Adverse Selection Borrowers project x i with probability p i C - - PowerPoint PPT Presentation

Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

Adverse Selection

CREDIT & MICROFINANCE

• Dr. Kumar Aniket

University of Cambridge Lecture 2

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BORROWER’S PROJECT & TYPE AND SO ON

• Borrower’s project

1 unit of capital − →    xi with probability pi . . . (1 − pi)

• Borrower type i = {s, f}

   ps (Safe type) pr (Risky type) . . . pr < ps

• Borrower’s type unobservable to lender

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ENVIRONMENT

⊙ Impoverished borrower i

• Risk neutral
• No wealth
• Reservation utility is ¯

u

• proportion of risky type r →

θ proportion of safe type s → (1 − θ)

⊙ Lender

• Risk neutral
• opportunity cost of capital ρ
• Lends in a competitive loan market

. . . lender’s zero profit condition

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FIRST BEST: PERFECT INFORMATION BENCHMARK

• If the lender knows borrower’s type (perfect information

environment) then the lender’s profit condition would be: ri = ρ pi i = r, s (L-ZPC)

. . . lender charges r and s different rate . . . risky type pays a higher interest rate

• Borrower i’s expected payoff

Ui(r) = payoffi(xi − ri) The borrower is risk neutral and thus only cares about her expected payoff.

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Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

pi ri piri = ρ ps ¯ p pr rs ¯ r rr θ

1 − θ

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SOCIALLY VIABLE PROJECT

Socially Viable Project

A project is social viable if the expected output is greater than the social cost, in this case, the opportunity cost of capital and reservation wage in this case. pixi ρ + ¯ u

• Under perfect information, all socially viable projects are

feasible.

– The lender would offer the borrowers contracts contingent

• n their type and all borrowers’projects would be funded.

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SECOND BEST: HIDDEN INFORMATION PROBLEM

If the lender is ignorant of the borrower’s type, he has the following two options. either lend to both type - Pooling Equilibrium . . . both type pay the same pooling interest rate ¯ p = θpr + (1 − θ)ps (loan repayment probability) ¯ r = ρ ¯ p (interest rate)

• r lend to only one type - Separating Equilibrium

. . . interest rate for the type left in the market . . . Which type do you think this will be? pr or ps (loan repayment probability) rr = ρ pr and rs = ρ prs (resp. interest rates)

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INTEREST RATE

With the zero profit condition, we only have to check for three interest rates: rs – separating equilibrium with only the safe types ¯ r – pooling equilibrium with both types rr – separating equilibrium with risky types . . . Timeline: Lender would choose the interest rate for the loan contract Borrowers would choose whether to self-select in the loan contract

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Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

IMPERFECT INFORMATION: ADVERSE SELECTION

• Stiglitz & Wiess (1981)

psxs = prxr = ˆ x . . . the expected project outputs (mean) are identical . . . the risky project has a greater spread around mean

• may lead to a problem of Under-investment

some safe type with socially viable projects, i.e., ˆ x = psxs ¯ u + ρ . . . driven out of the loan market

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PARTICIPATION CONSTRAINT: STIGLITZ & WIESS

Borrower’s Participation Constraint Ui(rj) = ˆ x − pir ¯ u i = r, s

ˆ x ¯ u Usafe Urisky r c Kumar Aniket 10/28 Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

PARTICIPATION CONSTRAINT: STIGLITZ & WIESS

Borrower’s Participation Constraint Ui(rj) = ˆ x − pir ¯ u i = r, s – Check participation constraint for both types at rs, ¯ r and rs. – Obtain lower threshold of ˆ x at which each type would self-select into the loan contract. Interest rate Safe type Risky type Us(r) = ˆ x − psr ¯ u Ur(r) = ˆ x − prr ¯ u rs = ρ

ps

ˆ x ρ + ¯ u ˆ x pr

ps ρ + ¯

u ¯ r = ρ

¯ p

ˆ x ps

¯ p ρ + ¯

u ˆ x pr

¯ p ρ + ¯

u rr = ρ

pr

ˆ x ps

pr ρ + ¯

u ˆ x ρ + ¯ u

Table: Self-selection condition at three interest rates in the Stiglitz Weiss

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UNDER-INVESTMENT: EXCLUSION OF THE SAFE TYPE

under-investment type s’s

Socially Viable Projects

Expected Output ρ + u

• ps

¯ p

• ρ + u

Figure: Safe type’s under-investment project range

Under-investment: Some safe agents with socially viable projects i.e., ¯ u + ρ < ˆ x < ¯ u + ps ¯ p ρ . . . unable to borrow.

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Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

IMPERFECT INFORMATION: ADVERSE SELECTION

• De Meza & Webb (1987)

psx > prx . . . projects have different mean . . . risky project has a lower mean

• may lead to a problem of Over-investment

risky type with projects which are not social viable (prx < ¯ u + ρ) may participate in the market at the pooling interest rate.

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PARTICIPATION CONSTRAINT: DE MEZA & WEBB

Borrower’s Participation Constraint Ui(r) = pi(xi − r) ¯ u i = r, s – Check participation constraint for both types at rs, ¯ r and rs. – Obtain lower threshold of ˆ x at which each type would self-select into the loan contract. Interest rate Safe type risky type Us(r) = psx − psr ¯ u Ur(r) = prx − prr ¯ u rs = ρ

ps

psx ρ + ¯ u prx pr

ps ρ + ¯

u ¯ r = ρ

¯ p

psx ps

¯ p ρ + ¯

u prx pr

¯ p ρ + ¯

u rr = ρ

pr

psx ps

pr ρ + ¯

u prx ρ + ¯ u

Table: Self-selection range at interest rates in the De Mezza Webb

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UNDER-INVESTMENT: DE MEZZA & WEBB

• ver-investment

type r’s

Socially Viable Projects

Expected Output

• pr

¯ p

• ρ + u

ρ + u Figure: Risky type’s over-investment project range

Over-investment: Risky type agents with projects that are not socially viable (¯ u + ρ > prx > ¯ u + pr

¯ p ρ) are able to

borrow (because they are cross-subsidised by the safe type borrowers).

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• ver-investment

type r’s under-investment type s’s

Socially Viable Projects

Expected Output

• pr

¯ p

• ρ + u

ρ + u ps

¯ r

• ρ + u

Figure: Under and Over investment Ranges

• Under-investment: Range of socially viable projects that are

not viable due to imperfect information ¯ u + ρ < ˆ x < ¯ u + ps ¯ p ρ

• Over-investment: Range of socially non-Viable projects that

are viable only due to imperfect information ¯ u + pr ¯ p ρ < prx < ¯ u + ρ

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INVESTMENT PROBLEM IN A ADVERSE SELECTION FRAMEWORK

⊙ Stiglitz & Webb Under-investment: Safe type unable to borrow for a range of socially viable projects because at high interest rates, only the risky types willing to borrow. ⊙ De Meza & Webb Over-investment: Risky type are able to borrow for a range of non socially viable projects because they are cross-subsidised by the safe type borrowers in a pooling equilibrium.

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GROUP LENDING WITH JOINT LIABILITY

Definition (Joint-Liability Group-Lending)

Lender lends to a group with the proviso that each borrower’s payoffs contingent on peer’s outcome.

• Joint-Liability Group-Contract: (r, c)

Definition (Joint Liability Payment: c)

Payment due if the borrower succeeds but her peer fails

Definition (Positive Assortative Matching)

Groups homogenous in the types of borrowers

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POSITIVE ASSORTATIVE MATCHING

Proposition (Positive Assortative Matching)

Joint Liability contracts lead to positive assortative matching. Uij(r, c) = pipj(xi − r) + pi(1 − pj)(xi − r − c) = pi(xi − r) − pi(1 − pj)c Urs(r, c) − Urr(r, c) = pr(ps − pr)c (1) Uss(r, c) − Usr(r, c) = ps(ps − pr)c (2) (2) > (1)

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POSITIVE ASSORTATIVE MATCHING AND SOCIAL OPTIMUM

Paper (Ghatak, 1999, 2000)

Joint Liability Group Lending leads to positive assortative matching solves the problems of under and over-investment.

Assumption (Socially Optimal Matching)

Positive assortative matching maximises the aggregate expected payoffs of borrowers over all possible matches Uss(r, c) − Usr(r, c) > Urs(r, c) − Urr(r, c) ((2) > (1)) Uss(r, c)+Urr(r, c) > Urs(r, c)+Urs(r, c) (rearranging)

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Prologue Stiglitz & Wiess De Meza & Webb Group Lending Ghatak Epilogue

INDIFFERENCE CURVES

Indifference Curve of borrower type i Uij(r,c) = pi(xi − r) − pi(1 − pj)c = ¯ k dc dr

• Uii=constant

= − 1 1 − pi s type’s indifference curve steeper

• −

1 1 − ps

• >
• −

1 1 − pr

• c

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INDIFFERENCE CURVES OF THE TWO TYPES

Interest rate r Joint Liability c

b

− 1 1 − ps

Safe borrower’s steeper IC

− 1 1 − pr

Risky borrower’s flatter IC

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LENDER’S PROBLEM

• Lender offers group contracts (rr, cr) and (rs, cs) which

maximise the borrower’s payoff subject to the following constraint”s: rrpr + cr(1 − pr)pr ρ ⇒ dc dr = − 1 1 − pr (L-ZPCr) rsps + cs(1 − ps)ps ρ ⇒ dc dr = − 1 1 − ps (L-ZPCs) Uii(ri, ci) ¯ u, i = r, s (PCi) xi ri + ci i = r, s (LLCi) Urr(rr, cr) Urr(rs, cs) (ICCrr) Uss(rs, cs) Uss(rr, cr) (ICCss)

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ABBREVIATIONS

L-ZPCi Lender’s Zero Profit Condition for type i PCi Participation Constraint for type i LLCi Limited Liability Constraint for type i ICCii Incentive Compatibility Constraint for group i, i

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SEPARATING EQUILIBRIUM IN GROUP LENDING

⊙ (L-ZPCs) and (L-ZPCr) cross at (ˆ r,ˆ c)

Proposition (Separating Equilibrium)

For any joint liability contract (r, c)

• i. if rs < ˆ

r, cs > ˆ c, then Uss(rs, cs) > Urr(rs, cs)

• ii. if rr > ˆ

r, cr < ˆ c, then Urr(rr, cr) > Uss(rr, cr)

• Safe groups prefer high joint liability payment low interest

rates

• Risky groups prefer low joint liability payments high

interest rate

• Different interest rates for different types – back to the

perfect information environment

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SEPARATING EQUILIBRIUM IN r-c SPACE

Interest rate r Joint Liability c

b b b

A B C D − 1 1 − ps

Safe borrower’s steeper IC

− 1 1 − pr

Risky borrower’s flatter IC

−1 LLC (ˆ r,ˆ c)

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CONTRACTS

Separating Contract

Safe: Segment BA Risky: Segment AC

Pooling Contract

(ˆ c,ˆ r) at A

Conditions: Projects sufficiently productive to satisfy the Limited Liability Condition (LLC) along respective contract segments.

Under-investment:

Bring back the safe borrowers with socially productive investment.

Over-investment:

Risky borrowers with socially productive investment drop out.

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CONCLUSIONS

In group lending – joint liability leads to positive assortative matching – the risky and safe group differ in the way they trade-off interest rates and joint liability payments – lender is able to discriminate between the risky and safe groups

• problem of under and over investment is solved

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