dsge models with financial frictions · idiosyncratic shock to an individual ﬁrm’s project...

Agency Costs, Net Worth and Business Cycle Fluctuations New Keynesian Model with Capital Financial Accelerator Model

DSGE Models with Financial Frictions

Simon Gilchrist1

1Boston University and NBER

September 2014


Overview

• OLG Model• New Keynesian Model with Capital• New Keynesian Model with Financial Accelerator


Introduction

• Bernanke and Gertler describe a general equilibrium model inwhich financial frictions cause fluctuations in output.

• The model is highly stylized and relies on an overlappinggenerations structure to obtain closed-form general equilibriumdynamics.

• Importantly, the contracting structure is developed from firstprinciples rather than ad hoc assumptions regarding therelationship between borrower balance sheets and economicactivity.

• It conveys the essence of the argument that financial marketdistortions create a powerful source of propagation via afinancial accelerator mechanism.


OLG Model

• Overlapping generations framework with two types of agents –households and entrepreneurs.

• Entrepreneurs borrow from financial intermediaries and facefrictions in capital markets owing to a costly-state-verificationproblem as described above.

• Households lend to financial intermediaries and hence serve asthe ultimate source of funds. In this sense, the financialintermediaries are a veil and can be ignored in the analysis.


• Households work when young, consume, and save forconsumption when old.

• Households can either lend to entrepreneurs or can save via astorage technology that yields gross return R.

• Assume that some fraction of households invest in the storagetechnology so that the household return on savings is R.Alternatively, one can view this as a simple open economymodel.


Savings of young

• A young household chooses savings Bt to solve:

max lnCyt + β lnCot+1

subject to Cyt = Wt −Bt, Cot+1 = RBt.• Household optimality implies that household savings is linear in

the current wage:Bt = bWt

where b denotes the savings rate.


Entrepreneurs

• Entrepreneurs work when young, earn a wage Wt and save topurchase kt+1 units of capital.

• Capital goods are produced one-for-one from consumptiongoods so that the price of capital in terms of foregoneconsumption is unity (Q = 1).

• Entrepreneurs also pay a fixed cost Ce which may be interpretedas a fixed amount of consumption when young.


Net Worth

• Entrepreneurs are risk neutral and, net of Ce, only consumewhen old. Entrepreneurial net worth is therefore

nt = Wt − Ce.

• This net worth is used to finance capital expenditures kt+1.


Production

• Given capital kt+1, an individual entrepreneur has access to aconstant returns to scale Cobb-Douglas production technology:

yt+1 = l1−αt+1 (θt+1ωkt+1)α

where θt+1 denotes aggregate productivity and ω denotes anidiosyncratic shock to an individual firm’s project return.

• Given kt+1 units of capital purchased at time t, entrepreneurshire labor in t+ 1 and earn the expost profits πt+1 which, giventhe form of production, is a linear function of capital

πt+1 = ωRkt+1kt+1

• Because ω can only be observed with cost µωRkt+1kt+1,entrepreneurs face a CSV contracting problem as outlined above.


Aggregate Economy• Let aggregate output be defined as

Yt =

∫ytdΦ(ω)

and define aggregate capital, Kt+1, in an analogous manner.(This ignores bankruptcy costs as resource drain).

• Total labor is in fixed supply and normalized to unity so that

Wt = (1− α)Yt

• Assume that shocks to the aggregate technology θt are iid and,without loss of generality, normalize the mean of θt such that theexpected aggregate return satisfies

EtRkt+1 = αKα−1

t+1

Thus, although individual entrepreneurs face constant returns toscale, the aggregate return on capital is a decreasing function ofthe aggregate stock of capital.


Benchmark Economy without Financial Frictions

• In the absence of frictions in financial markets, the expectedreturn on capital is equated to the risk free rate.

• Consequently, the aggregate capital stock is determined by theuser cost of capital:

K∗ =(αR

)1/(1−α)and is therefore constant.

• Thus iid shocks to technology have no persistent effects on theeconomy.


Model with Financial Frictions

• Entrepreneurial net worth is equal to wages earned when youngnet of consumption:

Nt = (1− α)Yt − Ce

• Aggregate capital next period is the sum of entrepreneurial networth and household savings:

Kt+1 = Nt +Bt

where Bt = b(1− α)Yt.• Entrepreneurial leverage is therefore

Kt+1

Nt= 1 +

b(1− α)Yt(1− α)Yt − Ce

and hence a decreasing function of current output.


Leverage

• With constant returns to scale, entrepreneurs leverage up to thepoint where the expected excess return on capital is equal to thepremium on external funds. As a result, capital expenditures aredetermined by available net worth:

EtRkt+1

R= s

(Kt+1

Nt

)where s() is derived from the contracting problem defined above.

• If K∗ < Nt, entrepreneurs need not borrow to finance desiredcapital. In this case s = 1, there is no premium on externalfunds, and the capital stock is equal to the first best.


Capital supply vs capital demand

• Assuming Nt < K∗ we have

αKα−1t+1 = Rs

(1 +

b(1− α)Yt(1− α)Yt − Ce

)where s() > 1.

• In this case, the premium on external funds is positive andaggregate capital is below first best.


Comments:

• The left-hand side of this equation can be interpreted as a capitaldemand equation. It is a downward sloping function of currentcapital.

• The right hand side can be interpreted as a capital supplyequation, i.e. the price at which the market can finance a givenlevel of capital depends on leverage – at higher leverage, defaultrisk is higher and hence the premium on external funds is higher.

• In this example household and entrepreneurial savings can besummarized by current output. As a result, the capital supplycurve does not depend on the current capital choice Kt+1.


Dynamics:

• With Ce > 0, an increase in the current level of technology θtleads to an increase in entrepreneurial net worth Nt that isproportionately larger than the increase in household savings.

• As output rises, leverage falls.• Given s′() > 0 , the premium on external finance falls.• This is represented as an outward shift in the capital supply

curve.• Next period’s capital will therefore increase. Furthermore, high

capital tomorrow implies higher output tomorrow. As a result,following a positive shock to technology in period t, leverageand the premium on external funds will be persistently belowsteady-state, and capital will be persistently above steady-state.


Intuition

• In Bernanke-Gertler, iid shocks to technology are propagatedthrough time via the financial accelerator mechanism.

• The essential ingredient necessary to obtain persistentprocyclical movements in output is that entrepreneurial net worthincreases more than household savings in response to an increasein current output.

• In the simple framework outlined above, this occurs because ofthe fixed entrepreneurial consumption requirement Ce.

• More generally, any mechanism that makes net worth moreprocyclical than savings result in an amplification andpropagation mechanism.

• Asset price movements are the most likely source ofprocyclicality in net worth.


Additional implications

• Any mechanism that transfers wealth from savers to borrowerswill have expansionary effects on the economy.

• Bernanke and Gertler use this insight to argue that debt-deflationwhich transfers net worth from entrepreneurs to households canhave persistent contractionary effects on economic activity.


Households:

• Households choose{Ct+i, Nt+i,

Bt+iPt+i

, Mt+i

Pt+i,Kt+i

}∞i=0

tomaximize

Et

∞∑i=0

βi

(1

1− γC1−γt+i +

am1− γm

(Mt+i

Pt+i

)1−γm− an

1

1 + γnN1+γnt+s

)

subject to

Ct =Wt

PtNt + Πt + TRt + ZtKt +Qt (Kt+1 − (1− δ)Kt)

+BtPt

+Mt−1Pt− Mt

Pt−(

1

1 + it

)Bt+1

Pt

where Wt/Pt denotes the real wage and Πt denotes profitsreceived from firms owned by households, Qt denotes the priceof capital and Zt denotes the rental rate on capital.


Household Optimality Conditions:

• The inter-temporal first-order conditions are:

C−γt = β(1 + it)Et

(PtPt+1

C−γt+1

)C−γt = βEt

(PtPt+1

C−γt+1

)+ am

(Mt

Pt

)−γmC−γt = βEt

([Zt+1 + (1− δ)Qt+1]

QtC−γt+1

)• The labor-leisure FOC is:

Wt

PtC−γt = anN

γnt


Household Optimality Conditions:

• Labor-leisure:Wt

PtC−γt = anN

γnt

• Consumption euler equation:

C−γt = Et

{βRt+1C

−γt+1

}where

Rt+1 =Zt+1 + (1− δ)Qt+1

Qt

• Money demand:

Mt

Pt= (am)

1γm

(1− 1

1 + it

)− 1γm

Cγγmt

with Rt+1 = (1 + it)PtPt+1


Final goods producers:• Firms in the final goods sector producer a homogenous good, Yt ,

using intermediate goods, Yt (z) according to the CESproduction function

Yt =

(∫ 1

0Yt(z)

ε−1ε dz

) εε−1

where Yt(z) denotes intermediate good z and ε > 1 is the priceelasticity of demand.

• The representative firm chooses inputs Yt(z) to solve

max

(∫ 1

0Yt(z)

ε−1ε dz

) εε−1

− EtPt

subject to

Et =

∫ 1

0Pt(z)Yt(z)dz

where Pt(z) denotes the price of intermediate good Yt (z).


Lagrangean Problem:• Substituting constraints and taking derivatives, the first order

condition is

ε

ε− 1

(∫ 1

0Yt(z)

ε−1ε dz

) εε−1−1

ε− 1

εYt (z)−

1ε =

Pt (z)

Pt

• Rearranging we get the demand for the intermediate good

Yt (z) =

(Pt (z)

Pt

)−εYt

• Using this demand curve in conjunction with the definition ofEt, it is straightforward to show that PtYt = Et for

Pt =

(∫ 1

0Pt(z)

1−εdz

)1/(1−ε)

where Pt represents the minimum cost of achieving one unit ofthe final goods bundle. We interpret Pt as the aggregate priceindex.


Intermediate goods producers:

• There is a continuum of monopolistically competitive firmsowned by consumers, indexed by z ∈ [0, 1] .

• Each intermediate good firm operates a CRS production functionand faces the demand curve for good z derived above.

• Nominal Rigidities: Calvo price setting.I With probability 1− θ firms reset their price in any given period.I Average price duration = 1

1−θ .


Cost Minimization• Firm z chooses inputs Nt (z) and Kt (z) to minimize:

Ct = minWt

PtNt (z) + ZtKt (z)

subject toYt(z) = AtNt(z)

αKt (z)1−α

• Cost minimization implies

Wt

Pt= MCtα

Yt (z)

Nt (z)

Zt = MCt (1− α)Yt(z)

Kt (z)

• Marginal Cost:Ct = MCtYt (z)

where

MCt =1

α1−α (1− α)α

(WtPt

)1−αZαt

At


Flexible Prices

• Firm z chooses Pt(z) to maximize

Πt(z) =Pt(z)

PtYt(z)−MCtYt(z)

subject to

Yt(z) =

(Pt(z)

Pt

)−εYt

• The firm solves

maxPt(z)

[(Pt(z)

Pt

)1−ε−MCt

(Pt(z)

Pt

)−ε]Yt


Price vs Marginal Cost

• First order conditions imply that the firm sets its relative price as

Pt(z)

Pt= (1 + µ)MCt

where(1 + µ) =

ε

ε− 1

denotes a constant markup over real marginal cost.


Nominal Price Rigidities:

• Firm solves:

maxP ∗t

Et

∞∑s=0

(Λt+sθ

s

[P ∗tPt+s

Y ∗t+s −MCt+sY∗t+s

])subject to

Y ∗t+s =

(P ∗tPt

)−εYt+s

• Write this as:

maxPt(z)

Et

∞∑s=0

(Λt+sθ

s

[(P ∗tPt+s

)1−ε−MCt+s

(P ∗tPt+s

)−ε]Yt+s

)


Optimal reset price

• The first-order-conditions imply

Et

( ∞∑s=0

Λt+sθs [P ∗t − (1 + µ)Pt+sMCt+s]

(P ∗t Y

∗t+s

Pt+s

))= 0

where1 + µ =

ε

ε− 1

• Optimal reset price is a weighted average of expected futuremarginal costs:

P ∗t = (1 + µ)Et

(∑∞s=0 Λt+sθ

sPt+sMCt+s

(P ∗t Y

∗t+s

Pt+s

))Et

(∑∞s=0 Λt+s

(P ∗t Y

∗t+s

Pt+s

))


Log-linearization of P ?t

• Log-linearize this equation:

p∗t = (1− βθ)Et∞∑s=0

βsθs [pt+s +mct+s]

which may be expressed as

p∗t = (1− βθ)mct + βθEtp∗t+1

or equivalently:

p∗t − pt = (1− βθ) mct + πt + βθEt(p∗t+1 − pt+1

)


Phillips Curve• Price index:

Pt =(θP 1−ε

t−1 + (1− θ)P ∗1−εt

)1/(1−ε)so

PtPt−1

=

(θ + (1− θ)

(P ∗tPt−1

)1−ε)1/(1−ε)

• Log linearize this equation around a constant price level we have

πt = (1− θ) (p∗t − pt−1)

• Combine with optimal reset price to obtain:

πt = κmct + βEtπt+1

where

κ =(1− θ) (1− βθ)

θ


Aggregation:• Total input demand satisfies

Nt =

∫ 1

0Nt(z)dz

Kt =

∫ 1

0Kt(z)dz

• Aggregate output is

Yt = At

(∫ 1

0

(Nt(z)

αKt (z)1−α)(ε−1)/ε

dz

)ε/(ε−1)= AtN

αt K

1−αt

(∫ 1

0

(Nt(z)

Nt

)(ε−1)/εdz

)ε/(ε−1)• The term in brackets reflects the (second-order) loss in output

owing to price dispersion.


Capital Production

• Capital accumulation is subject to adjustment costs:

Kt+1 = (1− δ)Kt + φ

(ItKt

)Kt

where φ(δ) = δ and φ′(δ) = 1.• Capital producers choose It to max

Qtφ

(ItKt

)− ItKt

• FOC implies

Qt =1

φ′(ItKt

)


Monetary Policy

• Monetary policy satisfies a Taylor rule:

1 + it =

(PtPt−1

)φπ ( YtY pt

)φy


Overview

• Bernanke, Gertler and Gilchrist develop a quantitative dynamicstochastic general equilibrium framework that embeds financialfrictions in an otherwise workhorse dynamic New Keynesianeconomy.

• Their framework allows one to quantitatively assess the strengthof the financial accelerator mechanism described byBernanke-Gertler and Kiyotaki and Moore.

• It also provides a basis for a quantitative evaluation of alternativemonetary policy rules.


Households:

• Households make consumption and labor supply decisions tomaximize the present value of utility subject to standardconstraints.

• Let Rt denote the risk-free rate. Household optimality conditionsimply a consumption Euler equation

U ′(Ct) = EtRt+1βU′(Ct+1)

and combined with firm’s hiring decisions the labor equation

U ′(Ct)1

µtαYt/Nt = v′(Nt)

where v′ (N) denotes the marginal disutility of labor.


Aggregate Capital Return

• Entrepreneurs in this economy purchase capital at period t,produce in period t+ 1 and then resell their capital at marketprice Qt+1. The required return on capital is

EtRkt+1 = Et

[(1/µt+1)Yt+1/Kt+1 + (1− δ)Qt+1

Qt

]where denotes the marginal profitability of capital.

• The term µt is the markup over marginal cost owing tomonopolistic-competition features of the New Keynesianframework.

• In the absence of financial frictions, the return on householdsavings is equal to the return on capital:

Rt+1 = Rkt+1


Tobin’s Q

• With adjustment costs to capital, Tobin’s Q is an increasingfunction of the rate of investment:

Qt =1

φ′(ItKt

)where φ′′() ≤ 0


Resource constraints

• Production::

Yt = AtNαt K

1−αt = Ct + It

• Capital Accumulation:

Kt+1 = (1− δ)Kt + φ

(ItKt

)Kt


Log-linearization

• Consumption-euler equation:

ct = −σEtrt+1 + Etct+1

• Return on capital:

rkt = (1− ν)(µt + yt − kt) + νqt − qt−1

• Markup:

µt = yt − γct − (1 + γn)nt

• Tobin’s Q:it − kt = ηqt


Log-linearization

• Production and resource constraints:

yt = at + αnt + (1− α)kt

yt =c

yct +

i

yit

kt+1 = (1− δ)kt + δit


Frictionless Model (RBC)

• With no nominal rigidities we impose:

µt = 0

• With no financial frictions we impose:

rt+1 = rkt+1


Nominal Rigidities and Monetary Policy

• The New Keynesian features of the model imply a relationshipbetween inflation and markups via the Phillips curve:

πt = −κµt + βEtπt+1

• Monetary policy specifies a Taylor type monetary policy rulewhere the nominal interest rate depends on inflation and output:

rnt = φππt + φyyt

• The Fisher equation determines the relationship between real andnominal rates:

rt+1 = rnt − Etπt+1


The Financial Sector:

• Entrepreneurs are long-lived but risk neutral.• Face an exogenous failure rate and hence discount the future

more than households.• Financial frictions based on the costly-state-verification model of

BGG.


Capital Return of Entrepreneur

• Continuum of risk-neutral entrepreneurs with exogenous birthand death rate γ.

• Entrepreneur with capital Kit has access to a technology thattransforms labor and capital services into wholesale outputgoods.

• Entrepreneur uses net worth Nit to buy capital Ki,t+1 at price Qtto be used in production at t+ 1.

• She then resells the capital at price ωi,t+1Qt+1.• The realized (gross) return on capital:

Rki,t+1 =ωit+1

[1

µt+1(1− α)

Yi,t+1

Ki,t+1+Qt+1(1− δ)

]Qt


Default and leverage• With constant returns to scale in production and monitoring

technologies, the contracting problem implies

EtRkt+1

Rt= ρ(ωt)

• Given a default barrier ω, capital expenditures are determined byavailable net worth:

QKi,t+1 =

[1 + λ

(Γ(ωt)− µG(ωt)]

1− Γ(ωt)

)]Ni,t

• Summing across entrepreneurs:

QKt+1 =

[1 + λ

(Γ(ωt)− µG(ωt)]

1− Γ(ωt)

)]Nt

where Nt =∑

iNit and Kt =∑

iKit


Leverage

• Let Nt+1 denote aggregate net worth.• Because all entrepreneurs face the same aggregate return on

capital, they face the same premium on external funds, hencesumming across entrepreneurs implies that this equation holds inthe aggregate.

• BGG obtain a relationship between the aggregate return oncapital and the aggregate degree of entrepreneurial leverage:

EtRKt+1

Rt= s

(QtKt+1

Nt

)where the function s() is derived from the costly stateverification contracting structure.


Entrepreneurial savings:

• Entrepreneurs discount the future more than households and arerisk neutral implies that they defer consumption until they exit.


Aggregate net worth

• Aggregating across entrepreneurs then implies the aggregate networth accumulation equation:

Nt = (1− γ)RktQt−1Kt

− (1− γ)(

(Rt−1 (Qt−1Kt −Nt−1) + µG(ωt)RktQt−1Kt)

)+ γW e

t

where γ is the death rate of entrepreneurs and γW et represents a

small exogenous transfer to new startups.• The first term in this expression is the aggregate return on

capital. The second term is the payment to bond holders(households) inclusive of compensation for default costs.


Aggregate net worth

• In equilibrium, the expected return on capital compensates forthe risk free rate plus the loss in monitoring. Rewriting we obtain

Nt = (1−γ)Rt−1Nt−1+(1−γ)(Rkt − Et−1Rkt

)Qt−1Kt+γW

et

• Thus surprise movements in the return on capital(Rkt − Et−1Rkt

)influence current net worth Nt. This is

analogous to the Kiyotaki-Moore result.• To fully describe model dynamics one must solve the full system

using standard numerical solution methods. Nonetheless, onecan get some idea of model dynamics from this last equation.


Log-linearization of financial sector

• External finance premium and net worth equation:

st = −χ (nt − qt − kt) + εfdt

nt =K

NrKt −

(K

N− 1

)(st−1 + rt−1 − πt) + θnt−1 + εnwt

• Allow for credit-supply shocks:I εfdt : disturbances to credit intermediation processI εnwt : disturbances to asset values that serve as collateral


Intuition:

• Consider a positive shock to the economy that leads to anincrease in the current value of assets in place Qt.

• For a given level of capital expenditures, a 1% increase in assetprices raises financing requirements QtKt+1 by 1%.

• Net worth will also increase owing to the rise in asset prices.• Around the steady-state we have that approximately

d lnNt

d lnQt=K

N> 1


Intuition:

• The percent change in net worth is roughly proportional to thesteady-state degree of leverage. As a result, holding other thingsfixed, surprise movements in asset prices cause net worth to riseproportionately more than the increase in capital expenditureshence leverage falls and the external finance premium decreases.

• This causes a further increase in asset prices and furtherreductions in the premium on external funds and hence anexpansion of investment spending.


The role of nominal price rigidities

• A positive shock that increases asset prices also leads to anincrease in marginal cost which is only partially transmitted tothe economy owing to nominal price rigidities. As a result,markups fall, and output expands. The reduction in markups andexpansion of output puts further upward pressure on asset pricesand thus strengthens the financial accelerator.

• The conduct of monetary policy also plays a key role: Arelatively weak response of monetary policy to expected inflationreduces the effect of increased investment demand on realinterest rates and causes a much larger increase in asset prices.


Quantitative results

• According to the calibration used in BGG, the financialaccelerator leads to a 30% amplification of the output in responseto increases in technology relative to a benchmark model withoutfinancial frictions. It also provides substantial amplification toshocks to monetary policy and other demand-side disturbances.

• A reallocation of wealth from households to entrepreneurs has astrong amplification mechanism: A reallocation that isequivalent to 1% of entrepreneurial net worth implies a 2% risein actual net worth owing to the fact that asset prices rise as networth expands.

• Such reallocation mechanisms allow one to consider the role ofdebt deflation as a source of business cycle dynamics. It alsoraises the possibility that the financial sector can serve as asource of economic volatility.

dsge models with financial frictions · idiosyncratic shock to an individual ﬁrm’s project...

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