Tirole’s Simple Credit Rationing Model Equity Multiplier...

IntroductionTirole’s Simple Credit Rationing Model

Equity MultiplierTheory of Standard Debt Contracts

Basic Assumptions (1)

An entrepreneur (borrower).An investment project requiring fixed investment I.The entrepreneur has cash on hand (or liquid securities)A < I.To implement the project the entrepreneur needs (borrows)I − A.

Robert J. Gary-Bobo




If undertaken the project either succeeds: verifiableincome R > 0.Or the project fails: yields zero income.Let p denote the probability of success.The project is subject to moral hazard.

Robert J. Gary-Bobo



Basic Assumptions(3)

The entrepreneur can exert high effort (work, take noprivate benefit), or zero effort (shirk, take a private benefit).Equivalently, the entrepreneur chooses between projectwith high or low probability of success.High effort yields p = pH ; low effort yields p = pL, withpL < pH . Denote ∆p = pH − pL.Low effort yields a private benefit B > 0 to theentrepreneur. (B can be interpreted as a disutility of effort).

Robert J. Gary-Bobo




Borrower and lenders (investors) are risk-neutral.For simplicity, there is no time preference: investors requirean interest rate equal to 0 (at least).The entrepreneur is protected by limited liability (incomecannot be negative).Competition among lenders drive interest and profit to zero.

Robert J. Gary-Bobo



Basic Assumptions(5)

The loan contract specifies how the profit is sharedbetween borrower and lenders.Limited liability implies that both sides receive zero in caseof failure.Profit sharing: R = Rb + Rl , where Rb is the the borrower’sshare, Rl is the lender’s share.Lender’s net payoff is Rl − (I − A) in case of success;−(I − A) in case of failure. The borrower’s payoff is thusRb − A in case of success and −A in case of failure.

Robert J. Gary-Bobo




The zero-profit constraint for lenders is pHRl = I − A.Assuming high effort, the rate of interest is ι, where

Rl = (1 + ι)(I − A) or 1 + ι =1

pH.

The nominal rate of interest ι reflects a default premium.We assume that the project is viable only if effort is high:that is,

pHR − I > 0 and pLR − I + B < 0.

No loan giving an incentive to low effort will be granted:either the lender or the borrower would lose money inexpectation.

Robert J. Gary-Bobo



The lender’s credit analysis (1)

The borrower’s tradeoff: obtain private benefit B but reduceprobability of success to pL.We have the following incentive compatibility constraint, IC:pHRb ≥ pLRb + B or

Rb ≥B

∆p.

The highest income that can be pledged to lenders isR − B/∆p. In expected terms: pH(R − B/∆p).

Robert J. Gary-Bobo




The lender’s individual rationality constraint, IR, is therefore

pH

(R − B

∆p

)≥ I − A.

Financing can be arranged only if

A ≥ I − pH

(R − B

∆p

).

We assume that I > pH(R − B/∆p).Otherwise, a lender with A = 0 could obtain credit. Theproject’s NPV is smaller than the minimum expected rentthat must be left to the borrower to satisfy IC.

Robert J. Gary-Bobo




The borrower must have enough assetsA ≥ A = I − pH(R − B/∆p) in order to be granted credit.If A < A, the project has positive NPV and yet is notfunded. The parties cannot find an agreement that bothinduces high effort and yield enough benefit to lenders.This is credit rationing. The borrower is ready to give moreof the return to the lender but the lender does not want togrant such a loan.If A ≥ A, the entrepreneur can secure financing (we have anecessary and sufficient condition for financing).

Robert J. Gary-Bobo




The entrepreneur offers the minimal claim Rl , such thatpHRl = I − A.The entrepreneur’s stake satisfies

Rb = R − Rl = R − I − ApH

≥ R − I − ApH

=B

∆p,

so that the entrepreneur chooses high effort.One only lends to the rich.If A ≥ A, the borrower’s utility isUb = pHRb − A = pH(R − Rl)− A = pHR − I: the borrowerreceives the entire social surplus of the investment.

Robert J. Gary-Bobo



Determinants of credit rationing

What are the determinants of credit rationing?A low amount of cash A.A high agency cost (or agency rent) B/∆p.Moral hazard is determined by the private benefit B andthe likelihood ratio ∆p/pH .The likelihood ratio measures how much the observableresult (success or failure) reveals the underlying choice ofeffort.

Robert J. Gary-Bobo



Do investors hold debt or equity?

Remark that the loan contract can be interpreted as debthere.This is because we have a two-outcome model.Interpretation as debt: the borrower must reimburse Rl orelse go bankrupt. In the case of reimbursement, theborrower keeps the residual R − Rl .

Robert J. Gary-Bobo



Costly State Verification Model

This model has been analyzed by Townsend (1979) andGale and Hellwig (1985).We derive the financial structure of the firm from anoptimization problem (and from primitive assumptions).Moral hazard here comes from the fact that theentrepreneur can divert (steal) income. There is no effortvariable here.Income is semi-verifiable: the lenders can perfectlyobserve income, provided that they incur an audit cost K .This cost is borne by lenders.

Robert J. Gary-Bobo



Costly State Verification Model (2)

The entrepreneur invests his(her)money A. Investmentcost is I.Investment yields a random income R distributed on[0,+∞), with density p(R). The entrepreneur observes Rwithout cost.Timing: 1. Loan agreement, I is sunk. 2. Income R isrealized. 3. Entrepreneur reports R̂. 4. Lender may decideto audit. 5. Reimbursement.We apply the Revelation Principle: there is no loss ofgenerality if we focus on revealing mechanisms, i.e.,contracts such that the entrepreneur has an incentive toreport the true income R̂ = R.

Robert J. Gary-Bobo



CSV Model (3): Definition of a Contract

A contract is a mapping giving a probability of no audity(R̂) ∈ [0,1] for each report R̂, ...and nonnegative rewards: w0(R, R̂) and w1(R, R̂) in theabsence or presence of an audit. The lender’s return Rldepends on R̂ only in the absence of audit:w0(R, R̂) = R − Rl(R̂).Define the entrepreneur’s expected reward under truthfulreporting w(R) = y(R)w0(R,R) + (1− y(R))w1(R,R).

Robert J. Gary-Bobo



CSV Model (4): Standard Debt Contract

A standard debt contract specifies a debt level D.There is no audit if debt D is repaid; an audit and noreward if D is not repaid.Formally, y(R) = 1 if R ≥ D and y(R) = 0 if R < D.w(R) = max{R − D,0}.

Robert J. Gary-Bobo



CSV Model (5): Optimal ContractWe maximize the expected income

Maximize∫ +∞

0w(R)p(R)dR,

with respect to {y(.),w0(., .),w1(., .)}, subject to theentrepreneur’s incentive constraint IC , i.e.,

w(R) = maxR̂{y(R̂)w0(R, R̂) + (1− y(R̂))w1(R, R̂)}

and the lender’s participation constraint, IR, that is,∫ +∞

0[R − w(R)− (1− y(R))K ]p(R)dR ≥ I − A.

and limited liability constraints w0(R,R) ≥ 0, w1(R,R) ≥ 0.

Robert J. Gary-Bobo



CSV Model (5b): Optimal Contract

Since IR will be binding at the optimum, we can susbsitute IR inthe objective function (the borrower’s expected profit) and wefind,∫ +∞

0w(R)p(R)dR = −K

∫ +∞

0(1−y(R))p(R)dR−(I−A)+E(R).

The objective is equivalent to minimizing the expected auditcost.

Robert J. Gary-Bobo



Analysis of the Optimal Contract (1)

Assume that audits are deterministic: y(R) = 0 or 1 for allR.Feasible values of R are divided into to regions: theno-audit region Q0 and audit region Q1 (A partition of[0,+∞)).Reimbursement R − w(R) must be constant over the noaudit region Q0.Proof. If reimbursement is higher for R′ than for R withR,R′ ∈ Q0 then, for R′ entrepreneur would prefer to reportR.Conclusion: Reimbursement must be constant, say D, ifR ∈ Q0, and Q0 ⊆ [D,+∞).

Robert J. Gary-Bobo



Analysis of the Optimal Contract (2)The reimbursement for R in Q1 cannot exceed D, for if itdid then R − w(R) > D and the entrepreneur would preferto report an income in Q0 to pay only D.Important Result : for any contract satisfying IC and IR,there exists a standard debt contract that does at least aswell for the entrepreneur.Prove this in 2 steps.Step 1. For any contract C, there exists a standard debtcontract C′ that pays out more to lenders at a smaller auditcost.Step 2. There exists a second standard debt contract C′′

that satisfies IR (lenders break even) and involving an evensmaller expected audit cost.

Robert J. Gary-Bobo



Analysis of the Optimal Contract (3)Proof of Step 1. Consider an arbitrary contract Csatisfying IC and IR, with regions Q0 (no audit) and Q1(audit). Let D be the repayment in Q0.Construct a standard debt contract in which repayment isD′ = D. Define new regions Q′

0 = [D,+∞) andQ′

1 = [0,D).The entrepreneur receives 0 in Q′

1.Since Q0 ⊆ Q′

0, the expected audit cost is weakly smaller(since Pr(Q0) ≤ Pr(Q′

0)).Expected repayment to lenders is weakly larger under C′:for Q ∈ Q0, repayment is the same, equal to D; forR ∈ Q0 ∩Q′

0 repayment is at most D under C and equal toD under C′; for R ∈ Q1 ∩Q′

1 the lender’s payoff is R − Kunder C′ and cannot be larger under C.

Robert J. Gary-Bobo



Analysis of the Optimal Contract (4)Proof of Step 2. Suppose that the standard debt contractC′ leaves a strictly positive expected profit to lenders.Then, by the intermediate value theorem, there existsD′′ < D such that the lenders’ payoff is just 0,

[1− P(D′′)]D′′ +

∫ D′′

0Rp(R)dR − P(D′′)K = I − A,

where P(D′′) = Pr(R ≤ D′′) is the cdf of p.Contract C′′ has a lower expected audit cost sinceD′′ < D′ = D, implying P(D′′)K < P(D′)K and leaves nosurplus to lenders (IR is an equality).We conclude that contract C′′ is preferred by the borrowerto the initial contract C.

Robert J. Gary-Bobo



Figure 1: Borrower’s reward

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Robert J. Gary-Bobo



Figure 2: Lender’s reward~

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CSV Model: Interpretation

The audit region Q1 is interpreted as bankruptcy region.When R < D, the entrepreneur fails to reimburse the debtand is declared bankrupt.The entrepreneur can withdraw nothing from the "cashregister" before the audit, but can fully withdraw theresidual income if there is no audit.Interpretation: the borrower can in fact steal the incomebut cannot consume it and must refund it if an audit takesplace.Alternative interpretation: the entrepreneur can, over time,transform hidden income into perks. Perks can be enjoyedif the firm is not shut down. During the bankruptcy process,the lenders recoup the value of the assets in the firm.

Robert J. Gary-Bobo

Date post:	18-Mar-2020
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Tirole’s Simple Credit Rationing Model Equity Multiplier...

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