Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory 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 Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Basic Assumptions (2)

If undertaken the project either succeeds: verifiable income 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 Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Basic Assumptions(3)

The entrepreneur can exert high effort (work, take no private benefit), or zero effort (shirk, take a private benefit). Equivalently, the entrepreneur chooses between project with high or low probability of success.

High effort yields p = pH ; low effort yields p = pL, with pL < pH . Denote ∆p = pH − pL. Low effort yields a private benefit B > 0 to the entrepreneur. (B can be interpreted as a disutility of effort).

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Basic Assumptions (4)

Borrower and lenders (investors) are risk-neutral. For simplicity, there is no time preference: investors require an interest rate equal to 0 (at least). The entrepreneur is protected by limited liability (income cannot be negative). Competition among lenders drive interest and profit to zero.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Basic Assumptions(5)

The loan contract specifies how the profit is shared between borrower and lenders. Limited liability implies that both sides receive zero in case of failure.

Profit sharing: R = Rb + Rl , where Rb is the the borrower’s share, 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 thus Rb − A in case of success and −A in case of failure.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Basic Assumptions (6)

The zero-profit constraint for lenders is pH Rl = I − A. Assuming high effort, the rate of interest is ι, where 1 Rl = (1 + ι)(I − A) or 1 + ι = . 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,

pH R − 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 in expectation.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts The lender’s credit analysis (1)

The borrower’s tradeoff: obtain private benefit B but reduce probability of success to pL. We have the following incentive compatibility constraint, IC: pH Rb ≥ pLRb + B or B R ≥ . b ∆p

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts The lender’s credit analysis (2)

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

 B  p R − ≥ I − A. H ∆p

Financing can be arranged only if

 B  A ≥ I − p R − . H ∆p

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts The lender’s credit analysis (3)

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts The lender’s credit analysis (4)

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

I − A I − A B Rb = R − Rl = R − ≥ R − = , pH pH ∆p so that the entrepreneur chooses high effort. One only lends to the rich. If A ≥ A, the borrower’s utility is Ub = pH Rb − A = pH (R − Rl ) − A = pH R − I: the borrower receives the entire social surplus of the investment.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts 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 and the likelihood ratio ∆p/pH . The likelihood ratio measures how much the observable result (success or failure) reveals the underlying choice of effort.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Do investors hold debt or equity?

Remark that the loan contract can be interpreted as debt here. This is because we have a two-outcome model.

Interpretation as debt: the borrower must reimburse Rl or else go bankrupt. In the case of reimbursement, the borrower keeps the residual R − Rl .

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Costly State Verification Model

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Costly State Verification Model (2)

The entrepreneur invests his(her)money A. Investment cost is I. Investment yields a random income R distributed on [0, +∞), with density p(R). The entrepreneur observes R without cost. Timing: 1. Loan agreement, I is sunk. 2. Income R is realized. 3. Entrepreneur reports Rˆ . 4. Lender may decide to audit. 5. Reimbursement. We apply the Revelation Principle: there is no loss of generality if we focus on revealing mechanisms, i.e., contracts such that the entrepreneur has an incentive to report the true income Rˆ = R.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts CSV Model (3): Definition of a Contract

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts 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 no reward 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 Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts CSV Model (5): Optimal Contract We maximize the expected income Z +∞ Maximize w(R)p(R)dR, 0

with respect to {y(.), w0(., .), w1(., .)}, subject to the entrepreneur’s incentive constraint IC , i.e., ˆ ˆ ˆ ˆ w(R) = max{y(R)w0(R, R) + (1 − y(R))w1(R, R)} Rˆ and the lender’s participation constraint, IR, that is,

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

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

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts CSV Model (5b): Optimal Contract

Since IR will be binding at the optimum, we can susbsitute IR in the objective function (the borrower’s expected profit) and we find, Z +∞ Z +∞ w(R)p(R)dR = −K (1−y(R))p(R)dR−(I−A)+E(R). 0 0 The objective is equivalent to minimizing the expected audit cost.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Analysis of the Optimal Contract (1)

Assume that audits are deterministic: y(R) = 0 or 1 for all R. Feasible values of R are divided into to regions: the no-audit region Q0 and audit region Q1 (A partition of [0, +∞)). Reimbursement R − w(R) must be constant over the no audit region Q0. Proof. If reimbursement is higher for R0 than for R with 0 0 R, R ∈ Q0 then, for R entrepreneur would prefer to report R. Conclusion: Reimbursement must be constant, say D, if R ∈ Q0, and Q0 ⊆ [D, +∞).

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Analysis of the Optimal Contract (2)

The reimbursement for R in Q1 cannot exceed D, for if it did then R − w(R) > D and the entrepreneur would prefer to 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 as well for the entrepreneur. Prove this in 2 steps. Step 1. For any contract C, there exists a standard debt contract C0 that pays out more to lenders at a smaller audit cost. Step 2. There exists a second standard debt contract C00 that satisfies IR (lenders break even) and involving an even smaller expected audit cost.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Analysis of the Optimal Contract (3) Proof of Step 1. Consider an arbitrary contract C satisfying 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 is 0 0 D = D. Define new regions Q0 = [D, +∞) and 0 Q1 = [0, D). 0 The entrepreneur receives 0 in Q1. 0 Since Q0 ⊆ Q0, the expected audit cost is weakly smaller 0 (since Pr(Q0) ≤ Pr(Q0)). Expected repayment to lenders is weakly larger under C0: for Q ∈ Q0, repayment is the same, equal to D; for 0 R ∈ Q0 ∩ Q0 repayment is at most D under C and equal to 0 0 D under C ; for R ∈ Q1 ∩ Q1 the lender’s payoff is R − K under C0 and cannot be larger under C.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Analysis of the Optimal Contract (4) Proof of Step 2. Suppose that the standard debt contract C0 leaves a strictly positive expected profit to lenders. Then, by the intermediate value theorem, there exists D00 < D such that the lenders’ payoff is just 0,

Z D00 [1 − P(D00)]D00 + Rp(R)dR − P(D00)K = I − A, 0 where P(D00) = Pr(R ≤ D00) is the cdf of p. Contract C00 has a lower expected audit cost since D00 < D0 = D, implying P(D00)K < P(D0)K and leaves no surplus to lenders (IR is an equality). We conclude that contract C00 is preferred by the borrower to the initial contract C.

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Figure 1: Borrower’s reward _ ~ o ~

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts Figure 2: Lender’s reward 1 ~ ~ 1 ~I - ~ ~ ~ ~ " ' ~ - D , f\) ~C) ~ ~~ ~------~----~~

Robert J. Gary-Bobo Introduction Tirole’s Simple Credit Rationing Model Equity Multiplier Theory of Standard Debt Contracts CSV Model: Interpretation

The audit region Q1 is interpreted as bankruptcy region. When R < D, the entrepreneur fails to reimburse the debt and is declared bankrupt. The entrepreneur can withdraw nothing from the "cash register" before the audit, but can fully withdraw the residual income if there is no audit. Interpretation: the borrower can in fact steal the income but cannot consume it and must refund it if an audit takes place. Alternative interpretation: the entrepreneur can, over time, transform hidden income into perks. Perks can be enjoyed if 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