a ﬁnancial accelerator or abalance sheet channel: a decrease in net worth increases the discounted return, which, inturn, decreases the net worth, and so on.
Equation (45) includes the uncertainty shocks, υt , which do not appear in the original
BGG (1999) model. In equation (45) the negative uncertainty shocks, υt < 0, raise thediscounted return to capital, st , or a wedge between a return to capital and risk free rates.
That is, the uncertainty shocks increase the cost of borrowing of consumption goods ﬁrms
relative to risk free rates. The negative shocks decrease the net worth and decrease thedemand for capital from equation (37). It results in a fall in the price of capital, investment
and output, generating business ﬂuctuations consistent with business cycle facts as I explore
quantitatively later.
The uncertainty shocks play exactly the same role as ﬁnancial shocks considered by Hall(2010), Gilchrist, Ortiz and Zakrajsek (2009) and Gilchrist and Zakrajsek (2010). They
25

introduce ﬁnancial shocks which change a wedge between the return to capital and therisk free rate in a reduced form manner. In my model the residual terms in equation (45)have a solid micro-foundation and a clear interpretation. The uncertainty shocks aﬀect the
distribution of the riskiness of entrepreneurial project. The uncertainty shocks change thedegree of asymmetric information exogenously and aﬀect the severity of adverse selection
endogenously, generating business ﬂuctuations.
3.3. Model-II: Adverse Selection in the Supply Side of Investment
I embed adverse selection in ﬁnancial markets, analyzed in Section 2, into the supply sideof investment as in Carlstrom and Fuerst (1997). In this model, entrepreneurs own net
worth and produce new capital goods. Entrepreneurs combine their own net worth andloans from intermediaries, purchase consumption goods and transform them into newly
produced capital goods. In contrast to Model-I, the entrepreneurs’ activities determine
the supply of investment. The intermediaries not only provide loans to entrepreneurs but
also purchase depreciated capital goods from entrepreneurs. In this model the uncertainty
shocks emerge nearly as shocks to the marginal eﬃciency of investment (MEI shocks).
Entrepreneurs and Intermediaries: Intermediaries take in deposits from households withrisk free rate Rt+1 . The intermediaries can earn return either by purchasing capital or by
investing in entrepreneurs and receiving newly produced capital from entrepreneurs. If theintermediaries purchase capital with one unit of consumption goods, they earn the rentalkrate of capital and sell the depreciated capital in the next period , so that return Rt+1
isgiven by (36). Because the intermediaries promise to pay Rt+1 to households, which maykkexceed risky return Rt+1
, the intermediaries sell contingent claims with return Rt+1
− Rt+1
to households. Consequently, the following arbitrage condition holds:][
U1 (Ct+1 , Lt+1 ) k
Rt+1 .
(46)
1 = Et βU1 (Ct , Lt )kIntermediaries pay the risk free rate to households by combining a return to capital, Rt+1,
kand a net return from contingent claims, Rt+1 − Rt+1 .
Intermediaries are willing to provide loans to entrepreneurs as long as they receiverisk-free rate Rt+1 tomorrow in return for one unit of consumption goods today. If an
entrepreneur provides 1/qt units of newly produced capital goods today in return for onek, or risk free rate Rt+1 if combined
unit of consumption goods, tomorrow it yields return Rt+1
with a net payment from contingent claims. A ﬁnancing problem between entrepreneurs
and intermediaries features an intra-period arrangement. The return, Rf , in the partial
equilibrium model in Section 2 corresponds to return 1/qt in this general equilibrium model.
An entrepreneur specializes in producing new capital goods from consumption goods.
The entrepreneur has linear technologies (projects) transforming one unit of consumption

26

goods into µt unit of new capital goods on average, where µt denotes investment shocks,
following a stochastic process:log(µt ) = ρµ log(µt−1 ) + ϵµ,t ,0 ≤ ρµ < 1.

(47)

As in the partial equilibrium model in Section 2, if the entrepreneur invests one unit ofconsumption goods in its project, the entrepreneur successfully produces θ(p)µt units of
new capital goods with probability p, where θ(p) ≡ 1/p. Riskiness p is private information
to the entrepreneur. In this model, an expected return on project is µt , which corresponds
to Re in the model in Section 2.
The ﬁnancing problem between entrepreneurs and intermediaries proceeds as in themodel in Section 2. At the beginning of time t, an entrepreneurs combines its own net
worth and the loan from intermediaries to ﬁnance the purchase of consumption goods. In
aggregate, the entrepreneur’s balance sheet is given by
It = Nt + Bte ,(48)

where It denotes the investment in terms of consumption goods and Bte denotes the loan
from intermediaries. Applying the result in Section 2, in equilibrium I can express theaggregate loan, Bte , as the right-hand-side of equation (22) with Re /Rf in (22) replaced byµt /(1/qt ) = qt µt . Then, I can rewrite (48) as][∗(1 − ϕ)qt µt pt − pt
Nt ,
(49)
It = 1 +
1 − (1 − ϕ)qt µt 1 − pt
where I have used the uniform distribution of the riskiness of project, given by (34). Threshold, p∗t , is determined by the zero proﬁt condition, (39) with st replaced by qt µt , which isessentially the same as in Model-I. The aggregate newly produced capital satisﬁes:
I¯t = µt It .

(50)

In contrast to Model-I, there is no investment adjustment cost in this model. Actually,
the ﬁnancial frictions embedded in the supply side of investment play a role similar toinvestment adjustment costs, as indicated by Carlstrom and Fuerst (1997).
In aggregate entrepreneurs produce new capital goods I¯t , given by (50), and pay (1/qt )Bte
units of capital goods to intermediaries. Entrepreneurs rent remaining capital goods to consumption goods ﬁrms and sell the depreciated capital to intermediaries in the beginning ofnext period. Then, the law of motion for aggregate net worth is given byk
Nt+1 = γ[rt+1+ (1 − δ)qt+1 ][I¯t − (1/qt )Bte ] + ξYt+1 ,(51)

where γ denotes the survival probability of entrepreneurs and ξYt+1 denote the lump-sumtransfers at time t + 1 from households for entrepreneurs who have failed in their projects
or for newly born entrepreneurs.
27Equilibrium of Model-II: Given the set of initial condition, {K0 , N0 , A−1 } and the pro-

cesses of shocks, a recursive competitive equilibrium consists of decision rules for the allocation, {Yt , Ct , It , Kt+1 , Lt , Nt+1 , ut , p∗t }, and pricing rules for the set of prices {rtk , wt ,qt , Rt+1 }, where the both rules are the functions of the states of the economy, satisfying:• Given the pricing rules, the decision rules satisfy the consumption goods producer’s
ﬁrst order conditions, (29) and (30), the household’s ﬁrst order conditions, (26) and(27), the condition for capital utilization rate, (31), and the arbitrage condition,(46). Also, the decision rules satisfy the optimality conditions of entrepreneurs andintermediaries, (49) and (39), and the law of motion for aggregate net worth, (51).
• All markets clear. That is, (32) and (33) hold, where Yt in (32) is given by (28) andI¯t in (33) is given by (50).
Uncertainty Shocks as Investment Shocks: Now I study the role of uncertainty shocks inthe model with adverse selection in the supply side of investment. I show that uncertainty
shocks emerge nearly as the MEI shocks which change the marginal eﬃciency of investment.
Two equations, (39) and (49) with st replaced by qt µt , determine the investment, It .
Log-linearizing those two equations I obtain:
( )
( )
1
χ4
ˆ
It =
(q̂t + µ̂t ) + N̂t +υt , χ3 , χ4 > 0,
(52)
χ3
χ3
where exact expressions for coeﬃcients χ3 and χ4 are given in Appendix. Then, the newly
produced capital, I¯t = µt It , after log-linearization, is given by
( )
[(
)
( ) ]
1
1 + χ3χ4ˆ¯It =q̂t + N̂t +
µ̂t +
υt .
(53)
χ3
χ3
χ3
Equation (53) makes clear that the uncertainty shocks, υt , emerge nearly as the MEI
shocks, µ̂t . Given q̂t and N̂t both the uncertainty shocks and the MEI shocks change the
newly produced capital. On the one hand, the uncertainty shocks change the severeness ofadverse selection and aﬀect the amount of loans and investment. On the other hand, the
MEI shocks aﬀect not only the amount of loans but also the eﬃciency of investment. After
log-linearization the two shocks aﬀect the newly produced capital in the same manner.
In this model both the uncertainty shocks and the MEI shocks appear in the two
equilibrium conditions: the condition for I¯t , (50) with It substituted out using (49), andthe law of motion for aggregate net worth, (51). Adjusted the magnitude of the two shocks,
the two shocks have the same eﬀect on the newly produced capital in equation (50). A
slight diﬀerence appears in the law of motion for aggregate net worth, (51). In (51), theeﬀects on I¯t of the two shocks are the same while the eﬀects on Bte of the two shocks are
diﬀerent. I will explore the diﬀerence between the uncertainty shocks and the MEI shocks
quantitatively in the next section.
428

4. Simulation
I log-linearize the two general equilibrium models, presented in Section 3, around steadystate and conduct simulations to explore the quantitative eﬀect of uncertainty shocks. In
the following I ﬁrst parameterize the two models. Next, I study impulse responses to uncertainty shocks for the two models. Then, I study mechanisms which amplify uncertainty
shocks and help uncertainty shocks generate business cycles. Finally, I conduct stochastic simulations and examine how much volatility of key macroeconomic variables can be
explained by uncertainty shocks.
4.1. Model Parameterization
Before setting parameters I specify the utility functional as follows:
U (C, L) = log(C) − ψ

L1+1/ν
,1 + 1/νψ, ν > 0.

I list the choice of parameters values in Table 1. Out of thirteen parameters, ﬁve {ϕ, p, γ, ξ, ρu }are speciﬁc to the models with credit frictions and the others are conventional except theelasticity of markups.
I begin with conventional parameters. The period of time is quarterly. I set a preference
discount factor as β = 0.993 so that the net real risk-free rate becomes 3 percent annual
rate in steady state. I set the coeﬃcient of the disutility of labor, ψ, in a way that the
average hours worked becomes unity in steady state. I choose conventional values for alabor supply elasticity (ν = 1), a capital income share (α = 0.36) and a capital depreciation
rate (δ = 0.025). I set the curvature of investment adjustment costs to S ′′ (1) = 1, which
locates at the lower range of estimated values in the DSGE literature, though there islittle guidance in the empirical literature about appropriate values. I choose this value
because the implied curvature is enough to generate the reasonable hump-shaped responsesof output and investment in Model-I. I set the parameter of capital utilization costs asχ ≡ a′′ (1)/a′ (1) = 5, which is consistent with estimated values in the DSGE literatureincluding Justiniano, et al (2009a,b). I set a markup in wages in steady state as λw = 1.2
and the elasticity of markup in wages as ω = 2. The elasticity is consistent with Gali,
Gertler and Lopez-Salido (2007) who report that a wage markup is slightly more than twice
as volatile as output. Also, they report that a contemporaneous correlation between a wage
markup and output is −0.83, consistent with a countercyclical markup in the models.
Next I set parameters speciﬁc to the models with credit frictions, {ϕ, p, γ, ξ, ρu }. I
set ξ = 0.001 which determines the amount of lump-sum transfers from households toentrepreneurs. As in Carlstrom and Fuerst (1997) I make the transfers small so that thetransfers do not add additional dynamics. I set the AR(1) coeﬃcient of uncertainty shocks
as ρu = 0.75, following Gilchrist and Zakrajsek (2010) who use VARs and estimate the29Table 1: Parameters Values
Parameter

β
α
′′
S (1)
λw
ϕ
γ
ρu

Description

Value

Parameter

discount factor

0.993
0.36
1
1.2
0.55
0.99
0.75

ν
δ
χ
ω
p
ξ

capital income shareadjustment costs
wage markup
moral hazard
survival rates
AR(1), uncertainty shocks

Description

Valuelabor supply elasticity

1
0.025
5
2
0.993
0.001

depreciation rate
capital utilization costs
elasticity of markup
parameter of F (·)
transfers

auto-correlation of ﬁnancial shocks equal to 0.75. As shown in Section 3 the uncertainty
shocks in Model-I turn out to be the same as ﬁnancial shocks after log-linearization.
I set remaining three parameters to hit following three targets in steady state: the leverage ratio of 1.5, the external ﬁnance premium of 2 percent annual rate, and the expected
equity premium of 1 percent annual rate. The target value of the leverage ratio is lower
than the literature13 , but it is consistent with U.S. Flow of Funds Accounts according towhich the debt net worth ratio of non-farm non-ﬁnancial corporate business is around 0.5,
implying a leverage ratio of 1.5.14 The target value of the external ﬁnance premium is close
to the average spread of corporate bonds with various credit quality relative to comparable
maturity Treasury yield from 1990 to 2008 of 1.92 percent, analyzed by Gilchrist, Yankov
and Zakrajsek (2009). The target value of the expected equity premium does not mean
to match any numbers but suggests that there exists a discrepancy between a return ofcapital and a risk-free rate due to credit frictions.
4.2. Responses to Uncertainty Shocks
I study responses to the uncertainty shocks in the two models presented in Section 3.
First, I draw the responses in Model-I and show that the uncertainty shocks generate theco-movement of variables consistent with business cycles. Second, I draw the responses inModel-II and show that the uncertainty shocks have similar quantitative implications for
key variables to the MEI shocks. Though Model-II exhibits ﬂuctuations consistent withbusiness cycles for key variables, the model fails to generate a co-movement for the priceof capital and the net worth.
13

For example, the leverage ratio in steady state is 2 in BGG (1999) and 4 in Gertler and Karadi (2010).
Ajello (2010) analyzes the U.S. public non-ﬁnancial companies included in Compustat and reports that33 percent of the capital expenditures of those ﬁrms is funded using ﬁnancial markets. While a leverage
ratio in general heavily depends on the type of borrowers and markets, Ajello (2010)’s ﬁnding implies thatleverage ratio of 1.5 is not low.
1430

Before proceeding to simulation result I would like to emphasize the unique nature of
the uncertainty shocks. The shocks have real eﬀects because of asymmetric informationand its by-product of adverse selection. With symmetric information the shocks would nothave any real eﬀects as I showed in Section 2.2.
Model-I: Figure 1 plots impulse response functions to the negative uncertainty shocks. In

period t = 0, an economy is in steady state. In period t = 1, the negative uncertainty
shocks hit the economy. I set the magnitude of the shocks in such a way that an externalﬁnance premium rises 1 percent annual rate from its steady state level at the impact of
the shocks. Except the external ﬁnancial premium the vertical axis of ﬁgures shows thepercent deviation of variable from its steady state.
Figure 1 shows that the uncertainty shocks in Model-I generate the co-movement ofvariables consistent with business cycles. In response to the negative uncertainty shocks
which increase the external ﬁnance premium by 1 percent annual rate, all variables decrease.
The output decreases about 0.4 percent with its bottom reached in four periods after theshocks. The output shows a hump-shaped response thanks to the CEE adjustment costs.
While the auto-correlation of shocks is relatively low, ρυ = 0.75, the output shows apersistent response: the output stays below a half of its bottom even after 20 periods.
The investment decreases about 1.2 percent with a hump-shaped response. The magnitude of the change is about three times as great as output. The consumption decreases
slightly in the initial periods and continue decreasing until eighteen periods after the shocks.
The hours show a response similar to the output. The capital utilization rate also decreases
with a hump-shaped response.
The notable feature of Model-I appears in the responses of the price of capital and thenet worth. The price of capital decreases about 0.3 percent at the impact of the shocks andquickly moves back to the steady state in seven periods. The net worth shows a response
similar to the price of capital, because the current net worth is mainly determined by thegross return on capital, which, in turn, is mainly determined by the current price of capital,
as is clear from equations (36) and (42). With the right co-movement of the price of capitaland the net worth, the uncertainty shocks in Model-I generate ﬂuctuations consistent withbusiness cycles.
Model-II: Next I proceed to responses to the uncertainty shocks in Model-II. Figure 2 plots

two impulse response functions: one to the negative uncertainty shocks (solid line) and theother to the negative MEI shocks (dashed line). I set the magnitude of the uncertainty
shocks as same as before. I set the magnitude of the MEI shocks in such a way that the
responses of output coincide with those to the uncertainty shocks at an impact. As a result,the magnitude of the MEI shocks is about one-third of the magnitude of the uncertainty
shocks, as shown in the right bottom panel in Figure 2.
Figure 2 makes three important observations. First, the uncertainty shocks fail to
31Output, Yt (%)
0

Investment, It (%)

Consumption, Ct (%)

0

0
−0.05

−0.1

−0.1

−0.5

−0.2

−0.15
−0.3
−0.4

−0.2

−1
0

10

20

0

Hours, Lt (%)

10

20

0

Capital Utilization, ut (%)

10

20

Price of capital, qt (%)
0

0
−0.1
−0.2

−0.1

−0.02

−0.3
−0.4

−0.2

−0.04

−0.5
0

10

20

Net worth, Nt (%)

0

10

20

10

20

Shock, υt (%)

Extenral finance premium, EFPt (APR diff)
0

1

0

0

−0.1

−0.1
0.5

−0.2

−0.2

−0.3

−0.3

−0.4
0

10

20

0

0

10

20

0

10

20

Figure 1: Impulse Responses to Uncertainty Shocks (Model I)
generate the co-movement of the price of capital and the net worth in Model-II. In thismodel the negative uncertainty shocks shift the supply curve of investment inward. As aresult, the price of capital increases, while the investment decreases. As shown in the lawof motion for net worth, (51), the current net worth is increasing in the price of capital.
An increase in the price of capital causes an increase in the net worth.
Second, the uncertainty shocks succeed in generating the co-movement of variables
except the price of capital and the net worth. In response to the negative uncertainty
shocks, an external ﬁnance premium increases slightly more than 1 percent annual rate.
The output decrease about 0.65 percent at the impact of the shocks and moves graduallyback to the steady state. The investment decreases about 2.2 percent whose magnitude isslightly more than three times as great as the output. The consumption decreases slightly
in the initial periods and continue decreasing, showing a very persistent response. The
hours show responses similar to the output and the capital utilization rate decreases too.
Third, the uncertainty shocks generate observationally equivalent responses to those to32

Output, Y (%)

Investment, It (%)

t

0
−0.2
−0.4

υ shockµ shock

−0.6
0

10

0

−0.5

−0.05

−1

−0.1

−1.5

−0.15

−2

−0.2
0

20

Hours, Lt (%)
0Consumption, Ct (%)

0

10

20

0

Capital Utilization, ut (%)

10

20

Price of capital, qt (%)

0
0.1

−0.2
−0.4

−0.05
0.05

−0.6
−0.8

−0.1
0

10

20

Net worth, Nt (%)

0
0

10

20

0

10

20

Shocks, υt and µt (%)

External finance premium, EFPt (APR diff)
0

1

−0.1

0.2

−0.2

0.5

0

−0.3
−0.2
0

10

20

0

0

10

20

0

10

20

Figure 2: Impulse Responses to Uncertainty Shocks (Model II)
the MEI shocks for all variables except the net worth and the external ﬁnance premium. As
argued in Section 3.3, the two shocks appear in two equations in Model II. On the one hand,the uncertainty shocks, after adjusted the magnitude of the shocks, have the same eﬀect
on newly produced capital, I¯t , as do the MEI shocks. On the other hand, the two shocks
have the diﬀerent eﬀect on the aggregate loan, Bte , and so do on the net worth. Because
the two shocks aﬀect the newly produced capital in the same manner, the responses ofall variables except the net worth and the external ﬁnance premium almost coincide withminor diﬀerences.
The minor diﬀerences reﬂect feedback eﬀect from the net worth, which shows diﬀerent
responses between the two shocks. In response to the uncertainty shocks the net worth
increases because the price of capital increases. In response to the MEI shocks the net
worth does not increase very much because the aggregate loan, Bte , is greater than that in
response to the uncertainty shocks, while the eﬀect on newly produced capital, I¯t , is almost

33

the same.15 As a result, from the law of motion for net worth, (51), the net worth does notincrease relative to that in response to the uncertainty shocks.
The third observation of the equivalence between the uncertainty shocks and the MEI
shocks provides a rationale for the ﬁnding by Justiniano, et al (2009b). They estimate a
DSGE model and ﬁnd that the MEI shocks serve as the most important shocks driving the
U.S. business cycles. They interpret the MEI shocks as something related with ﬁnancial
factors. They show empirically that their estimated MEI shocks are negatively correlatedwith an external ﬁnance premium, though their estimated model abstracts from ﬁnancial
factors and does not have an external ﬁnance premium. In Model-II, imperfect ﬁnancial
markets materialize the uncertainty shocks and the negative uncertainty shocks increase
an external ﬁnance premium, consistent with the ﬁnding by Justiniano, et al (2009b). The
result obtained here suggests that the uncertainty shocks combined with imperfect ﬁnancial
markets can be a candidate for the source of the MEI shocks.
4.3. Amplification Mechanisms
The previous analysis on responses to the uncertainty shocks shows that both Model-I andModel-II generate the co-movement of key variables: output, investment, consumption andhours. Here I study mechanisms behind the success in generating the co-movement. Set
aside the adverse selection in ﬁnancial markets, the two models have additional features
relative to the standard real business cycle model: a countercyclical markup in wages andvariable capital utilization rates. The two features amplify the uncertainty shocks and help
generate the co-movement of key variables.
I show analytically that both a countercyclical markup in wages and variable capitalutilization rates are essential in generating the co-movement of key variables. The equation
governing a co-movement between hours and consumption is the intra-temporal optimality
condition of households, (27), which equates wages to a markup over the marginal rateof substitution between consumption and hours. After substituting out wages wt , capitalutilization rates ut , markup λw,t and output Yt using (30), (31), (25) and (28) respectively,
I log-linearize equation (27) and obtain a relationship between consumption and hours:
[
(
)
]
α
(ω + 1)αχĈt = (ω + 1)(1 − α) 1 +
K̂t ,
(54)
− (1 + 1/ν) L̂t +
χ+1−α
χ+1−α
where ω ≥ 0 denotes the elasticity of markup and χ > 0 denotes the elasticity of capital
15

From equations (48)-(50), on the one hand, the uncertainty shocks aﬀect the newly produced capital,¯
It , only through an eﬀect on the aggregate loan, Bte . On the other hand, the MEI shocks aﬀect the newly
produced capital not only through an eﬀect on the aggregate loan but also through the change in themarginal eﬃciency of investment. Because I scale the magnitude of the investment shocks in such a waythat the eﬀect on the newly produced capital becomes the same, the eﬀect on the aggregate loan to the
MEI shocks becomes smaller than that of the uncertainty shocks.

34

utilization costs. I put the derivation of equation (54) in Appendix. If ω = 0 a countercyclical markup in wages vanishes. If χ = ∞ variable capital utilization rates vanish.
In order to understand the role of a counter-cyclical markup in wages and variable
capital utilization rates, I consider the case of the standard business cycle model in which
the two are shut oﬀ: ω = 0 and χ = ∞. In this case, a coeﬃcient on hours becomes negative,−(α + 1/ν) < 0, because the two parameters, α and ν, have to satisfy 0 < α < 1 andν > 0. A co-movement between consumption and hours depends mainly on a coeﬃcient on
hours, because the capital moves very slowly. Therefore, without a counter-cyclical markup
in wages and variable capital utilization rates, both Model-I and Model-II would fail togenerate a co-movement between the two variables. Intuitively, in recessions hours decrease
and wages increase without neutral technological shocks and variable capital utilization
rates. As a result of substitution from hours to consumption, consumption increases withouta counter-cyclical markup in wages. This result reﬂects a famous co-movement problem:
only neutral technology shock can easily generate the co-movement among key variables ina real business cycle framework, ﬁrst pointed out by Barro and King (1984).
Equation (54) shows that a coeﬃcient on hours in equation (54) is increasing in the
degree of a counter-cyclical markup in wages (ω) and is increasing in the degree of variable
capital utilization rates (1/χ). Intuitively, in recessions a labor market becomes less competitive due to an increase in markup. Also capital utilization rates decrease, mitigating
an increase in wages due to a decrease in hours. If the eﬀect of a counter-cyclical markup
supported by variable capital utilization rates dominates the substitution eﬀect, consumption decreases. Under the baseline parameters values, a coeﬃcient on hours is positive
and that’s why both Model-I and Model-II succeeds in generating a co-movement between
hours and consumption.
The mechanisms generating a co-movement between hours and consumption also serveas ampliﬁcation mechanisms. In response to the negative uncertainty shocks, an decreasein the aggregate demand is enhanced by a decrease in consumption as well as a decrease
in investment, resulting in an ampliﬁed response of output.
To see the eﬀect of the ampliﬁcation mechanisms, Figure 3 plots impulse response functions to the negative uncertainty shocks with various degree of a counter-cyclical markup:
from baseline ω = 2 to no variable markup ω = 0. The ﬁrst raw of Figure 3 plots the responses of output and consumption in Model-I and the second raw of Figure 3 plots thosein Model-II.
The responses in the two models exhibit high sensitivity to the value of ω aroundbaseline ω = 2. When ω drops to ω = 1.8, the output decreases less by about one-fourthand the consumption increases initially in Model-I, while in Model II the output decreases
less by about a half and consumption increases initially too.

35

Model I, Output, Yt (%)

Model I, Consumption, Ct (%)

0
0.1
0.05

−0.1

0
−0.2

−0.05
ω=0
ω=1.5
ω=1.8
ω=2

−0.3
−0.4
0

5

10

15

20

−0.1
−0.15
−0.2

25

0

Model II, Output, Yt (%)

5

10

15

20

25

Model II, Consumption, Ct (%)

0

0.1

−0.1

0.05

−0.2

0

−0.3

−0.05

−0.4

−0.1

−0.5
−0.15
−0.6
−0.2
0

5

10

15

20

25

0

5

10

15

20

25

Figure 3: Impulse Responses to Uncertainty Shocks with Various Values of ω
4.4. Business Fluctuations
I have shown that the uncertainty shocks drive business cycles for key variables in bothModel-I and Model-II. Still, the result is qualitative rather than quantitative, focusing ona co-movement among key variables. Now I take the uncertainty shocks slightly more
seriously and ask the following question: how much volatility can the uncertainty shocks
explain for key variables in the U.S.?
In order to answer the question I conduct a typical business cycle stochastic simulation.I
assume that the uncertainty shocks, υu,t , follow an AR(1) process with its disturbance following a normal distribution with mean zero. I simulate the time series of the disturbancesand generate the time series of key variables from a log-linearized model. Using the artiﬁcial
data with the same size as sample data, I calculate the key statistics in the same manneras I do for the sample data. I repeat this process for one-thousand times and report the
average values of the statistics.16
A central issue in conducting the simulation lies in the value of the standard deviation of
For some series of the disturbances, threshold p∗t exceeds unity, which violates condition p∗t < 1. I
exclude those series and repeat the process until I obtain one-thousand series of artiﬁcial data satisfyingp∗t < 1 for all t.
1636the uncertainty shocks. Because the uncertainty shocks are not observable I should guess areasonable value for the standard deviation. I set the value in such a way that the simulated
data hits the target value of 0.77 percent for the standard deviation of an external ﬁnancepremium. The value coincides with the premium of sample data, deﬁned by a diﬀerence
between Moody’s BAA corporate bond yield versus the U.S. 10-year Treasury yield.
I use the above procedure to set the standard deviation of the uncertainty shocks forthe following reason. According to Gilchrist, Yankov and Zakrajsek (2009) and Gilchristand Zakrajsek (2010) who analyze various external ﬁnance premiums between the U.S.
corporate bonds yield and the U.S. Treasury yield of comparable maturity, the averageof the standard deviation of the premiums is above 2 percent. The value of 0.77 percent
used in the simulation is less than a half of the standard deviation implied by the data
analyzed by them. Also, Gilchrist and Zakrajsek (2010) use VARs and show that morethan 40 percent of the volatility of excess return on capital, which is closely related withan external ﬁnance premium, is explained by ﬁnancial shocks which turn out to be thesame as the uncertainty shocks in Model-I.17 Albeit this is little better than a guess it is
not unrealistic that about 40 percent (≈ 0.77/2) of the volatility of the average premium
attributes to the uncertainty shocks.
Table 2 shows the result of the stochastic business cycle simulations. The upper left in
Table 2 shows standard deviations and cross correlations for sample data from 1987 to 2010,
detrended by Hodrick-Prescott ﬁlter with smoothing parameter 1600 except for premiums.
For comparison I use another premium deﬁned by a diﬀerence between high-yield B-rated
corporate bonds from the Merrill Lynch’s High Yield Master ﬁle versus AAA corporatebond yields of comparable maturity, in addition to a premium deﬁned by a diﬀerence
between BAA corporate bonds yield versus the U.S. 10-year Treasury yield. The upper
right and the lower left in Table 2 show the corresponding average statistics calculated
from one-thousand artiﬁcial data series with the same sample size in Model-I and Model-II
respectively.
The simulation result, reported in Table 2, reveals three ﬁndings. First, both in ModelI and in Model-II the uncertainty shocks generate the signiﬁcant ﬂuctuations of output,
consumption, investment and hours. In Model-I and in Model-II the uncertainty shocks
explain about 20 percent and 30 percent of the volatility of the data respectively for output, investment and hours, while the shocks explain about 10 percent of the volatility for
consumption, respectively.
Second, both in Model-I and in Model-II the uncertainty shocks generate a co-movement
among output, consumption, investment and hours. For investment and hours both Model-I
17

Gilchrist and Zakrajsek (2010) uses terminologies, an external ﬁnance premium and an excess bondpremium (an excess return on capital), interchangeably. For clariﬁcation I stick to the following terminolok
gies in this paper. Here, an excess return to capital refers st ≡ Et Rt+1
/Rt+1 , in Model-I, and an externalﬁnance premium refers EF Pt deﬁned by equation (41) in Model-I.

37

Table 2: Cyclical Behavior of the U.S. and the Model Economy
Data (1987-2010)
SD (%)
Variable

Model I

Cross-Correlation of Output with

x(-1)

x

x(+1)

SD (%)[ratio to Data]

Cross-Correlation of Output with

x(-1)

x

x(+1)

Output

1.12

0.87

1.00

0.87

0.24 [ 0.22 ]

0.89

1.00

0.89

Consumption

0.80

0.68

0.87

0.85

0.08 [ 0.10 ]

0.12

0.38

0.60

Investment

4.67

0.86

0.93

0.80

0.82 [ 0.18 ]

0.91

0.97

0.79

Hours

1.79

0.74

0.88

0.90

0.35 [ 0.19 ]

0.90

0.99

0.85

Premium, BAA-Treasury

0.77
2.17

-0.50
-0.48

-0.39
-0.29

-0.20
-0.03

0.77 [ 1.00 ] -0.65

-0.50

-0.11

Premium, High-AAA

Model II
Cross-Correlation of Output with

VariableSD (%)[ratio to Data]

x(-1)

x

x(+1)

Output0.39 [ 0.34 ] 0.55

1.00

0.55

Consumption

0.08 [ 0.10 ] -0.06

0.31

0.55

Investment

1.26 [ 0.27 ] 0.58

0.99

0.49

Hours

0.57 [ 0.32 ] 0.56
0.77 [ 1.00 ] -0.55

1.00
-0.87

0.52
-0.44

Premium

Note: The data of output, consumption, investment,
hours are taken logs and are detrended by HodrickPrescott filer with smoothing parameter value 1600.
The simulated numbers are the average of thestatistics calculated in the same manner as the data
for the artificial data generated from the model for1000 times.

and Model-II succeeds in reproducing high contemporaneous correlation of output. Model-Ialso succeeds in reproducing high cross correlation of output with investment and hours,
while Model-II reproduces smaller cross correlation relative to sample data. The success ofModel-I reﬂects its persistence mechanisms. The CEE adjustment costs generate a humpshaped response of various variables and make responses persistent. For consumption
both Model-I and Model-II reproduce mild contemporaneous correlation of output relative
to sample data. While Model-I reproduces positive cross correlations, Model-II fails to
reproduce positive cross-correlation of output with lagged consumption. Relatively poor
performance on consumption reﬂects a co-movement problem in a real business framework,
though the problem is mitigated by ampliﬁcation mechanisms embedded in the two models.
Third, the uncertainty shocks succeed in reproducing the predictive power of an externalﬁnance premium in Model-I, while it is not the case in Model-II. According to the two data
series of premiums, the lagged premium shows a higher correlation with the current output(−0.50 and −0.48 respectively) more than do the current premiums (−0.39 and −0.29
respectively) and the lead premiums (−0.20 and −0.03 respectively). On the one hand,
Model-I shows a similar pattern; −0.65, −0.50 and −0.11 for the correlation of output withthe lagged, the current and the lead premium respectively, while the degree of correlations
is slightly higher relative to sample data. On the other hand, Model-II does not show sucha pattern: a premium is highly contemporaneously correlated with output. The diﬀerence
between Model-I and Model-II has to do with a diﬀerence in persistence mechanisms. In
38

Model-I the response of output is hump-shaped thanks to the CEE investment adjustment
costs, while in Model-II the response of output is the same as that of the premium. Thisobservation implies that a hump-shaped response, generated by persistence mechanisms, iscrucial for the model to reproduce the predictive power of premiums.

5. Extensions
In the previous section I showed that the uncertainty shocks in Model-I generate a comovement among variables including the price of capital and the net worth, consistent
with the U.S. business cycles. In this section I focus on Model-I and extend the model in
two ways.18
First, I compare the uncertainty shocks with risk shocks considered by CMR (2010) andclarify a distinction between the two shocks. Second, I introduce unrealized uncertainty
shocks about the riskiness of project, which change only ex-ante uncertainty about theriskiness of project but do not change the true distribution of the riskiness of projectex-post. I interpret the unrealized uncertainty shocks as shocks aﬀecting intermediaries’
animal spirits or intermediaries’ mood (pessimism and optimism). I show analytically thatthe unrealized uncertainty shocks are more powerful than the uncertainty shocks.
5.1. Comparison with Risk Shocks
The uncertainty shocks in Model-I share similar quantitative implications with risk shocks
considered by CMR (2010). However, the two shocks are conceptually diﬀerent. In the following I explain the risk shocks brieﬂy and provide two examples to shed light on conceptual
diﬀerences between the two shocks
CMR (2010) builds a DSGE model incorporating BGG (1999) and introduce shocks
to the standard deviation of idiosyncratic productivity of entrepreneurs, which they callrisk shocks. In their model entrepreneurs own, trade and rent out capital as in Model-I.
All entrepreneurs are identical ex-ante. The entrepreneurs make a debt contract with intermediaries, borrow funds and purchase capital. At the beginning of the next period an
idiosyncratic shock hits the entrepreneurs and a realized return constitutes private information to the entrepreneurs. The intermediaries observe the return if they use costly monitoring technologies. Under the debt contract an entrepreneur who cannot pay a promised
return goes bankrupt. The bankrupt entrepreneur is monitored and taken all assets by anintermediary.
As presented in Appendix, three equations summarize credit frictions in CMR (2010):(i) a balance sheet equation, (35), (ii) an optimality condition relating excess returnk/Rt+1 and the threshold of realized productivity under which entrepreneursst ≡ Et Rt+1
18

The extension for Model-II is straight forward and its implications are similar to Model-II.

39

go bankrupt, and (iii) an intermediary’s zero proﬁt condition. Let ur,t denote the risk
shocks to the standard deviation of idiosyncratic productivity. Log-linearizing the threeequations results in the following relationship:()
ŝt = −χ1,r N̂t − q̂t − K̂t+1 − χ2,r Et ur,t+1 .
(55)
Under conventional parameters values, coeﬃcients satisfy χ1,r , χ2,r > 0. Given that therisk shocks are persistent, for example ur,t = ρr ur,t−1 + ϵr,t with 0 < ρr < 1, the risk shocks
emerge as ﬁnancial shocks as do the uncertainty shocks. Equation (55) share the samestructure with its counterpart of the uncertainty shocks, (45). This observation makes
clear that the two shocks have similar quantitative implications.
Now I discuss conceptual distinctions between the uncertainty shocks and the risk
shocks. The uncertainty shocks concern the degree of uncertainty about the riskiness ofproject, while the risk shocks concern the riskiness of project. In other words, the riskinessof project itself does not matter to the uncertainty shocks, while it does matter to the risk
shocks. What matters to the uncertainty shocks is the degree of uncertainty (asymmetric
information) about the riskiness, not the riskiness itself.
In order to make clear distinctions between the two shocks I provide two examples.
As shown in CMR (2010) an increase in risk, caused by the risk shocks, decreases output
and vice-versa. Contrary to the risk shocks, the ﬁrst example shows that an increase inrisk, caused by the uncertainty shocks, increases output. The second example shows thata decrease in risk, caused by the uncertainty shocks, decreases output. While those results
sound counter-intuitive, the results clarify distinctions between the two shocks.
Instead of the distribution of riskiness, given by (34), suppose that the distribution
stays uniform but the uncertainty shock appears in the upper bound of the support:Ft (p) =p−p,p̄t − pp̄t ≡ p̄e−υt ,0 < p < p̄ < 1.

(56)

As before the negative uncertainty shock, υt < 0, increases the degree of uncertainty ofriskiness. The diﬀerence from the baseline distribution, (34), appears in the riskiness. The
negative uncertainty shocks decrease the overall riskiness of project, because an increase inp̄t with p ﬁxed implies that there are more projects with less riskiness. (Remember thatthe riskiness of project is measured by 1/p).
Using new distribution (58), I derive a log-linearized equation summarizing the eﬀectof uncertainty shocks, similar to equation (45), as follows:()
ŝt = −χ1 N̂t − q̂t − K̂t+1 − χ3 υt , χ1 , χ3 > 0,(57)
where coeﬃcient χ1 is the same as in equation (45). For the derivation of equation (57) see
Appendix.
40

As the ﬁrst example, consider the negative uncertainty shocks, υt < 0. The negative
uncertainty shocks in this case increase p̄t and decrease the over all riskiness of project.
However, equation (57) implies that the negative uncertainty shocks increase an excess
return to capital and decrease output as we saw in the previous section. Next as thesecond example, consider the positive uncertainty shocks, υt > 0. The positive uncertainty
shocks decrease p̄t and increase the over all riskiness of project. However, equation (57)
implies that the positive uncertainty shocks decrease an excess return to capital and increase
output. Unlike the risk shocks in CMR (2010), a decrease (increase) in riskiness does notnecessarily result in an increase (decrease) in output.
The above two examples make a stark contrast between the uncertainty shocks and therisk shocks. Still, the negative uncertainty shocks have to be accompanied with an increasein riskiness in order to generate counter-cyclical external ﬁnance premiums as observedin the sample data.19 In that sense, the uncertainty shocks and the risk shocks sharea similar empirical implication: both the two shocks are in some extent captured by anincrease in riskiness. Bloom (2009) and Bloom, et al (2010) document that uncertainty,
measured by various second moments, is counter-cyclical and propose a model in whicha change in uncertainty drives business cycles. In my context, uncertainty referred bythem can be interpreted as riskiness. The models proposed in this paper provide anothermechanisms through which a change in riskiness (accompanied by a change in uncertainty
about riskiness) drives business cycles.
5.2. Unrealized Uncertainty Shocks
The previous subsection makes clear that the heart of the uncertainty shocks lies in ex-ante
uncertainty about the riskiness of project. This observation suggests that shocks to exante uncertainty, not necessarily realized ex-post, can have similar eﬀect to the uncertainty
shocks. I call the shocks as unrealized uncertainty shocks, which change only ex-ante
uncertainty about the riskiness of project but do not change the true distribution of theriskiness of project ex-post. I interpret the unrealized uncertainty shocks as shocks aﬀecting
intermediaries’ animal spirits or intermediaries’ mood (pessimism and optimism), becausea change in ex-ante uncertainty is not realized ex-post. It turns out that the unrealized
uncertainty shocks are more powerful than the uncertainty shocks.
I extend Model-I by introducing the unrealized uncertainty shocks. As in Model-I, I
assume that the distribution, perceived by intermediaries, of riskiness of project is uniform,
19

In practice, an increase in riskiness, caused by the uncertainty shocks, has to accompany with a decreasein output as in Model-I, because an increase in riskiness is reﬂected to an increase in an external ﬁnancepremium. As shown in the data in Table 2, an external ﬁnance premium is counter-cyclical. Pro-cyclicalexternal ﬁnance premium in the above two examples is at odd with the data.

41

given by
Ft (p) =p − pt1 − pt,pt = peυt +ηt ,0 < p < 1,(58)

where ηt denotes the unrealized uncertainty shocks while υt denotes the uncertainty shocks.
Intermediaries perceive that the distribution is given by (58) in making a contract withentrepreneurs, but the true distribution turns out to be (58) with ηt = 0. Then, theintermediaries provide a menu of loan contracts based on distribution (58). Consequently,
a threshold above which entrepreneurs do not get funded is given by zero proﬁt condition(39) with pt given by pt = peυt +ηt . The aggregate loan is given by individual loan (17)integrated over net worth and p according to distribution (58) with ηt = 0:[]1−ϕ) p∗ − peυt
2ϕ−11
(p∗t ) ϕ ( ∗ 2ϕ−1tυBt =(pt ) ϕ − (pe t ) ϕ
−
Nt ,
(59)
2ϕ − 1 1 − peυt
1 − peυtwhere I have assumed ϕ ̸= 1/2. The aggregate loan, given by equation (59), coincides withthe aggregate loan, implied by equation (37) in Model-I, if and only if there is no unrealized uncertainty shocks: ηt = 0. With the unrealized uncertainty shocks, the zero proﬁt
condition does not hold ex-post. I assume that the intermediary’s proﬁts are transfered tohouseholds in a lump-sum manner.
As in Model-I, I log-linearize the zero proﬁt condition, equation (59) and balance sheet
equation (35), and rearrange those equations to obtain the following equation:()
ŝt = −χ1 N̂t − q̂t − K̂t+1 − χ2 υt − χ4 ηt , 0 < χ2 < χ4 ,(60)
where coeﬃcients χ1 and χ2 are the same as in equation (45) and the expression for coeﬃcient χ4 is given in Appendix. Equation (60) shows that the unrealized uncertainty shocks,
ηt , and the uncertainty shocks, υt , aﬀect the economy in the same manner. The negative(unrealized) uncertainty shocks raise an excess return to capital, ŝt , and cause recessions
by aggravating the degree of adverse selection in ﬁnancial markets. More importantly,
equation (60) shows that the unrealized uncertainty shocks have greater eﬀect on ŝt than
do the uncertainty shocks.
To understand why the unrealized uncertainty shocks have greater eﬀect than do theuncertainty shocks, consider two economies where one is hit by the uncertainty shocks only,υt < 0, and the other is hit by the unrealized uncertainty shocks only, ηt = υt . Because
the magnitude of the shocks is the same, intermediaries in the two economies provide thesame menu of loan contracts, characterized by the same threshold, p∗t . Given threshold p∗t ,the aggregate loan in the ﬁrst economy with υt < 0 is greater than in the second economy
with υt = 0, according to equation (59). The type-p entrepreneurs receive loans for p = pt
to p∗t in the ﬁrst economy, while they do so for p = p > pt to p∗t in the second economy.
Therefore, fewer entrepreneurs receive loans and a drop in the aggregate loan is greater inthe second economy than in the ﬁrst economy.
42

Output (%)

External finance premium (APR diff)

01
Unrealized uncertainty shock
Uncertainty shock

−0.1

0.8

−0.2
0.6
−0.3
0.4
−0.4
0.2

−0.5
−0.6

0

5

10

15

20

0

25

0

5

10

15

20

25

Figure 4: Impulse Responses to Unrealized Uncertainty Shocks (Model I)
To explore a quantitative diﬀerence between the unrealized uncertainty shocks and theuncertainty shocks, I plot impulse responses of the two shocks in Figure 4. In plotting
the responses I assume that the unrealized uncertainty shocks follow the same stochastic
process as the uncertainty shocks and I use the same magnitude of the initial shocks for thetwo shocks. The left panel in Figure 4 shows that the response of output to the unrealized
uncertainty shocks is about 50 percent greater than the response to the uncertainty shocks.
The right panel in Figure 4 shows that the response of an external ﬁnance premium tothe unrealized uncertainty shocks is less than the response to the uncertainty shocks. The
result suggests that given a rise in an external ﬁnance premium, the unrealized uncertainty
shocks generate much larger drop in output. For example, given four percent rise in thepremium, the unrealized uncertainty shocks generate 3.2 percent drop in output at bottom
while the uncertainty shocks generate only 1.6 percent drop in output. The magnitude of
3.2 percent is slightly smaller than but comparable with a drop in output observed in thegreat recession in the U.S.

6. Conclusion
In this paper I build a dynamic model in which imperfect ﬁnancial markets materialize
uncertainty shocks. I model imperfect ﬁnancial markets by introducing asymmetric information on the riskiness of project and an agency problem. Asymmetric information
causes adverse selection in ﬁnancial markets, while an agency problem limits the amountof borrowing. I solve a static optimal contracting problem between intermediaries andentrepreneurs and embed it into dynamic general equilibrium models.
In a dynamic general equilibrium framework I consider the eﬀect of uncertainty shocks
which change the degree of uncertainty about the riskiness of project. On the one hand,the uncertainty shocks emerge as ﬁnancial shocks if I embed credit frictions in the demandside of capital (Model-I). On the other hand, the uncertainty shocks emerge as shocks to43

the marginal eﬃciency of investment (MEI shocks) if I embed credit frictions in the supply
side of investment (Model-II). The result suggests that the uncertainty shocks and themechanisms studied here serve as micro-foundations for ﬁnancial shocks and investment
shocks, which have received a lot of attention as a source of business cycles and the causesof a ﬁnancial crisis (Hall ,2010, Gilchrist, et al, 2009, and Gilchrist and Zakrajsek, 2010 for
ﬁnancial shocks, and Justiniano, et al, 2009a, 2009b for investment shocks).
In a quantitative analysis, I show that the uncertainty shocks generate signiﬁcant ﬂuctuations consistent with business cycles for standard variables both in Model-I and Model-II.
Ampliﬁcation mechanisms embedded in a counter-cyclical markup in wages and variable
capital utilization rates play a crucial role in generating business cycles. A diﬀerence between the two models appears in the response of the price of capital and net worth. For
those two variables, while the uncertainty shocks generate right co-movement in ModelI, the shocks generate wrong co-movement in Model-II. The result suggests either thatthe uncertainty shocks in the supply side of investment is not signiﬁcant or that Model-II
needs other mechanisms which solve the co-movement problem of the price of capital andnet worth.
In this paper I focused on mechanisms through which shocks to uncertainty in ﬁnancial
markets generate business cycles, and did not discuss policy issues. In order to derive policy
implications it is essential to model a counter-cyclical markup in wages in a micro-founded
manner. One candidate is to introduce nominal wage rigidities. Introducing nominal
rigidities opens a door to discussing both conventional and unconventional monetary policy.
My another paper, Ikeda (2011), introduces nominal rigidities and discusses the policy
issues.

44

Appendix
Proof of Proposition 1: The proposition states that there exists a unique symmetric equilibrium
if a solution to condition (18) uniquely exists. The proposition consist of an equilibrium part anda uniqueness part. First I show the equilibrium part. The candidate for an equilibrium satisﬁes(IC1), (IC2) and (PCe) so that entrepreneurs choose their best contract oﬀered by intermediaries.
The intermediary’s zero proﬁt condition holds so that intermediaries have no incentive to exit themarket. What remains to be shown is that there is no proﬁtable deviation from the candidate foran equilibrium in the ﬁrst stage.
In Proposition 1, it is assumed that there exits a unique solution to condition (18). Thisimplies that an intermediary’s proﬁts, Vn (p), are positive for p < p∗ , and negative for p > p∗ . To
see this, note that distribution F (p) has full support on [p, p̄]. From the expression for ω(p), (19),
I obtain ω(p) > 0. From the expression for Vn (p), (18), an intermediary’s proﬁts are positive forp close to p. The assumption of unique p∗ implies that Vn (p) > 0 for p < p∗ and Vn (p) < 0 forp > p∗ .
Suppose that there exists the i-th intermediary’s proﬁtable deviation, {Xni (p), Bni (p)}, withthreshold pi , which satisfy (IC1), (IC2) and (PCe). There are three cases: (i) pi < p∗ , (ii) pi = p∗and (iii) pi > p∗ . In case (i), Lemma 1 and pi < p∗ implies that Xni (p) < Xn (p) for all p ≤ pi . An
entrepreneur with p ≤ pi prefers schedule {Xn (p), Bn (p)} to {Xni (p), Bni (p)}, because the type-p
entrepreneurs proﬁts are increasing in Xn (p′ ) for p′ ≥ p, as in (13). Therefore, no entrepreneur
would choose {Xni (p), Bni (p)}. The deviated schedule results in zero proﬁts and cannot be an
proﬁtable deviation.
In case (ii), an only diﬀerence between {Xn (p), Bn (p)} and {Xni (p), Bni (p)} lies in the inequalityin (IC2). For {Xn (p), Bn (p)}, the inequality holds with equality for p ≤ p∗ . For {Xni (p), Bni (p)},there exist some p’s such that the inequality does not hold with equality. Without loss of generality
I assume that {Xni (p), Bni (p)} satisﬁes a property that the inequality holds with equality for p
close to p. Because ω(p) > 0 for p close to p and an intermediary’s proﬁts are given by (18), a
deviated schedule which does not satisfy the property earns lower proﬁts than another schedule
whose only diﬀerence from the former is that it satisﬁes the property. In addition, without lossof generality, I assume that for a deviated schedule there exists p′ ∈ (p, pi ) such that (IC2) holdswith equality in an interval between p′ and threshold pi . Otherwise, the argument in (i) applies
and no entrepreneur would choose the deviated schedule.
From the above argument, the deviated schedule satisﬁes that Xni (p) = Xn (p) for p in aninterval between p′ and pi = p∗ , and Xni (p) < Xn (p) for p < p′ (or p ≤ p′ ). If all intermediaries
stayed in the market, the i-th intermediary would succeed in attracting only safe entrepreneurs
with p in an interval between p′ and pi , because entrepreneurs with p in the interval chooseintermediaries randomly. An intermediary’s proﬁts made from the type-p entrepreneur is given
by
(
)
∫ ∗
Rf
Rf p
f
pXn (p) − R Bn (p) = pXn (p) 1 − e − eXn (p)dp,
RR p
where I have substituted out Bn (p) using (14). Since Rf < Re , the proﬁts made from the typep entrepreneur is positive for p close to p∗ . This implies that the i-th intermediary would be45

able to earn positive proﬁts by choosing p′ close to p∗ if the other intermediaries stayed in themarket. However, after observing {Xni (p), Bni (p)}, the other intermediaries best response is toleave the market, otherwise they would end up with negative proﬁts. Therefore, given that thei-th intermediary stays in the market, it faces distribution F (p): it has to make an contract not
only with safe entrepreneurs but also with risky entrepreneurs.
I show that the deviated schedule results in negative proﬁts in case (ii). The above argument
about Xni (p) in case (ii) implies that for any small value δ > 0 there exists an interval withmeasure δ such that the inequality in (IC2) holds with equality for p lower than the interval. I
construct a new schedule X̃n (p) whose only diﬀerence from Xni (p) is that there exists p̃ in theinterval such that X̃n (p̃) − Xni (p̃) = ϵ for small value ϵ > 0, and that the inequality in (IC2) holdswith equality for p < p̃. Also, I construct another schedule X̃ni (p) whose only diﬀerence from
Xni (p) is that the inequality in (IC2) holds with equality for p < p̃ in the interval. Then, from(15) I have: for p ≤ p̃{∫ p̃ [}]
(1 − ϕ)iiX̃n (p) − X̃n (p) dp + ∆(δ) ,(61)
X̃n (p) − X̃n (p) =
ϕp
p
∫ p∗
where ∆(δ) ≡ p̃ [X̃n (p)−X̃ni (p)]dp. Since X̃n (p) = X̃ni (p) for p higher than the interval, ∆(δ) → 0
as δ → 0. Solving (61) for X̃n (p)−X̃ni (p) with the terminal condition, X̃n (p̃)−X̃ni (p̃) = ϵ, I obtain20X̃n (p) = X̃ni (p) +

( )1
p̃ ϕ
ϵ,
p

for p ≤ p̃. Then from the expression for the intermediary’s proﬁts, (18), I can express the diﬀerenceof intermediary’s proﬁts as∫ p∗

ω(p)[X̃n (p) − X̃ni (p)]dp =

p

( )1
p̃ ϕ
˜
ω(p)
ϵdp + ∆(δ),
p
p

∫ p̃

∫ p∗
˜
˜
where ∆(δ)≡ p̃ ω(p)[X̃n (p) − X̃ni (p)]dp. Similar to ∆(δ), ∆(p)
→ 0 as δ → 0. Because δ isiarbitrary, X̃n (p) earns more proﬁts than X̃n (p) if∫ p̃

1ω(p)(1/p) ϕ dp > 0.

(62)

p

This has to be the case since p̃ < p∗ and the intermediary’s proﬁts, Vn (p) given by (18), arepositive for p < p∗ .
Note that Xni (p)− X̃ni (p) → 0 as δ → 0 for all p by construction. This and the above argument
imply that new schedule X̃n (p) earns more proﬁts than the deviated schedule, Xni (p). Comparedwith Xni (p) new schedule X̃n (p) has more p’s such that constraint (IC2) holds with equality for
p. Repeating this argument I reach a conclusion that the candidate for an equilibrium, Xn (P ),which satisﬁes (IC2) with equality for all p ≤ p∗ , earns more proﬁts than the deviated schedule,
Given ϵ > 0 and δ > 0 such that ϵ > [(1 − ϕ)/(ϕp̃)]∆(δ), there exists ϵ̃(δ) such that ϵ = ϵ̃(δ) + [(1 −ϕ)/(ϕp̃)]∆(δ). As δ → 0, ∆(δ) → 0 and ϵ̃(δ) → ϵ so that X̃n (p̃) − X̃ni (p̃) = ϵ.
2046

Xni (p). Since Xn (p) earns zero proﬁts, Xni (p) results in negative proﬁts. Therefore, the deviatedschedule cannot be a proﬁtable deviation.
In case (iii), suppose that there exists a proﬁtable deviation, {Xni (p), Bni (p)}, with pi > p∗ .
Without loss of generality I assume that constraint (IC2) holds with equality for p ≤ p∗ in Xni (p),because it is proﬁtable to do so from the argument in case (ii). Then, the diﬀerence of proﬁts
between the two schedules is given by
∫ p̄
∫ p̄
i
ω(p)[−Xni (p)] > 0.
(63)
ω(p)[Xn (p) − Xn (p)]dp =p∗p

In the equality I have used the condition implied by the zero proﬁt condition:

∫ p∗
p

1

ω(p) (1/p) ϕ dp =0. In the strict inequality I have used ω(p) < 0 for p > p∗ .21 The inequality in (63) implies thatthe deviated schedule, Xni (p), cannot be a proﬁtable deviation. This completes the proof that thecandidate for an equilibrium is actually an equilibrium.
Now I shall show the uniqueness of the equilibrium. As before I assume that the solution
to the zero proﬁt condition, (18), uniquely exists. For any schedule {B̃n (p), X̃n (p)} satisfying
conditions (IC1), (IC2), (PCe) and (8) yet diﬀering from equilibrium schedule {Bn (p), Xn (p)} ,I can construct the i-th intermediary’s proﬁtable deviation. Let {Bni (p), Xni (p)} be such that it
satisﬁes (IC1), (IC2) with equality, (PCe) and such that threshold pi satisﬁes p∗ − pi = ϵ forsmall value ϵ > 0. Under this deviated schedule the i-th intermediary’s proﬁts become positive
if the i-th intermediary faces distribution F (·) up to a constant scaling factor. For small enoughϵ the deviated schedule becomes close to the equilibrium schedule and becomes more attractive
to entrepreneurs than {B̃n (p), X̃n (p)}. Then the deviated schedule succeeds in earning positive
proﬁts. This completes the proof of the uniqueness of the equilibrium.
Derivation of Equation (45): I derive equation (45) using equations (37) and (39) in two steps.
First, log-linearizing equation (39) I obtain:p̂∗t = θp∗ ,s ŝt + υt ,withθp∗ ,s ≡

(64)

(
2ϕ−1
2ϕ−1 )
ϕ− 2ϕ−1(p∗ ) ϕ − p phi(1 − 2/s)(p∗ )2ϕ−1ϕ+ (p/s)(p∗ )

− 1−ϕ
ϕ

> 0,

Second, log-linearizing equation (37) I obtain()[1 − (1 − ϕ)s]qKp∗
1 − p∗ p∗ŝt =(q̂t + K̂t+1 − N̂t ) + [1 − (1 − ϕ)s] − ∗
p̂ +
υt .
B
p −p t1 − p p∗ − p
Substituting out for p̂∗t using (64) I obtain:()
ŝt = −χ1 N̂t − q̂t − K̂t+1 − χ2 υt ,21

Otherwise the zero proﬁt condition would not hold at p∗ .

47

where{[1 − (1 − ϕ)s]p∗θp∗ ,sp∗ − p}−1[1 − (1 − ϕ)s]qK
> 0,
B
{
}−1 [
(
)]
[1 − (1 − ϕ)s]p∗
1 − (1 − ϕ)s
1 − p∗∗
χ2 = 1 +θp∗ ,sp −p> 0.
p∗ − pp∗ − p
1−p

χ1 ≡

1+

(65)
(66)

This completes the derivation of equation (45). 
Derivation of Equation (52): I derive equation (52) using equations (49) and (39) with st
replaced by qt µt . Note that the two equations are almost the same as those used in deriving
equation (45). Then, χ3 and χ4 in equation (52) are given by χ1 , given by (65), and χ2 , given by(66), respectively, where s and qK are replaced by q and I respectively. 
Derivation of Equation (54): Taking into account the functional form of utility function I
combine equation (27) and equation (30) and obtain:(1 − α)(ut Kt )α L−α= wt = λw,t ψLt Ct .
t
1/ν

Substituting out for λw,t using equation (25) I obtain:(1 − α)(ut Kt )α L−α= λw (Yt /Y )−ω ψLt Ct .
t
1/ν

Substituting out for Yt using equation (28) I obtain:(1 − α)Y ω /(λw ψ) = [(ut Kt )α L1−α]−ω−1 Lt
t

1+1/ν

Ct .

Log-linearizing this equation results in:−(ω + 1)α(ût + K̂t ) + [−(ω + 1)(1 − α) + 1 + 1/ν]L̂t + Ĉt = 0.
Log-linearizing equation (31) gives a relationship between the return to capital and capital utilization rates: r̂tk = χût . Substituting out for r̂tk using the log-linearized equation of equation(29), I express capital utilization rates as:ût =From the above two equations I obtain:[(Ĉt = (ω + 1)(1 − α) 1 +1−α(−K̂t + L̂t ).
χ+1−α

α
χ+1−α

)

]
− (1 + 1/ν) L̂t +(ω + 1)αχ
K̂t .
χ+1−α

This completes the derivation of equation (54). Equations Relating to Risk Shocks: Let ωt denote an idiosyncratic productivity shock,following ωt ∼ Ft , where Ft denotes a c.d.f. with mean unity and standard deviation σt =σeur,t . Let ω̄t denote a threshold under which entrepreneurs go bankrupt. Deﬁne Γt (ω̄t ) and

48

∫ ω̄
Gt (ω̄t ) as Γt (ω̄t ) ≡ Gt (ω̄t ) + ω̄t [1 − Ft (ω̄t )] and Gt (ω̄t ) ≡ 0 t ω̄t dFt (ω̄t ) respectively. Except the
introduction of the risk shock, the setup and the notation is exactly the same as BGG (1999)
and Christiano and Ikeda (2010, Section 6) who provide a simple analysis on BGG (1999). The
equations characterizing the demand for capital consists of three equations:qt Kt+1 = Nt + Bt ,[]Γ′t+1 (ω̄t+1 ) {[Γt+1 (ω̄t+1 ) − µGt+1 (ω̄t+1 )]st+1 − 1}′
0 = Et [1 − Γt+1 (ω̄t+1 )]st+1 +,Γ′t+1 (ω̄t+1 ) − µG′t+1 (ω̄t+1 )0 = [Γt+1 (ω̄t+1 ) − µGt+1 (ω̄t+1 )]st+1 (Nt + Bt ) − Bt ,where qt denotes the price of capital, Kt+1 denotes the capital stock, Nt denotes the net worth,k /R
Bt denotes the loan, and st ≡ Et Rt+1
t+1 as deﬁned in Model-I, and parameter 0 < µ < 1determines monitoring costs. Readers who are interested in the model’s set up and in deriving
those three equations may refer BGG (1999, Appendix A) or Christiano and Ikeda (2010, Section
6). I log-linearizing those three equations and substitute out for B̂t from the ﬁrst equation using
ˆ t+1 using the second equation. As a result, I obtainthe third equation, and substitute out for ω̄equation (55). In practice, log-linearization is done numerically because of the complexity of theequations.
Derivation of Equation (57): As I derived equation (45) I log-linearized two equations (37)and (39) with distribution (34) replaced by (58). With new distribution (58) those two equations
become:[](1 − ϕ)st p∗t − p
qt Kt+1 = 1 +Nt ,1 − (1 − ϕ)st p̄t − pt()]
2ϕ−1 ]
p ϕ [ ∗ − 1−ϕ
2 [ ∗ 2ϕ−1
ϕ
− 1−ϕ
1−
(pt ) ϕ − p ϕ −(pt ) ϕ − p ϕ .
0=2ϕ − 1st
st 1 − ϕLog-linearizing those two equations I obtain
ŝt =

(
)
[1 − (1 − ϕ)s]qKp∗p̄(q̂t + K̂t+1 − N̂t ) + [1 − (1 − ϕ)s] − ∗p̂∗t −υt ,Bp −pp̄ − pp̂∗t = θp∗ ,s ŝt ,where coeﬃcient θp∗ ,s > 0 is the same as in equation (64). Substituting out for p̂∗t I obtain:()
ŝt = −χ1 N̂t − q̂t − K̂t+1 − χ3 υt ,where coeﬃcient χ1 > 0 is given by (65) and coeﬃcient χ3 is given by
{
χ3 =

1+

[1 − (1 − ϕ)s]p∗θp∗ ,sp∗ − pThis completes the derivation of equation (57). 

49

}−1

p̄
> 0.
p̄ − pDerivation of Equation (60): Log-linearizing equations (35), (59) and (39) with pt = peυt +ηt
I obtainq̂t + K̂t+1 − N̂t = θq,p∗ p̂∗t − θq,υ υt ,p̂∗t = θp∗ ,s ŝt + υt + ηt ,
where
(
)
1−ϕ B
N 2p∗ − p+
θq,p∗ ≡
> 0,
ϕ
qK
qK 1 − p)[( ) 1−ϕp
N 1 p∗ ϕθq,υ ≡− 1 > 0,1 − p qK ϕ pand θp∗ ,s is given by (64). Solving those two equations for st , I obtain
)
) (
(
θq,υ
1
1
1
N̂t − q̂t − K̂t+1 − 1 −
υt −
ηt .
ŝt = −
θq,p∗ θp∗ ,sθq,p∗ θp∗ ,sθp∗ ,sBecause this equation should coincide with equation (45) when ηt = 0, it must be the case that1/(θq,p∗ θp∗ ,s ) = χ1 and (1 − θq,υ /θq,p∗ ) (1/θp∗ ,s ) = χ2 . Let χ4 ≡ 1/θp∗ ,s . Then the above equation
implies that χ4 > χ2 . 50

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