A Theory of Credit Scoring and CompetitivePricing of Default Risk

Satyajit Chatterjee Dean Corbae Victor Rios-Rull

Fed. Res. Bank of Philadelphia

University of Texas

University of Minnesota and CAERP

September 22, 2007

What We Do

1. Provide a theory of unsecured consumer credit where:

1.1 borrowers have the legal option to default1.2 defaulters are not exogenously excluded from future borrowing1.3 there is free entry of lenders1.4 lenders cannot collude to punish defaulters

2. Use the theory to address the welfare consequences of something likethe Fair Credit Reporting Act.

Legal Environment

I A Ch.7 bankruptcy permanently discharges net debt(liabilities-assets above statewide exemption levels).

I A filer is ineligible for a subsequent Ch. 7 discharge for 6 years(instead forced into Ch. 13 which is a 3-5 year repayment schedulefollowed by discharge).

I The Fair Credit Reporting Act requires credit bureaus to excludethe filing from credit reports after 10 years (and all other adverseitems after 7 years).

Consumption-Smoothing in Practice

I Unsecured consumer debt is 10% of aggregate consumption andthere is 1 bankruptcy filing per 75 households.

I Lenders assess creditworthiness of borrowers using FICO creditscores (higher score, higher creditworthiness)

I Over 75% of mortgage lenders and 80% of the largest financialinstitutions use FICO scores.

Inputs into FICO Scores

FICO scores are calculated from data in the individual’s credit report infive basic catogories:

I Payment history (35%) – includes adverse public records

I Amounts owed (30%)

I Length of credit history (15%)

I Credit limits (10%) and types of credit used (10%)Ignores:

I Race, color, national origin, sex, and marital status (prohibited bylaw)

I Age, assets, salary, occupation, and employment history.

Key Properties of FICO Scores

1. Low score raises interest rate

2. Default lowers score, credit access with flag is restricted, removalraises it

3. Increasing indebtedness lowers score

4. Scores are mean reverting

Evidence for Properties

PF1:

FICO Score Auto Loan Mortgage720-850 4.94% 5.55%700-719 5.67% 5.68%675-699 7.91% 6.21%620-674 10.84% 7.36%560-619 15.14% 8.53%500-559 18.60% 9.29%

Source: http://www.myfico.com

I PF2/PF4: Musto (2004, Journal of Business)/Fisher,Filer,Lyons(2004)

I Removal of Bankruptcy flag substantially raises credit scoresI Bankruptcy lowers availability of credit

I PF3: On its website Fico advises to keep balances low and pay offdebt.

Environment

I Bewley model of precautionary savings (unobservable idiosyncraticearnings shocks).

I 2 types of people: type affects preferences, is unobservable, andfollows a Markov process.

I Borrowers can default on their loans.

I Interest rates on loans depend on loan size and on theintermediary’s assessment that a person is of a given type.

I Assessments about type are updated according to Bayes’ rule.

I Free entry of intermediaries.

Can this Environment deliver PF1-4?

I Crucial inference problem for the intermediary: Is a defaulter or aborrower a low risk type with a low earnings realization or a high risktype?

Relevant Prior Work

I Bankruptcy: Athreya (2002, JME ), Chatterjee, et.al. (forthcomingECMTA), Livshits, et.al. (2007,AER)

I Reputation and Signalling: Cole, et.al. (1995, IER), Chatterjee,et.al. (forthcoming JET )

People

I Unit measure of two types of agents it ∈ {g, b} where type affectspreferences βi. Agents switch types with probability δi > 0.

I Each period an agent’s endowment et is an i.i.d draw from adistribution η with compact support E ⊂ R++.

I Type, earnings, and cons. are private information, but defaultdecision dt ∈ {0, 1} and asset choice `t+1 ∈ L are observable.

I An agent’s history of observed actions at beginning of period t isgiven by (`t, h

Tt ) where hT

t = (dt−1, `t−1, dt−2, ..., `t+1−T , dt−T )

Intermediaries

I Competitive credit industry accepts deposits `t > 0 and makes loans`t < 0 to people.

I Let σ(`t+1, dt, `t, hTt ) be the probability that a person is of type g

conditional on history (`t, hTt ) and choices (`t+1, dt) - Call σ a

person’s end-of-period type score.

I Assume the period t price of a loan of size `t+1 ∈ L made to anindividual with history (`t, h

Tt ) is given by

q(`t+1, σ(`t+1, 0, `t, hTt )) ≥ 0.

Timing

I Enter period with credit history (`t, hTt )

I Type and Earnings shock (it, et) realized

I In state (it, et, `t, hTt ) choose whether to default ( dt+1)

I If don’t default, choose next period assets ( `t+1)

Recursive Formulation of HH Problem

vi(e, `, h; q, σ) = maxd∈{0,1}

vdi (e, `, h; q, σ)

where

v0i (e, `, h; q, σ) = max

(c,`′)∈B(e,`,h;q,σ) 6=∅ui(c)

+βi(1− δi)∫

E

vi(e′, `′, h′; q, σ) η(de′)

+βiδi

∫E

v−i(e′, `′, h′; q, σ) η(de′)

with B(e, `, h; q, σ) = {c ≥ 0, `′ ∈ L | c+q(`′, σ(`′, 0, `, h))·`′ ≤ e+`}and

v1i (e, `, h; q) = ui(e)

+ βi(1− δi)∫

E

vi(e′, 0, h′; q, σ) η(de′)

+ βiδi

∫E

v−i(e′, 0, h′; q, σ) η(de′)

where h′ = λT (`, d, h) = (d, `, ..., `+2−T , d+1−T ).

I Solution induces decision rules di(e, `, h; q, σ) and `′i(e, `, h; q, σ).I Let Di(`, h; q, σ) = {e | di(e, `, h; q, σ) = 1} ⊆ E denote the set of

earnings for which an individual of type i and history (`, h) defaultson a loan of size `.

I Then η(Di(`, h; q, σ)) is the fraction of type i agents who default ona loan of size ` in history h.

I Let Ei(`′, `, h; q, σ) = {e | `′i(e, `, h; q, σ) = `′} ⊆ E as the set ofearnings for which an individual of type i in history (`, h) chooses `′.

Intermediary’s ProblemI An agent’s beginning-of-next period type score given by

Ψ(`′, d, `, h) = (1− δg)σ(`′, d, `, h) + δb [1− σ(`′, d, `, h)] .

I Fraction of agents in history (`, h) expected to default on a loan ofsize `′ tomorrow given by

p(`′, `, h; q, σ) ={

η(Dg(`′, h′; q, σ)) ·Ψ(`′, 0, `, h)+η(Db(`′, h′; q, σ)) · (1−Ψ(`′, 0, `, h))

}I Profit on a loan or deposit, denoted π(`′, `, h; q, σ), is given by:

π(`′, `, h; q, σ)

=

{[1−p(`′,`,h;q,σ)](−`′)

(1+r) − q(`′, σ(`′, 0, `, h))(−`′) if `′ < 0q(`′, σ(`′, 0, `, h))`′ − (1 + r)−1`′ if `′ ≥ 0

I An intermediary solves the linear problem

maxα(`′,`,h)≥0

∑`′,(`,h)

π(`′, `, h; q) · α(`′, `, h)

where α(`′, `, h) ≥ 0 is the measure of (`′, `, h) type contracts soldby the intermediary

Equilibrium Conditions

A steady state equilibrium is a list of decision rules {`′∗, d∗}, prices q∗,scoring function (beliefs) σ∗, and a distribution µ∗ :{g, b} × L×H→ [0, 1] which satisfy:

1. Given q∗, σ∗, Di(`, h; q∗, σ∗) and Ei(`′, `, h; q∗, σ∗) are consistentwith hh optimization.

2. Given σ∗, the zero profit condition implies q∗must satisfy:

q∗(`′, σ∗(`′, 0, `, h)) =

{[1−p(`′,`,h;q∗,σ∗)]

(1+r) `′ < 0(1 + r)−1 `′ ≥ 0

(1)

3. Scoring function σ∗(`′, d, `, h) satisfies Bayes’ Rule whereverpossible.

4. µ∗ reproduces itself:

µ∗(i′, `′, h′)

=

∑i,`,h

∫e

δ(i′|i)18>><>>: `′∗i (e,`,h;q∗,σ∗)=`′,

λT (`,d∗i (e,`,h;q∗,σ∗),h)=h′

9>>=>>;dη(e)µ∗(i, `, h)

Updating Type ScoresI Intermediaries update assessments of agent type according to

σ(`′, d, `, h) =Pr(`′, d|g, `, h) Pr(g|`, h)

Pr(`′, d|g, `, h) Pr(g|`, h) + Pr(`′, d|b, `, h) Pr(b|`, h)

I Two sorts of observable and mutually exclusive events:I Person of type i with history (`, h) defaults on a loan of size ` :

Pr(0, 1|i, `, h) = η(Di(`, h; q, σ))

I Person of type i with history (`, h) does not default and chooses `′ :

Pr(`′, 0|i, `, h) = η(Ei(`′, `, h; q, σ))

I Current type score given by

Pr(i|`, h) =µ(i, `, h)∑

j∈{g,b} µ(j, `, h)

I With T = ∞, h∞ is an infinite-dimensional state variable. But interms of t + 1, Ψ is equivalent to µ′ so replace h∞ by s ∈ [0, 1].

When is a Type Score a Credit Score?

1. Rates fall with creditworthiness; for σ1 > σ2 and `′ < 0

q∗(`′, σ1) ≥ q∗(`′, σ2).

2. Scores fall after bankruptcy; for ` < 0

Ψ(0, 1, `, s) < s ∀s.

3. Scores fall with indebtedness; for `′ < 0 and `′ < `

Ψ(`′, 0, `, s) < s ∀s.

4. Mean reversion

σ < γ =⇒ Ψ > σand σ > γ =⇒ Ψ < σ.

How Can We Get This?

If type characteristics are such that

Dg(`, s; q∗, σ∗) ⊆ Db(`, s; q∗, σ∗)∀`, s,

Example 2 - T = ∞ with Nearly Myopic Type b

agents

Parameterization

I βg = 0.9, βb = 0.05, r = 0.066, δg = 0.1, δb = 0.4, L = {−x, 0, x}where x = 2.5, uniform distn over earnings from [0, 21]

I Specification of off-equilibrium-path beliefs:

I If Di(−x, s) and/or Ei(`′, `, s) are empty ∀i, then denominator of

Bayes law is 0. Assign σ(`′, d, `, s) = s.

Example 2 - Equilibrium Scoring Function(Beliefs)

I In the case where an agent defaults, σ∗(0, 1,−x, s) < s for all s(Fig13a)

I All other cases below:

Table 3.state/decision (`′, d) = (−x, 0) (`′, d) = (0, 0) (`′, d) = (x, 0)

(`, s) =(−x, s) σ∗(−x, 0,−x, s) = s (o-e-p) σ∗(0, 0,−x, s) = s (o-e-p) σ∗(x, 0,−x, s) = 1(`, s) =(0, s) σ∗(−x, 0, 0, s) < s σ∗(0, 0, 0, s) > s σ∗(x, 0, 0, s) = 1(`, s) =(x, s) σ∗(−x, 0, x, s) = 0 σ∗(0, 0, x, s) > s σ∗(x, 0, x, s) = 1

0.4 0.5 0.6 0.7 0.8 0.9

0

0.2

0.4

0.6

0.8

1

Fig 13.a: σ∗(l′=0,d=1,l=−x,s) vs. σ∗(l′=x,d=0,l=−x,s)

score (s)

σ

σ∗(0,1,−x,s)

σ∗(x,0,−x,s)s

0.4 0.5 0.6 0.7 0.8 0.9

0.4

0.5

0.6

0.7

0.8

0.9

Fig 13.b: Ψ(l′=0,d=1,l=−x,s) vs. Ψ(l′=x,d=0,l=−x,s)

score (s)

Ψ

Ψ(0,1,−x,s)Ψ(x,0,−x,s)s

Example 2 - Type b behavior

I Fig6a,b (` = −x). All type b default (sacrifice reputation).

I Fig6c (` = 0). Except for very high earnings, type b borrow.

I Those with lower scores try to build a reputation since this mayassure him lower future interest rates. Fig8 plots how futurereputation is affected.

I Fig6d (` = x). Type b with low earnings borrow and those with highearnings dissave (the latter mimics type g behavior so is reputationbuilding).

0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

Fig 6.a: Bad Type Default Decision at (s,−x,e)

endowment (e)

Db(s,−x,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 6.b: Bad Type Asset Accumulation Decision at (s,−x,e)

endowment (e)

Apb(s,−x,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 6.c: Bad Type Asset Accumulation Decision at (s,0,e)

endowment (e)

Apb(s<=0.64,0,e)Apb(s∈[0.64,0.9],0,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 6.d: Bad Type Asset Accumulation Decision at (s,x,e)

endowment (e)

Apb(s∈[0.4,0.74],0,e)Apb(s∈(0.74,0.87],0,e)Apg(s∈(0.87,0.90],0,e)

Example 2 - Type g behavior

I Fig 11a.(` = −x). Default decision exhibits score dependence.

I A type g agent defaults for low earnings (i.e. e ∈ [0, 16.3]) and doesnot default for high earnings (i.e. e ∈ (16.6, 21]) independent ofscore.

I For intermediate earnings whether or not he defaults depends on hisscore (those with higher score are more willing to run down theirreputation while those with a lower score build their reputation sinceσ∗(x, 0,−x, s) = 1 in Fig13.

I Fig 11b. (` = −x).If a type g agent with debt does not default, hechooses to save. Building precautionary balances is another reasonfor not defaulting besides a rise in score.

I Fig11c (` = 0). At e = 0, type g borrows, for low earnings he staysout of the asset market, and at high earnings he saves.

I Fig11d (` = x). At low earnings he dissaves and at high earnings hecontinues to save (`′ = x).

I Given these decision rules for type b and g, it is clear whyEi(`′ ∈ {−x, 0}, ` = −x, s) = ∅, o-e-p beliefs apply in Table 3.

0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

Fig 11.a: Good Type Default Decision at (s,−x,e)

endowment (e)

Dg(s<=0.55,−x,e)Dg(s∈(0.55,0.74],−x,e)Dg(s>0.74,−x,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 11.b: Good Type Asset Accumulation Decision at (s,−x,e)

endowment (e)

Apg(s<=0.55,−x,e)Apg(s∈(0.55,0.74],−x,e)Apg(s>0.74,−x,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 11.c: Good Type Asset Accumulation Decision at (s,0,e)

endowment (e)

Apg(s<=0.64,0,e)Apg(s∈(0.64,0.79],0,e)Apg(s>=0.79,0,e)

0 2 4 6 8 10 12 14 16 18 20

−2

−1

0

1

2

Fig 11.d: Good Type Asset Accumulation Decision at (s,x,e)

endowment (e)

Apg(s<=0.61,x,e)Apg(s>=0.63,x,e)

Example 2 - Prices

I Price function in Fig15.

I However, since no agents in this economy with ` = −x choose`′i(e,−x, s) = −x, the price function q∗(−x, σ∗(−x, 0,−x, s)) iso-e-p.

0.4 0.5 0.6 0.7 0.8 0.9

0.4

0.5

0.6

0.7

0.8

0.9

1Fig 15.a: Ψ for Borrowers with Different Initial Assets

score (s)

Ψ(lp

=−

x,0,

l,s)

Ψ(lp=−x,0,l=−x,s)Ψ(lp=−x,0,l=0,s)Ψ(lp=−x,0,l=x,s)

0.4 0.5 0.6 0.7 0.8 0.90.05

0.1

0.15

0.2

0.25Fig 15.b: Equilibrium Borrowing Prices for Different Initial Assets

score (s)

q(−

x,s)

q(−x,σ(lp=−x,0,l=−x,s))q(−x,σ(lp=−x,0,l=0,s))q(−x,σ(lp=−x,0,l=x,s))

Example 2 - Distribution of Scores

I Invariant Distribution in Fig16.

I 9% of the population are borrowers in 1st or 2nd decile.

I Agents in highest decile hold positive assets (a little over 52%) or noassets (26%).

I 11% have no assets in 4th through 7th decile. These agents recentlydefaulted or chose `′ = 0.

0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

s

Mea

sure

of A

gent

s

Fig 16.a: Fraction of Agents Across type Scores at the Stationary Distribution

Fraction of Agents with l=−xFraction of Agents with l=0Fraction of Agents with l=x

0.4 0.5 0.6 0.7 0.8 0.90

0.2

0.4

0.6

0.8

1

s

Mea

sure

of A

gent

s

Fig 16.b: Total Measure Across type Scores at the Stationary Distribution

Total Measure

Example 2 - Default Likelihoods

I Fig 17 plots p(−x, `, s; q∗, σ∗), the fraction of individuals in history(`, s) expected to default on a loan of size `′ tomorrow for different` over s

I Even though some type g find it optimal to default at higher score,this doesn’t translate into perverse relation between score anddefault.

0.4 0.5 0.6 0.7 0.8 0.90.7

0.75

0.8

0.85

0.9

Fig 17.a: Equilibrium Probability of Default

score (s)

p(−

x,σ(

lp=

−x,

0,l,s

))

p(−x,σ(lp=−x,0,l=−x,s))p(−x,σ(lp=−x,0,l=0,s))p(−x,σ(lp=−x,0,l=x,s))

0.4 0.5 0.6 0.7 0.8 0.9

0

0.2

0.4

0.6

0.8

1

Fig 17.b: Equilibrium Probability of Default Conditional on Score

score (s)

∆(−

x,s)

Prob Default ∆(−x,s)Positive Measure

Example 3 - Restricting Info on Adverse

Events (T = 1)

I The Fair Credit Reporting Act requires credit bureaus to exclude abankruptcy filing from credit reports after 10 years (and all otheradverse items after 7 years). We model this as a restriction on T.

I When T = 1, there are 4 possible (`t, hT=1t ) = (`t, dt−1) histories at

any date t.

I For the finite T case, we define beginning-of-period type scores

sT = pr(i = g|`, hT ) =µ(i, `, hT )∑

j∈{g,b} µ(j, `, hT ).

I Parameterization of beliefs: In o-e-p events take σ(`′, d, `, h1) =s1.

Example 3 - Scores

Table 4. Beginning-of-period Score Values sT=1 = p(g|`, hT=1)state,history sT=1 = p(g|`, hT=1)

(`,d−1) =(−x, 0) 0.45(`,d−1) =(0, 1) 0.60(`,d−1) =(0, 0) 0.88(`,d−1) =(x, 0) 0.90

Example 3 - Equilibrium Scoring Function

(Beliefs)

I When an agent defaults, σ∗(0, 1,−x, 0) = 0.37. From the set ofbeginning-of-period scores given by Table 4, we see that an agentstarting in history/state (`,d−1) =(−x, 0) has sT=1 = 0.45, so thatdefault lowers an agent’s end-of-period score.

Table 5. Equilibrium σ Function: σ∗(`′, d, `, h) for T = 1, βb > 0(history,state) decision (`′, d) = (−x, 0) (`′, d) = (0, 0) (`′, d) = (x, 0)

(d−1, `) =(0,−x) 0.45 = sT (o-e-p) 0.45 = sT (o-e-p) 1

(d−1, `) =(1, 0) 0.06 < 0.60 = sT 1 1

(d−1, `) =(0, 0) 0.29 < 0.88 = sT 1 1

(d−1, `) =(0, x) 0 0.94 > 0.9 = sT 1

Example 3 - Type b behavior

I Fig16a,b ((`, d−1) = (−x, 0)). All type b in debt default.

I Fig16c ((`, d−1) ∈ {(0, 0), (0, 1)}). Whether or not they defaulted toget to ` = 0, agents take the same action. Even though there was alot of score dependence for the ` = 0 case when T = ∞ in Fig6c,these results are identical since we are just picking 2 scores out ofFig6c.

I Fig16d ((`, d−1) = (x, 0)). Type b borrows if e ∈ [0, 3] and dissavesif e ∈ (3, 21]. Identical to Fig6d.

I Bottom line: Except for few histories, agents act the same atimplied T = 1 scores as in T = ∞ case.

Example 3 - Type g behavior

I Fig17a,b ((`, d−1) = (−x, 0)). For e ∈ [0, 16.5] the agent defaultsand does not default for e ∈ (16.5, 21]. When he doesn’t default, hehas same decision rule as an agent in a T = ∞ env. in Fig10a,bsince s1 = 0.45 in (0, 0) thereby correspnding to the low score plot.

I Fig 17c ((`, d−1) ∈ {(0, 0), (0, 1)}). Whether or not they defaultedto get to ` = 0, agents take the same actions as the T = ∞ case inFig10c.

I Fig17d ((`, d−1) = (x, 0)). Type g dissaves at low earnings andsaves at high earnings

I Bottom line: Except for few histories, agents act the same atimplied T = 1 scores as in T = ∞ case.

Example 3 - Prices

Table 6. Equilibrium borrowing prices and Ψ Function forT = 1, βb > 0

state decision q(−x, 0, `, d−1) Ψ(−x, 0, `, d−1) s1

(`,d−1) =(0, 1) 0.089 0.430 0.60(`,d−1) =(−x, 0) 0.128 (o-e-p) 0.620 0.45(`,d−1) =(0, 0) 0.112 0.547 0.71(`,d−1) =(x, 0) 0.082 0.400 0.90

I The negative relationship between sT=1 and interest rates remain forpeople with the same asset position ` = 0 when s1 ∈ {0.60, 0.88}

Example 3 - Distribution of Scores

I Invariant Distribution in Fig20.

I Compared with T = ∞, the fractions of the population holdinggiven amounts of assets are very similar but...

I The 5% of pop. who are borrowers in 2nd decile in T = 1 case werein both 1st and 2nd decile in T = ∞ case.

I The 12% of pop. who have no assets are only in 4th decile in T = 1case were in 4th through 7th decile in T = ∞.

0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

s

Mea

sure

of A

gent

s

Fig 20.a: Fraction of Agents Across type Scores at the Stationary Distribution h1

Fraction of Agents with l=−xFraction of Agents with l=0Fraction of Agents with l=x

0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

s

Mea

sure

of A

gent

s

Fig 20.b: Total Measure Across type Scores at the Stationary Distribution h1

Total Measure

Example 3 - Musto’s Jump

I Compute the changes in average score following a removal of thebankruptcy flag from one’s record to compare it to Figure 1 inMusto.

I When (`,d−1) =(0, 1), s1 = 0.5864.

I The average score after the default leaves the credit record (nextperiod for T = 1) is given by∑

i,(`′,0)⊃(0,1) η(Ei(`′, 0, 0, 1))Ψ(`′, 0, 0, 1)µ∗(i, 0, 1) = 0.7031 where

(`′, 0) ⊃ (0, 1) denotes continuation histories that emanate from ahistory with (`, d−1) = (0, 1).

I In this case, the jump in score is 27.25%. Musto found that forindividuals in the highest pre-default quintile of credit scores, theyjumped ahead of 19% of households after the score left their record.

Welfare Consequences of Legal Restrictions

I In a world of incomplete markets and private information, flagremoval may provide insurance to impatient agents in our frameworkthat competitive intermediaries may not be able to provide.

I Hence extending the length of time that bankruptcy flags remain oncredit records may not necessarily raise ex-ante welfare.

I Question: how much would an agent in history (i, `, hT ) be willingto pay forever to be in a regime where T = ∞?

Welfare Consequences of Legal Restrictions

I For each (i, `, hT ) we compute compensating consumptionvariations λ(i, `, hT ) that satisfy

v(i, `, h∞;∞) = Ei

[ ∞∑t=0

βti

[(1 + λ(i, `, hT ))ct(i, `, hT ;T )

]1−γ

1− γ

]= (1 + λ(i, `, hT ))1−γv(i, `, hT ;T )

I Welfare number given by

W (T ) =∑

i,`,hT

λ(i, `, hT )µ(i, `, hT ).

I W (T = 1) = 0.00065). This small aggregate welfare gain, however,hides the fact that not all agents would be willing to pay to get ridof the restriction.

0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

s

Mea

sure

of A

gent

s

Fig 21: Fraction of Agents Across type Scores Total Measure

T=1T=∞

Welfare Consequences of Legal Restrictions

Table 7. Distribution and Compensating Consumption Variations

` h1 = d−1 µ(g, `, h1) µ(b, `, h1) s λ(g, `, h1) λ(b, `, h1)

−x 0 0.060661 0.073284 0.45288 3.26E-05 -1.09E-060 1 0.071997 0.048713 0.5842 3.88E-05 -6.27E-050 0 0.19371 0.025377 0.8737 0.00023682 3.11E-05x 0 0.47363 0.052626 0.9 0.00038914 -1.99E-05

I Good Types willing to pay, bad types must be compensated exceptfor those in state (0, 0) since they get mixed in with good types whoborrow.

To Do

I Theory

I Other types of non-Markov equilibria?

I Data

I A serious moment matching exercise