macroeconomic models with financial frictionsjesusfv/macrofinancial.pdfthe purchased capital is...

Macroeconomic Models

with Financial Frictions

Jesús Fernández-Villaverde

University of Pennsylvania

May 31, 2010

Jesús Fernández-Villaverde (PENN) Macro-Finance May 31, 2010 1 / 69

Motivation I

Traditional macro models embody a Modigliani-Miller environment.

For example, in the standard RBC and NKE models(Smets-Wouters/CEE):

1 It does not even matter who owns the capital (household, firm, aninvestment fund).

2 Asset market is pretty boring: complete markets for households andfirms.

3 Money is introduced in a rather ad hoc way through MIU or CIA.

Therefore, there is little room to think about finance-related policy.

Motivation II

This was always a nagging worry of macroeconomists, who suspectedthey were missing important mechanisms.

However, relatively little progress was made.

After the Great Recession of 2007-2010, there seems to be littleexcuse not to analyze the interactions between the macroeconomyand the financial markets in much more detail.

A lot of recent work.

Unfortunately, our understanding is still limited.

Illustrate how we can incorporate financial frictions in standard macromodels.

Think about limitations of current approaches.

Think about implications for policy:

1 Fiscal and monetary.

2 Systemic regulator.

3 Financial supervision.

An Analytic Framework

How can we analyze formally some of these issues?

We want a model:

1 With financial markets.

2 Where the Modigliani-Miller theorem does not hold.

3 Where we have mechanisms that resemble some of the channelsemphasized by observers of the recent market turbulences.

4 Where we can perform quantitative analysis.

StrategiesSeveral approaches:

1 Models of constraints on borrowing:

1 Costly state verification.

2 Costly enforcement model.

2 Models of financial intermediation.

3 Models of leverage cycles.

Because of time constraints, I will focus on models of constraints onborrowing. Why?

1 Historical reasons.

2 Foundation of more recent models.

A Model with Costly-State VerificationTradition of financial accelerator of Bernanke, Gertler, and Gilchrist(1999), Carlstrom and Fuerst (1997), and Christiano, Motto, andRostagno (2009).

Elements:

1 Information asymmetries between lenders and borrowers⇒costly stateverification (Townsend, 1979).

2 Debt contracting in nominal terms: Fisher effect.

3 Changing spreads.

We will calibrate the model to reproduce some of the basicobservations of the U.S. economy.

Flowchart of the Model

Households

Representative household:

∑t=0

βtedt{u (ct , lt ) + υ log

dt is an intertemporal preference shock with law of motion:

dt = ρddt−1 + σd εd ,t , εd ,t ∼ N (0, 1).

Why representative household? Heterogeneity?

Asset Structure

The household saves on three assets:

1 Money balances, mt .

2 Deposits at the financial intermediary, at , that pay an uncontingentnominal gross interest rate Rt .

3 Arrow-Debreu securities (net zero supply in equilibrium).

Therefore, the household’s budget constraint is:

ct +atpt+mt+1pt

= wt lt + Rt−1at−1pt

+mtpt+ Tt +zt + tret

where:tret = (1− γe ) nt − w e

Optimality Conditions

The first-order conditions for the household are:

edtu1 (t) = λt

λt = βEt

{λt+1

RtΠt+1

}−u2 (t) = u1 (t)wt

Asset pricing kernel:

SDFt = Etβλt+1λt

and standard non-arbitrage conditions.

The Final Good Producer

Competitive final producer with technology

yt =(∫ 1

ε−1ε

it di) ε

ε−1.

Thus, the input demand functions are:

yit =(pitpt

)−ε

yt ∀i ,

Price level:

pt =(∫ 1

0p1−εit di

) 11−ε

Intermediate Goods Producers

Continuum of intermediate goods producers with market power.

Technology:yit = eztkα

it−1l1−αit

wherezt = ρzzt−1 + σz εz ,t , εz ,t ∼ N (0, 1)

Cost minimization implies:

mct =(

11− α

)1−α (1α

)α w1−αt r α

kt−1lt

1− α

Sticky Prices

Calvo pricing: in each period, a fraction 1− θ of firms can changetheir prices while all other firms keep the previous price.

Then, the relative reset price Π∗t = p∗t /pt satisfies:

εg1t = (ε− 1)g2tg1t = λtmctyt + βθEtΠε

t+1g1t+1

g2t = λtΠ∗t yt + βθEtΠε−1t+1

(Π∗t

Π∗t+1

)g2t+1

Given Calvo pricing, the price index evolves as:

1 = θΠε−1t + (1− θ)Π∗1−ε

Capital Good Producers I

Capital is produced by a perfectly competitive capital good producer

It buys installed capital, xt , and adds new investment, it , to generatenew installed capital for the next period:

xt+1 = xt +(1− S

[itit−1

where S [1] = 0, S ′ [1] = 0, and S ′′ [·] > 0.The period profits of the firm are:

(1− S

[itit−1

)− qtxt − it = qt

(1− S

[itit−1

])it − it

where qt is the relative price of capital.

Capital Good Producers II

Discounted profits:

∑t=0

βtλtλ0

(1− S

[itit−1

])it − it

)Since this objective function does not depend on xt , we can make itequal to (1− δ) kt−1.

First-order condition of this problem is:

(1− S

[itit−1

]− S ′

[itit−1

]itit−1

)+ βEt

λt+1λt

qt+1S ′[it+1it

] (it+1it

and the law of motion for capital is:

kt = (1− δ) kt−1 +(1− S

[itit−1

Entrepreneurs I

Entrepreneurs use their (end-of-period) real wealth, nt , and a nominalbank loan bt , to purchase new installed capital kt :

qtkt = nt +btpt

The purchased capital is shifted by a productivity shock ωt+1:1 Lognormally distributed with CDF F (ω) and2 Parameters µω,t and σω,t3 Etωt+1 = 1 for all t.

Therefore:

Etωt+1 = eµω,t+1+12 σ2ω,t+1 = 1⇒ µω,t+1 = −

σ2ω,t+1

This productivity shock is a stand-in for more complicated processessuch as changes in demand or the stochastic quality of projects.

Entrepreneurs II

The standard deviation of this productivity shock evolves:

log σω,t = (1− ρσ) log σω + ρσ log σω,t−1 + ησεσ,t , εσ,t ∼ N (0, 1).

The shock t + 1 is revealed at the end of period t right beforeinvestment decisions are made. Then:

log σω,t − log σω = ρσ (log σω,t−1 − log σω) + ησεσ,t

⇒ σ̂ω,t = ρσσ̂ω,t−1 + ησεσ,t

More general point: stochastic volatility.

Entrepreneurs III

The entrepreneur rents the capital to intermediate goods producers,who pay a rental price rt+1.

Also, at the end of the period, the entrepreneur sells theundepreciated capital to the capital goods producer at price qt+1.

Therefore, the average return of the entrepreneur per nominal unitinvested in period t is:

Rkt+1 =pt+1pt

rt+1 + qt+1 (1− δ)

Debt ContractCostly state verification framework.

For every state with associated Rkt+1, entrepreneurs have to either:

1 Pay a state-contingent gross nominal interest rate R lt+1 on the loan.

2 Or default.

If the entrepreneur defaults, it gets nothing: the bank seizes itsrevenue, although a portion µ of that revenue is lost in bankruptcy.

Hence, the entrepreneur will always pay if it ωt+1 ≥ ωt+1 where:

R lt+1bt = ωt+1Rkt+1ptqtkt

If ωt+1 < ωt+1, the entrepreneur defaults, the bank monitors theentrepreneur and gets (1− µ) of the entrepreneur’s revenue.

Zero Profit ConditionThe debt contract determines R lt+1 to be the return such that bankssatisfy its zero profit condition in all states of the world:

[1− F (ωt+1, σω,t+1)]R lt+1bt︸︷︷︸Revenue if loan pays

+(1− µ)∫ ωt+1

0ωdF (ω, σω,t+1)Rkt+1ptqtkt︸︷︷︸

Revenue if loan defaults

= stRtbt︸︷︷︸Cost of funds

st = 1+ es+s̃t is a spread caused by the cost of intermediation suchthat:

s̃t = ρs s̃t−1 + σs εs ,t , εs ,t ∼ N (0, 1).

For simplicity, intermediation costs are rebated to the households in alump-sum fashion.

External finance premium.

Optimality of the Contract

This debt contract is not necessarily optimal.

However, it is a plausible representation for a number of nominal debtcontracts that we observe in the data.

Also, the nominal structure of the contract creates a Fisher effectthrough which changes in the price level have an impact on realinvestment decisions.

Importance of working out the optimal contract.

Characterizing the Contract I

Define share of entrepreneurial earnings accrued to the bank:

Γ (ωt+1, σω,t+1) = ωt+1 (1− F (ωt+1, σω,t+1)) + G (ωt+1, σω,t+1)

where:

G (ωt+1, σω,t+1) =∫ ωt+1

0ωdF (ω, σω,t+1)

Thus, we can rewrite the zero profit condition of the bank as:

Rkt+1stRt

[Γ (ωt+1, σω,t+1)− µG (ωt+1, σω,t+1)] qtkt =btpt

which gives a schedule relating Rkt+1 and ωt+1.

Characterizing the Contract II

Now, define the ratio of loan over wealth:

$t =bt/ptnt

=qtkt − ntnt

=qtktnt− 1

and we get

Rkt+1stRt

[Γ (ωt+1, σω,t+1)− µG (ωt+1, σω,t+1)] (1+ $t ) = $t

that is, all the entrepreneurs, regardless of their level of wealth, willhave the same leverage, $t .

A most convenient feature for aggregation.

Balance sheet effects.

Problem of the EntrepreneurMaximize its expected net worth given the zero-profit condition of thebank:

max$t ,ωt+1

R kt+1Rt(1− Γ (ωt+1, σω,t+1)) +

[R kt+1stRt

[Γ (ωt+1, σω,t+1)− µG (ωt+1, σω,t+1)]− $t1+$t

] After a fair amount of algebra:

EtRkt+1Rt

(1− Γ (ωt+1, σω,t+1)) = Etηtntqtkt

where the Lagrangian multiplier is:

ηt =stΓω (ωt+1, σω,t+1)

Γω (ωt+1, σω,t+1)− µGω (ωt+1, σω,t+1)

This expression shows how changes in net wealth have an effect onthe level of investment and output in the economy.

Death and Resurrection

At the end of each period, a fraction γe of entrepreneurs survive tothe next period and the rest die and their capital is fully taxed.

They are replaced by a new cohort of entrepreneurs that enter withinitial real net wealth w e (a transfer that also goes to survivingentrepreneurs).

Therefore, the average net wealth nt is:

nt = γe1

[(1− µG (ωt , σω,t ))Rkt qt−1kt−1 − st−1Rt−1

bt−1pt−1

]+ w e

The death process ensures that entrepreneurs do not accumulateenough wealth so as to make the financing problem irrelevant.

The Financial Intermediary

A representative competitive financial intermediary.

We can think of it as a bank but it may include other financial firms.

Intermediates between households and entrepreneurs.

The bank:

1 Lends to entrepreneurs a nominal amount bt at rate R lt+1,

2 But recovers only an (uncontingent) rate Rt because of default and the(stochastic) intermediation costs.

3 Thus, the bank pays interest Rt to households.

The Monetary Authority Problem

Conventional Taylor rule:

(Rt−1R

)γΠ(1−γR )(yty

)γy (1−γR )

exp (σmmt )

through open market operations that are financed through lump-sumtransfers Tt .

The variable Π represents the target level of inflation (equal toinflation in the steady-state), y is the steady state level of output,and R = Π

β the steady state nominal gross return of capital.

The term εmt is a random shock to monetary policy distributedaccording to N (0, 1).

AggregationUsing conventional arguments, we find expressions for aggregatedemand and supply:

yt = ct + it + µG (ωt , σω,t ) (rt + qt (1− δ)) kt−1

yt =1vteztkα

t−1l1−αt

where vt =∫ 10

(pitpt

)−εdi is the ineffi ciency created by price

dispersion.

By the properties of Calvo pricing, vt evolves as:

vt = θΠεtvt−1 + (1− θ)Π∗−ε

We have steady state inflation Π. Hence, v̂t 6= 0 and monetary policyhas an impact on the level and evolution of measured productivity.

Equilibrium Conditions IThe first-order conditions of the household:

edtu1 (t) = λt

λt = βEt{λt+1Rt

Πt+1}

−u2 (t) = u1 (t)wtThe first-order conditions of the intermediate firms:

εg1t = (ε− 1)g2tg1t = λtmctyt + βθEtΠε

t+1g1t+1

g2t = λtΠ∗t yt + βθEtΠε−1t+1

(Π∗t

Π∗t+1

)g2t+1

kt−1 =α

1− α

wtrtlt

mct =(

11− α

)1−α (1α

)α w1−αt r α

Equilibrium Conditions II

Price index evolves:

1 = θΠε−1t + (1− θ)Π∗1−ε

Capital good producers:

(1− S

[itit−1

]− S ′

[itit−1

]itit−1

)+βEt

λt+1λt

qt+1S ′[it+1it

] (it+1it

kt = (1− δ) kt−1 +(1− S

[itit−1

Equilibrium Conditions III

Entrepreneur problem:

Rkt+1 = Πt+1rt+1 + qt+1 (1− δ)

qtRkt+1stRt

[Γ (ωt+1, σω,t+1)− µG (ωt+1, σω,t+1)] =qtkt − ntqtkt

EtRkt+1Rt

(1− Γ (ωt+1, σω,t+1)) =(Etst

1− F (ωt+1, σω,t+1)

1− F (ωt+1, σω,t+1)− µωt+1Fω (ωt+1, σω,t+1)

)ntqtkt

R lt+1bt = ωt+1Rkt+1ptqtkt

qtkt = nt +btpt

nt = γe1

[(1− µG (ωt , σω,t ))Rkt qt−1kt−1 − st−1Rt−1

bt−1pt−1

]+ w e

Equilibrium Conditions IV

The government follows its Taylor rule:

(Rt−1R

)γΠ(1−γR )(yty

)γy (1−γR )

exp (σmmt )

Market clearing

yt = ct + it + µG (ωt , σω,t ) (rt + qt (1− δ)) kt−1

yt =1vteztkα

t−1l1−αt

vt = θΠεtvt−1 + (1− θ)Π∗−ε

Equilibrium Conditions V

Stochastic processes:

dt = ρddt−1 + σd εd ,t

zt = ρzzt−1 + σz εz ,t

st = 1+ es+s̃t

s̃t = ρs s̃t−1 + σs εs ,t

log σω,t = (1− ρσ) log σω + ρσ log σω,t−1 + ησεσ,t

CalibrationUtility function:

u (ct , lt ) = log ct − ψl1+ϑt

1+ ϑψ: households work one-third of their available time in the steadystate and ϑ = 0.5, inverse of Frisch elasticity.Technology:

α δ ε S ′′ [1]0.33 0.023 8.577 14.477

Entrepreneur:

µ σω w e s0.15 2.528 n

n−k ≈ 2 25bp.

For the Taylor rule, Π = 1.005, γR = 0.95, γΠ = 1.5, and γy = 0.1are conventional values.For the stochastic processes, all the autoregressive are 0.95.

Computation

We can find the deterministic steady state.

We linearize around this steady state.

We solve using standard procedures.

Alternatives:

1 Non-linear solutions.

2 Estimation using likelihood methods.

How Can We Use the Model?

Christiano, Motto, and Rostagno (2003): Great depression.

Christiano, Motto, and Rostagno (2008): Business cycle fluctuations.

Fernández-Villaverde and Ohanian (2009): Spanish crisis of2008-2010.

Fernández-Villaverde (2010): fiscal policy.

Figure: IRFs of Output to Different Fiscal Policy Shocks

A Model with Costly Enforcement

We keep the basic structure as before, except the financial marketfriction is the cost of enforcing contracts.

Structure:

1 Borrower may decide to renege on debt.

2 If that is the case, the lender can only recover the fraction (1− λ) ofthe gross return Rkt+1ptqtkt where:

(1− λ)Rkt+1 < Rt

and the borrower keeps the rest, λRkt+1ptqtkt .

Costly Enforcement Model II

Value of project:

Vt = Rkt+1ptqtkt − Rt (qtkt − nt )

Incentive constraint:Vt ≥ λRkt+1ptqtkt

Since the constraint must be binding:

Rkt+1ptqtkt − Rt (qtkt − nt ) = λRkt+1ptqtkt ⇒

qtkt =1

1− (1− λ)R kt+1Rtptnt

Costly Enforcement Model III

Advantage: much easier to handle than costly state verificationmodel.

Disadvantage: no default in equilibrium, no spreads.

When to use each of them?

A Model with Financial Intermediation

Previous models have a very streamlined financial intermediationstructure.

Many of the events of the 2007-2010 recession were aboutbreakdowns in intermediation.

Kiyotaki and Gertler (2010)incorporate now a richer financialintermediation sector.

In particular, we will deal with liquidity.

To keep the presentation simple, I will get rid of nominal rigidities.

Also, this will facilitate comparison with a neoclassical framework.

Environment

Island model: continuum of islands of measure 1.

In each island, there is a firm that produces the final good withcapital (not mobile) and labor (mobile across islands) and aCobb-Douglas production function.

Then, by equating the capital-labor ratio across island, aggregateoutput is:

yt = Atkαt l1−αt

Liquidity Risk

Each period, investment opportunities arrive randomly to a fractionπi of islands. There is no opportunity in πn = 1− πi .

Investment opportunities are i.i.d. across time and islands.

Only firms in islands with investment opportunities can accumulatecapital.

Then:kt = (1− δ) kt + it

and:yt = ct + it + gt

Representative Household

Preferences:

∑t=0

(log ct − ψ

l1+ϑt

Continuum of members of measure one with perfect consumptioninsurance within the family.

A fraction 1− f are workers and f are bankers.

Workers work and send wages back to the family.

Bankers run a bank that sends (non-negative) dividends back to thefamily.

In each period, a fraction (1− σ) of bankers become workers and afraction (1− σ) f

1−f of workers become bankers. Why?

Representative Household

Household can save in: deposits in the banking sector:

1 Deposits at the financial intermediary, at , that pay an uncontingentnominal gross interest rate Rt .

2 Arrow-Debreu securities (net zero supply in equilibrium).

The budget constraint is then:

ct + at = wt lt + Rt−1at−1 + Tt +zt

The first-order conditions for the household are:

1ct= λt

λt = βEtλt+1Rt

ψlϑt ct = wt

Asset pricing kernel:

SDFt = Etβλt+1λt

and standard non-arbitrage conditions.

Banks are born with a small initial transfer from the family.

Initial equity is increased with retained earnings. Dividends are onlydistributed when the bank dies.

Banks are attached to a particular island, which in this period may beh = {i , n} .

Banks move across islands over time to equate expected rate ofreturn:

1 Before moving they sell their loans.

2 This allows us to forget about distributions.

Balance Sheet

Net worth: nht .

Besides equity, banks raise funds in a national financial market:

1 Retail market: from the households, at at cost Rt . Before investmentshock is realized.

2 Wholesale market: from each other, bht at cost Rbt . After investmentshock is realized.

Then, bank lend to non-financial firms in their island sht . Noenforcement problem (we can think about sht as equity).

Flow-of-Funds

Balance sheet constraint:

qht sht = n

ht + at + b

Evolution of net worth:

nht =[zt + (1− δ) qht

]st−1 − Rt−1at−1 − Rbt−1bt−1

Objective function of bank:

Vt = Et

∑i=0(1− σ) σi−1βt

λt+iλt

Value function: maximized objective function:

V(sht , b

ht , at

)= maxVt

Financial Friction

We need some financial friction to make the intermediation probleminteresting.

Diversion of funds to family:

θ(qht s

ht −ωbht

)and close down.

Then, the incentive constraint is:

V(sht , b

ht , at

)≥ θ

(qht s

ht −ωbht

)The Lagrangian associated with the previous constraint is λht and:

λht = πiλit + πnλnt

Guess of Value Functions

We guess that value function is linear in states:

V(sht , b

ht , at

)= νstsht − νbtb

ht − νatat

Interpretation of coeffi cients.

Substituting the balance sheet constraint, two equivalent expressions:

V(sht , b

ht , at

(νst − νbtq

)sht − (νat − νbt ) at + νbtn

= νstsht − νbtbht − νat

(qht s

ht − nht − bht

The FOC are:

sht :(

qht− νbt

)= λht θ (1−ω)

at : (νbt − νat )(1+ λ

)= θωλ

λht : νatnht ≥(

θ −(

qht− νat

))qht s

ht − (θω− (νbt − νat )) bht

Interpretation:

1 Marginal value of assets is higher than marginal cost of interbankborrowing if λht > 0 and ω < 1.

2 Marginal cost of interbank borrowing is higher than cost of deposits if

λht > 0 and ω > 0.

3 Balance sheet effect: equity in bank must be suffi ciently high inrelation with assets and interbank borrowing.

Aggregation

Total bank net worth:nht = n

hot + n

Total net worth of old banks:

nhot = σπh{[zt + (1− δ) qht

]st−1 − Rt−1at−1

}where we have net out the interbank loans.

Total net worth of new banks:

nhyt = ξ{[zt + (1− δ) qht

]st−1 − Rt−1at−1

}Aggregate balance sheet constraint:

at = ∑h=i ,n

(qht s

ht − nht

Equilibrium

Three cases:

1 ω = 1 (frictionless interbank market). The interbank and loan ratesare the same. They are bigger than the deposit rate if banks areconstrained (only one aggregate constrain holds).

2 ω = 1 (symmetric frictions). The interbank and deposit rate are thesame. The returns on loans if banks on non-investing islands areconstrained.

3 ω ∈ (0, 1). The interbank rate lies between the return on loans andthe deposit rate.

Policy Experiments

1 Lending facilities: the central bank lends directly to banks that arebalance sheet constrained.

2 Liquidity facilities: the central bank discounts loans. Banks can divertless funds from the central bank than from the regular interbankmarket.

3 Equity injections: the Treasury transfers wealth to banks.

Sargent and Wallace (1983)

Andrew Wiles

On Doing ResearchPerhaps I can best describe my experience of doing mathematics in termsof a journey through a dark unexplored mansion. You enter the first roomof the mansion and it’s completely dark. You stumble around bumpinginto the furniture, but gradually you learn where each piece of furniture is.Finally, after six months or so, you find the light switch, you turn it on,and suddenly it’s all illuminated. You can see exactly where you were.Then you move into the next room and spend another six months in thedark. So each of these breakthroughs, while sometimes they’re momentary,sometimes over a period of a day or two, they are the culmination of– andcouldn’t exist without– the many months of stumbling around in the darkthat proceed them.

Conclusions

Integrating macro with finance is one of the great challenges ahead ofus.

While there has been some progress, we are still exploring the firstrooms of the mansion.

We discussed today the introduction of financial frictions.

Other key aspect is asset pricing.

But this is a whole new ball game...

macroeconomic models with financial frictionsjesusfv/macrofinancial.pdfthe purchased capital is...

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