Notes on Financial Frictions Under Asymmetric Information and Costly

State Verification

Incorporating Financial Frictions into a Business Cycle Model

• General idea:– Standard model assumes borrowers and lenders

are the same people..no conflict of interest

– Financial friction models suppose borrowers and lenders are different people, with conflicting interests

– Financial frictions: features of the relationship between borrowers and lenders adopted to mitigate conflict of interest.

Discussion of Financial Frictions

• Simple model to illustrate the basic costly state verification (csv) model. – Original analysis of Townsend (1978), Bernanke-

Gertler.

• Integrating the csv model into a full-blown dsge model.– Follows the lead of Bernanke, Gertler and Gilchrist

(1999).– Empirical analysis of Christiano, Motto and Rostagno

(2003,2009).

Simple Model• There are entrepreneurs with all different levels of

wealth, N. – Entrepreneur have different levels of wealth because they

experienced different idiosyncratic shocks in the past.

• For each value of N, there are many entrepreneurs.

• In what follows, we will consider the interaction between entrepreneurs with a specific amount of N with competitive banks.

• Later, will consider the whole population of entrepreneurs, with every possible level of N.

Simple Model, cont’d• Each entrepreneur has access to a project with

rate of return,

• Here, is a unit mean, idiosyncratic shock experienced by the individual entrepreneur after the project has been started,

• The shock, , is privately observed by the entrepreneur.

• F is lognormal cumulative distribution function.

0

dF 1

1 Rk

Banks, Households, Entrepreneurs

HouseholdsBank

entrepreneur

entrepreneurentrepreneur

entrepreneur

entrepreneur

Standard debt contract

~ F , 0

dF 1

• Entrepreneur receives a contract from a bank, which specifies a rate of interest, Z, and a loan amount, B.– If entrepreneur cannot make the interest

payments, the bank pays a monitoring cost and takes everything.

• Total assets acquired by the entrepreneur:

• Entrepreneur who experiences sufficiently bad luck, , loses everything.

total assetsA

net worthN

loansB

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Cutoff,

• Cutoff higher with:– higher leverage, L– higher

gross rate of return experience by entrepreneur with ‘luck’,

1 Rk total assetsA

interest and principle owed by the entrepreneurZB

1 Rk A ZB

Z1 Rk

BNAN

Z1 Rk

leverage LAN

1AN

Z1 Rk

L 1L

Z/1 Rk

• Expected return to entrepreneur, over opportunity cost of funds:

Expected payoff for entrepreneur

opportunity cost of fundsFor lower values of , entrepreneur receives nothing‘limited liability’.

1 Rk A ZBdF

N1 R

• Rewriting entrepreneur’s rate of return:

• Entrepreneur’s return unbounded above– Risk neutral entrepreneur would always want to

borrow an infinite amount (infinite leverage).

1 Rk A ZBdF

N1 R

1 Rk A 1 Rk AdF

N1 R

dF 1 Rk

1 R L

Z1 Rk

L 1L L

Z1 Rk

• Rewriting entrepreneur’s rate of return:

• Entrepreneur’s return unbounded above– Risk neutral entrepreneur would always want to

borrow an infinite amount (infinite leverage).

1 Rk A ZBdF

N1 R

1 Rk A 1 Rk AdF

N1 R

dF 1 Rk

1 R L

Z1 Rk

L 1L L

Z1 Rk

• Rewriting entrepreneur’s rate of return:

• Entrepreneur’s return unbounded above– Risk neutral entrepreneur would always want to

borrow an infinite amount (infinite leverage).

1 Rk A ZBdF

N1 R

1 Rk A 1 Rk AdF

N1 R

dF 1 Rk

1 R L

Z1 Rk

L 1L L

Z1 Rk

• Rewriting entrepreneur’s rate of return:

• Entrepreneur’s return unbounded above– Risk neutral entrepreneur would always want to

borrow an infinite amount (infinite leverage).

1 Rk A ZBdF

N1 R

1 Rk A 1 Rk AdF

N1 R

dF 1 Rk

1 R L

Z1 Rk

L 1L L

Z1 Rk Gets smaller with L

Larger with L

• Rewriting entrepreneur’s rate of return:

• Entrepreneur’s return unbounded above– Risk neutral entrepreneur would always want to

borrow an infinite amount (infinite leverage).

1 Rk A ZBdF

N1 R

1 Rk A 1 Rk AdF

N1 R

dF 1 Rk

1 R L

Z1 Rk

L 1L L

Z1 Rk

1.5 2 2.5 3 3.5 4 4.5 5 5.5

1

1.2

1.4

1.6

1.8

2

2.2

leverage

Exp

ect

ed

re

turn

fo

r e

ntr

ep

ren

eur

Expected entrepreneurial return, over opportunity cost, N(1+R)

In our baseline parameterization, risk spread = 1.0063,return is monotonically increasingin leverage

1.5 2 2.5 3 3.5 4 4.5 5 5.5

1

1.2

1.4

1.6

1.8

2

2.2

leverage

Exp

ect

ed

re

turn

fo

r e

ntr

ep

ren

eur

Expected entrepreneurial return, over opportunity cost, N(1+R)

Z/(1+R) = 1.0063Z/(1+R) = 1.5

Baseline parameters

More leverage locally reduces expected returnwith high risk spread.

High leverage always preferredeventually linearly increasing

• If given a fixed interest rate, entrepreneur with risk neutral preferences would borrow an unbounded amount.

• In equilibrium, bank can’t lend an infinite amount.

• This is why a loan contract must specify both an interest rate, Z, and a loan amount, B.

• Need to represent preferences of entrepreneurs over Z and B.– Problem, possibility of local decrease in utility with

more leverage makes entrepreneur indifference curves ‘strange’ ..

1.5 2 2.5 3 3.5

1.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

Z/(

1+

R),

ris

k sp

rea

d

Leverage (i.e., Assets/Net Worth)

Entrepreneurial indifference curves

Indifference Curves Over Z and B Problematic

Utility increasing

Downward-sloping indifference curves reflect local fall in net worth with rise in leverage whenrisk premium is high.

Solution to Technical Problem Posed by Result in Previous Slide

• Think of the loan contract in terms of the loan amount (or, leverage, (N+B)/N) and the cutoff,

1 Rk A ZBdF

N1 R

dF 1 Rk

1 R L

2 3 4 5 6 7

1

2

3

4

5

6

7

-

ba

r

leverage

Indifference curve, (leverage, - bar) space

L AN N B

N

Utility increasing

Solution to Technical Problem Posed by Result in Previous Slide

• Think of the loan contract in terms of the loan amount (or, leverage, (N+B)/N) and the cutoff,

1 Rk A ZBdF

N1 R

dF 1 Rk

1 R L

2 3 4 5 6 7

1

2

3

4

5

6

7

-

ba

r

leverage

Indifference curve, (leverage, - bar) space

L AN N B

N

Utility increasing

Banks• Source of funds from households, at fixed

rate, R

• Bank borrows B units of currency, lends proceeds to entrepreneurs.

• Provides entrepreneurs with standard debt contract, (Z,B)

Banks, cont’d• Monitoring cost for bankrupt entrepreneur

with

• Bank zero profit conditionfraction of entrepreneurs with

1 F

quantity paid by each entrepreneur with ZB

quantity recovered by bank from each bankrupt entrepreneur

1 0

dF 1 Rk A

amount owed to households by bank

1 RB

Bankruptcy cost parameter

1 Rk A

Banks, cont’d• Monitoring cost for bankrupt entrepreneur

with

• Bank zero profit conditionfraction of entrepreneurs with

1 F

quantity paid by each entrepreneur with ZB

quantity recovered by bank from each bankrupt entrepreneur

1 0

dF 1 Rk A

amount owed to households by bank

1 RB

Bankruptcy cost parameter

1 Rk A

Banks, cont’d• Simplifying zero profit condition:

• Expressed naturally in terms of

1 F ZB 1 0

dF 1 Rk A 1 RB

1 F 1 Rk A 1 0

dF 1 Rk A 1 RB

1 F 1 0

dF 1 R

1 RkB/NA/N

1 R1 Rk

L 1L

,L

Banks, cont’d• Simplifying zero profit condition:

• Expressed naturally in terms of

1 F ZB 1 0

dF 1 Rk A 1 RB

1 F 1 Rk A 1 0

dF 1 Rk A 1 RB

1 F 1 0

dF 1 R

1 RkB/NA/N

1 R1 Rk

L 1L

,L

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

2

4

6

8

10

12

14

-

ba

r

leverage

Bank zero profit condition, in (leverage, - bar) space

•Free entry of banks ensures zero profits

• zero profit curve represents a ‘menu’ of contracts, , that can be offered in equilibrium.

•Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage.

,L

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

2

4

6

8

10

12

14

-

ba

r

leverage

Bank zero profit condition, in (leverage, - bar) space

•Free entry of banks ensures zero profits

• zero profit curve represents a ‘menu’ of contracts, , that can be offered in equilibrium.

•Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage.

,L

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

2

4

6

8

10

12

14

-

ba

r

leverage

Bank zero profit condition, in (leverage, - bar) space

•Free entry of banks ensures zero profits

• zero profit curve represents a ‘menu’ of contracts, , that can be offered in equilibrium.

•Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage.

,L

1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3

2

4

6

8

10

12

14

-

ba

r

leverage

Bank zero profit condition, in (leverage, - bar) space

•Free entry of banks ensures zero profits

• zero profit curve represents a ‘menu’ of contracts, , that can be offered in equilibrium.

•Only the upward-sloped portion of the curve is relevant, because entrepreneurs would never select a high value of if a lower one was available at the same leverage.

,L

Some Notation and Results• Let

• Results:

G

expected value of , conditional on

0

dF , 1 F G ,

G dd

0

dF

Leibniz’s rule F

1 F F G 1 F

Moving Towards Equilibrium Contract• Entrepreneurial utility:

dF 1 Rk

1 R L

1 G 1 F 1 Rk1 R L

share of entrepreneur return going to entrepreneur

1 1 Rk1 R L

Moving Towards Equilibrium Contract, cn’t

• Bank profits:

share of entrepreneurial profits (net of monitoring costs) given to bank

1 F 1 0

dF 1 R

1 RkL 1L

G 1 R1 Rk

L 1L

L 11 1 Rk

1 R G

Equilibrium Contract• Entrepreneur selects the contract is optimal,

given the available menu of contracts.

• The solution to the entrepreneur problem is the that solves:

log

profits, per unit of leverage, earned by entrepreneur, given

dF 1 Rk

1 R

leverage offered by bank, conditional on

11 1 Rk

1 R G

log

higer drives share of profits to entrepreneur down (bad!)

1 log 1 Rk1 R

higher drives leverage up (good!)

log 1 1 Rk1 R G

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.0128

1.013

1.0132

1.0134

1.0136

1.0138

utili

ty

- bar

entrepreneurial utility as a function of - bar only

1 2 3 4 5 6 7 8 9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

utili

ty

- bar

Equilibrium Contracting Problem Not Globally Concave, But Has Unique Solution Characterized by First Order Condition

Close up of the objective, in neighborhood of optimum. Locally concave.

Non-concave part

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

1.0128

1.013

1.0132

1.0134

1.0136

1.0138

utili

ty

- bar

entrepreneurial utility as a function of - bar only

1 2 3 4 5 6 7 8 9

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

utili

ty

- bar

Equilibrium Contracting Problem Not Globally Concave, But Has Unique Solution Characterized by First Order Condition

Close up of the objective, in neighborhood of optimum

Non-concave part

Computing the Equilibrium Contract• Solve first order optimality condition uniquely for the

cutoff, :

• Given the cutoff, solve for leverage:

• Given leverage and cutoff, solve for risk spread:

elasticity of entrepreneur’s expected return w.r.t.

1 F 1

elasticity of leverage w.r.t.

1 Rk1 R 1 F F

1 1 Rk1 R G

L 1

1 1 Rk1 R G

risk spread Z1 R 1 Rk

1 R LL 1

Result• Leverage, L, and entrepreneurial rate of

interest, Z, not a function of net worth, N.

• Quantity of loans proportional to net worth:

• To compute L, Z/(1+R), must make assumptions about F and parameters.

L AN

N BN

1 BN

B L 1N

1 Rk1 R , , F

The Distribution, F

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

dens

ity

Log normal density function, E = 1, = 0.82155

Results for log-normal• Need: G

0

dF , F

Can get these from the pdf and the cdf of the standard normaldistribution.

These are available in most computational software, like MATLAB.

Also, they have simple analytic representations.

Results for log-normal• Need: G

0

dF , F

0

dF

change of variables, x log 1

x 2

logexe

x Ex 2

2 x2 dx

E 1 requires Ex 12 x2

1 x 2

logexe

x 12 x2

2

2 x2 dx

combine powers of e and rearrange 1

x 2

loge

x 12 x2

2

2 x2 dx

change of variables, vx 1

2 x2

x 1

x 2

log 12 x2

x xexp

v22 xdv

prob v log 1

2 x2

x x cdf for standard normal

Results for log-normal• Need: G

0

dF , F

0

dF

change of variables, x log 1

x 2

logexe

x Ex 2

2 x2 dx

E 1 requires Ex 12 x2

1 x 2

logexe

x 12 x2

2

2 x2 dx

combine powers of e and rearrange 1

x 2

loge

x 12 x2

2

2 x2 dx

change of variables, vx 1

2 x2

x 1

x 2

log 12 x2

x xexp

v22 xdv

prob v log 1

2 x2

x x cdf for standard normal

Results for log-normal• Need: G

0

dF , F

0

dF

change of variables, x log 1

x 2

logexe

x Ex 2

2 x2 dx

E 1 requires Ex 12 x2

1 x 2

logexe

x 12 x2

2

2 x2 dx

combine powers of e and rearrange 1

x 2

loge

x 12 x2

2

2 x2 dx

change of variables, vx 1

2 x2

x 1

x 2

log 12 x2

x xexp

v22 xdv

prob v log 1

2 x2

x x cdf for standard normal

Results for log-normal• Need: G

0

dF , F

0

dF

change of variables, x log 1

x 2

logexe

x Ex 2

2 x2 dx

E 1 requires Ex 12 x2

1 x 2

logexe

x 12 x2

2

2 x2 dx

combine powers of e and rearrange 1

x 2

loge

x 12 x2

2

2 x2 dx

change of variables, vx 1

2 x2

x 1

x 2

log 12 x2

x xexp

v22 xdv

prob v log 1

2 x2

x x cdf for standard normal

Results for log-normal• Need: G

0

dF , F

0

dF

change of variables, x log 1

x 2

logexe

x Ex 2

2 x2 dx

E 1 requires Ex 12 x2

1 x 2

logexe

x 12 x2

2

2 x2 dx

combine powers of e and rearrange 1

x 2

loge

x 12 x2

2

2 x2 dx

change of variables, vx 1

2 x2

x 1

x 2

log 12 x2

x xexp

v22 xdv

prob v log 1

2 x2

x x cdf for standard normal

Results for log-normal, cnt’d• The log-normal cumulative density:

• Differentiating (using Leibniz’s rule):

F 0

dF 1

x 2

loge

x 12 x2

2

2 x2 dx

F ; 1

12

exp

log 1

2 2

2

2

1 Standard Normal pdf

log 12 2

‘Test’ of the Model• Obtain the following for each firm from a

micro dataset:

• Using definition of F, risk spread, first order condition associated with optimal contract and zero profit condition of banks, can compute:

• Test the model: do the results look sensible?

probability of default (from rating agency)

F ,

firm leverageL ,

interest rateZ

ex ante mean return on firm investment projectRk ,

ex ante idiosyncratic uncertainty ,

monitoring costs ,

cutoff productivity

• Levin, Natalucci, Zakrajsek, ‘The Magnitude and Cyclical Behavior of Financial Market Frictions’, Finance and Economics Discussion Series, Federal Reserve Board, 2004-70.

A jump in spreads occurred here, interpreted bythe model as a jump (in part) of bankruptcy costs.

Changes in idiosyncratic volatility not very important

400 1 Rk1 R 1

400 1 Rk1 R 1

High values consistent with highbankruptcy costs.

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

dens

ity

Impact on lognormal cdf of doubling standard deviation

Effect of Increase in Risk, • Keep

• But, double standard deviation of Normal underlying F.

0

dF 1

Doubled standard deviation

Increasing standard deviation raisesdensity in the tails.

1.08 1.1 1.12 1.14 1.16 1.18

1

2

3

4

5

6

7

8

9

leverage, qK/N

risk

spre

ad (

AP

R)

Effect of a 5% jump in

Risk spread= 2.67Leverage = 1.12

Risk spread=2.52Leverage = 1.13

Risk spread = , Leverage = (B+N)/N 400 Z1 R 1

Entrepreneur Indifference curve

Zero profit curve

Issues With the Model• Strictly speaking, applies only to ‘mom and pop grocery

stores’: entities run by entrepreneurs who are bank dependent for outside finance.– Not clear how to apply this to actual firms with access to equity

markets.

• Assume no long-run connections with banks.

• Entrepreneurial returns independent of scale.

• Overly simple representation of entrepreneurial utility function.

• Ignores alternative sources of risk spread (risk aversion, liquidity)

• Seems not to allow for bankruptcies in banks.

Incorporating BGG Financial Friction into Neoclassical Model

Model with Financial Frictions

Firms

household

Entrepreneurs

Labor market

Capital Producers

L

C I

K

Model with Financial Frictions

Firms

household

Entrepreneurs

Labor market

banks

Capital Producers

Loans

K’

Equations of the Model• Aggregate resource constraint:

• Households:

• Capital producers:

bought by households ct

bought by capital producersI t

monitoring costs of banks

0

t dF 1 R tk kt yt

t 0

tuct , ct bt 1 1 R t 1 bt wtl t

uc,t uc,t 11 R t , l t 1

kt 1 1 kt I t

Equations, ctn’d• Entrepreneurs:

– First order condition associated with optimization problem.

– Zero profit condition of banks.

– Law of motion of aggregate net worth.Nt 1 time t earnings of entrepreneurs net of interest on previous period’s bank loans T t

1 T t,

~fraction of entrepreneurs that survive, 1 ~fraction of entrepreneurs born

T t~small transfer made to all entrepreneurs

Conclusion• We’ve reviewed one interesting model of

financial frictions.

• Needs a lot more work!