7. Numerical Examples: Hart (1995)

There is an important difference between being observable and being contractible:

• Being contractible requires being both observable and verifiable.
• Verifiability is crucial for the enforcement of a contract.
• In a society with a good legal system, it is easy for a court to verify the truth, so the contract terms can be enforced.
• However, in some cases it is extremely hard to find an authority to be the judge, or extremely hard to make the judgment.
  • For example, temperature is easy to measure, but it is hard to judge "good weather" — what is "good weather"? Nobody can tell.
• In non-verifiable scenarios, some (not necessarily all) contract terms are not enforceable. We call this an incomplete contract problem.

The core of non-verifiability is that no commitment ex-ante is credible. So the agents have to align their incentives at some efficiency cost, and expect to renegotiate rationally based on their bargaining power at some point. We illustrate the deep issues through several numerical examples and papers below.

This section mainly focuses on the examples in Chapter 5 of Hart (1995) on debt contracts — debt is a good financing instrument in the incomplete-contract world because it has some court-enforceable features, such as liquidation in default. The four numerical examples below are cooked-up numbers to illustrate the deep ideas.

There are three periods in this economy:

• \(t=0\):
• The risk-neutral entrepreneur \(E\) has a project requiring an investment of \(I\).
• \(E\) has initial wealth \(W\), which we assume cannot be diverted for other purposes than investing in the project.
• \(E\) borrows \(B\ge I-W\) from the risk-neutral creditor \(C\) and promises to repay \(F\) (face value) at \(t=1\).
• \(E\) could borrow more than needed, i.e. \(B>I-W\). Here \(B\) is the actual cash flow \(E\) received from \(C\) at \(t=0\); \(F\) is the promised face value.
• \(t=1\): cash flow \(X_1\) and liquidation value \(L\) are realized (\(L\) is the market value/technology of liquidating assets at \(t=1\), positive only at \(t=1\)).
• The firm could be liquidated by any fraction \(\theta\in[0,1]\) for \(\theta L\) amount of cash at \(t=1\).
• If the firm is liquidated at \(t=1\), the game is over, and there is no \(t=2\) cash flow.
• As long as the face value \(F\) is repaid in full at \(t=1\), \(C\) has no rights to declare liquidation.
• If \(F\) is not repaid in full, \(C\) could liquidate the firm with enforceability from the court:
  • if \(L>F\), the firm only needs to be liquidated by a fraction \(\tfrac{F}{L}\) to repay \(F\);
  • if \(L\le F\), the firm is completely liquidated for \(L\).
• \(t=2\): the realized cash flow is \(X_2\) for the complete firm.
• If the firm was liquidated by fraction \(\theta\) at \(t=1\), the cash flow is \((1-\theta)X_2\).
• The liquidation value in this period is 0. So \(C\) would never wait until this period, since \(X_2\) will never fall into \(C\)'s pockets due to non-verifiability.

We further assume a zero discount rate and an inefficient liquidation technology, i.e.

$$L

and that \(E\) has all the bargaining power so that \(C\) always just breaks even. Bargaining power is defined mathematically as follows.

For this example, assume

$$I=90,\quad W=30,\quad L=60,\quad B=60,\quad X_1=50,\quad X_2=100$$

No extra borrowing takes place: the amount the firm borrows at \(t=0\) is exactly equal to the required amount, i.e. \(B=I-W=60\). At \(t=1\), \(E\) has two choices: default or pay.

• If \(E\) defaults:
• the firm is completely liquidated, and \(C\) gets paid \(L=60\);
• \(E\) can hide \(X_1=50\) in his pocket;
• \(E\) gets 0 at \(t=2\);
• so the total value for \(E\) is \(50\).
• If \(E\) doesn't default:
• \(C\) gets paid \(F=60\);
• \(E\) liquidates \(\tfrac16\) of the firm to get \(\tfrac16 L=10\) (since the \(X_1=50\) cash is 10 short of repaying \(F=60\), the liquidation fraction \(\theta\) satisfies \(\theta L=10\Rightarrow\theta=\tfrac16\));
• \(E\) uses the liquidation revenue together with \(X_1=50\) to repay \(F=60\);
• \(E\) gets \(\tfrac56\times100=83.33\) at \(t=2\);
• so the total value for \(E\) is \(83.33\).
• Since $83.33>50$, \(E\) will not default.

Even though the outcomes are perfectly deterministic and observable, \(E\) still needs to liquidate \(\tfrac16\) of the firm inefficiently. The core reason for this necessary inefficient liquidation is that \(E\) cannot make a credible commitment to \(C\) to pay the 10 at \(t=2\) (\(t=2\) cash flow is non-verifiable, so \(C\) cannot get it).

For this example, assume

$$I=90,\quad W=30,\quad L=50,\quad B=60,\quad X_1=100,\quad X_2=100$$

No extra borrowing takes place: \(B=I-W=60\). This is a positive-NPV project (\(\text{NPV}=200-90=110\)). However, we will finally show that \(C\) will not lend to \(E\), so the project will not be taken.

Suppose the face value is \(F\). Note \(F\ge B\) always holds for \(C\) to participate. At \(t=1\), \(E\) has two choices: default or pay.

• If \(E\) defaults:
• the firm is completely liquidated, and \(C\) gets paid \(L=50\);
• \(E\) can hide \(X_1=100\) in his pocket;
• \(E\) gets 0 at \(t=2\);
• so the total value for \(E\) is \(100\).
• If \(E\) doesn't default:
• since we assumed \(E\) has all the bargaining power, \(E\) can offer \(C\) a take-it-or-leave-it offer of \(L+\varepsilon\), which is just \(50\).
  • This offer is a renegotiation, or a soft budget constraint, which we will discuss in Example 4.
• \(C\) would have to accept this offer since it can't do better;
• therefore \(E\) pays \(C\) 50, and the total value for \(E\) is $100+100-50=150$.
• Since $150>100$, \(E\) will not default.
• \(C\) knows this is what will happen. Since \(50

Even though the outcomes are profitable, \(E\) still cannot raise the money from \(C\). The core reason for this under-investment is that \(E\) cannot make a credible commitment to \(C\) to pay back more than 50 at \(t=2\).

In Example 1 with a deterministic outcome, \(C\) is always repaid in full since \(F=B\). More generally, for a deterministic outcome, borrowing more (\(B>I-W\)) is weakly dominated (indifferent to borrowing exactly enough, \(B=I-W\)). This is true because: in the solvent case, \(E\) pays back the extra debt to \(C\) one-to-one at \(t=1\), so \(E\) cannot benefit; in the bankruptcy case, \(E\) cannot get anything, so \(E\) cannot benefit from extra borrowing either. These two cases prove that extra borrowing is weakly dominated by borrowing exactly enough.

But in the stochastic outcome case, we will show that borrowing more could be strictly dominant for \(E\) because of liquidity transfer between states. For this example, assume \(I=20\) and \(W=10\):

• With probability \(\tfrac12\), state 1 (efficient liquidation): \(X_1^1=0,\quad X_2^1=20,\quad L^1=20\)
• With probability \(\tfrac12\), state 2 (inefficient liquidation): \(X_1^2=0,\quad X_2^2=40,\quad L^2=10\)

Since there is no cash flow at \(t=1\) and strictly positive cash flow at \(t=2\), \(E\) will not intentionally choose to default at \(t=1\).

7.4.1 Case 1: Just Borrow Enough

Suppose \(E\) borrows just enough, \(B=I-W=10\); and suppose \(F=10\), which (as shown later) makes \(C\) break even. At \(t=1\):

• State 1: \(E\) has zero cash flow, but has to pay 10 to \(C\). Since \(L^1=20\), \(E\) liquidates \(\tfrac12\) of the firm (\(\theta L^1=10\Rightarrow\theta=\tfrac12\)). This liquidation is efficient since \(L^1=X_2^1=20\), so \(E\) is indifferent between liquidating and holding towards \(t=2\).
• State 2: \(E\) has zero cash flow, but has to pay 10 to \(C\). Since \(L^2=10\), \(E\) liquidates the whole firm (\(\theta=1\)). This liquidation is inefficient since \(10=L^2
• For \(C\): the payoff is 10 regardless of state, so the break-even condition implies \(F=B=10\).
• For \(E\): the payoff is

$$\frac12\times\left(\underbrace{\tfrac12\times20}_{\text{state 1}}\right)+\frac12\times\left(\underbrace{0}_{\text{state 2}}\right)=5$$

• Total payoff:

$$\underbrace{10}_{C}+\underbrace{5}_{E}=15$$

7.4.2 Case 2: Over-Borrowing

Suppose \(E\) borrows more than enough, \(B=15>I-W=10\); for now suppose \(F=20\), which (as shown later) makes \(C\) break even. At \(t=1\):

• State 1: \(E\) has zero cash flow, but has 5 from over-borrowing, and has to pay 20 to \(C\). So \(E\) is short of 15 at \(t=1\). Since \(L^1=20\), \(E\) liquidates \(\tfrac34\) of the firm (\(\theta L^1=15\Rightarrow\theta=\tfrac34\)). This liquidation is efficient (\(L^1=X_2^1=20\)), \(E\) indifferent.
• State 2: \(E\) has zero cash flow, but has 5 from over-borrowing, and has to pay 10 to \(C\) (footnote: even though the face value is 20, the final payment to \(C\) is always 10 since 10 is the outside option of \(C\) from liquidation, and \(E\) has all the bargaining power). So \(E\) is short of 5 at \(t=1\). Since \(L^2=10\), \(E\) liquidates \(\tfrac12\) of the firm (\(\theta L^2=5\Rightarrow\theta=\tfrac12\)). This liquidation is inefficient (\(10=L^2
• For \(C\): the payoff is

$$\frac12\times\left(\underbrace{20}_{\text{state 1}}\right)+\frac12\times\left(\underbrace{10}_{\text{state 2}}\right)=15$$

which verifies \(C\)'s break-even condition (\(C\) inputs \(B=15\)).

• For \(E\): the payoff is

$$\frac12\times\left(\underbrace{\tfrac14\times20}_{\text{state 1}}\right)+\frac12\times\left(\underbrace{\tfrac12\times40}_{\text{state 2}}\right)=12.5$$

• Total payoff:

$$\underbrace{15}_{C}+\underbrace{12.5}_{E}=27.5$$

7.4.3 Comparing Just-Borrowing and Over-Borrowing

Note that in Case 2 of over-borrowing, \(C\) gives \(E\) 5 more and takes back 5 more than Case 1, so comparing the total payoff between Case 1 and Case 2 makes no sense. But comparing the payoffs to \(E\) in Case 1 and Case 2 makes sense since \(E\) has exactly the same initial wealth and doesn't break even in both cases.

We can see that \(E\) is strictly better off (12.5 vs 5) when over-borrowing. The reason is:

• \(E\) has a different marginal value of liquidity in the two states; whereas \(C\) always breaks even (a constant marginal value of liquidity).
• So over-borrowing allows \(E\) to transfer liquidity from the efficient-liquidation state (\(L^1=X_2^1\), where \(E\) doesn't care about liquidity and so would like to liquidate \(\tfrac14\) more) to the inefficient-liquidation state (\(L^2

Up to now, all discussions are based on the soft budget constraint: \(C\) would accept renegotiation before liquidation, so \(E\) can always squeeze all the surplus from \(C\) by giving \(C\) a take-it-or-leave-it offer. However, in a hard budget constraint scenario, \(C\) can commit to ignoring whatever \(E\) says and always liquidate if \(F\) is not repaid in full (a credible threat). In this example, we will see that the hard budget constraint is a double-edged sword, which can both cause inefficiency and increase debt capacity. For this example, assume:

• With probability \(\tfrac12\), state 1: \(X_1^1=100,\quad X_2^1=100,\quad L^1=80\)
• With probability \(\tfrac12\), state 2: \(X_1^2=40,\quad X_2^2=100,\quad L^2=30\)

7.5.1 Case 1: Soft Budget Constraint

\(E\) could always bargain with \(C\) to make the payment never higher than the liquidation value \(L\). Suppose \(F=80\):

• in state 1, \(C\) gets repaid by 80 since \(F=L^1=80\);
• in state 2, \(C\) gets repaid by 30 since \(L^2=30\) and we assumed \(E\) has all the bargaining power.

Then, imposing \(C\)'s break-even condition, the maximum amount of borrowing (i.e. the debt capacity) is

$$B_{\max}^{\text{soft}}=\frac12\times80+\frac12\times30=55$$

7.5.2 Case 2: Hard Budget Constraint

\(E\) could do nothing other than paying back to \(C\) by \(F\) in full or being liquidated. \(C\) ignores all words from \(E\), so no renegotiation is possible, and \(E\) loses all the bargaining power. Suppose \(F=100\):

• in state 1, \(C\) gets repaid by 100 since \(F=X_1^1=100\) (\(E\) has enough cash and repays in full to avoid liquidation);
• in state 2, \(C\) gets repaid by 30 since \(L^2=30So there is efficiency loss from the hard budget constraint.

Then, imposing \(C\)'s break-even condition, the debt capacity is

$$B_{\max}^{\text{hard}}=\frac12\times100+\frac12\times30=65$$

7.5.3 Comparing Soft and Hard Budget Constraint

First, note that

$$B_{\max}^{\text{hard}}>B_{\max}^{\text{soft}}$$

which means \(E\) can raise more money under the hard budget constraint, since there will be less incentive concerns of \(C\) about possible renegotiation. But there is an efficiency cost of the hard budget constraint when state 2 realizes: the two parties could avoid the efficiency loss under the soft budget constraint by renegotiating, but can't do it with the hard budget constraint. So there is a trade-off between soft and hard budget constraints:

• if \(E\) has a project that requires a lot of cash for investment, then debt capacity is more important, which means the hard budget constraint is better for \(E\);
• if \(E\) does not require a higher level of cash, then the soft budget constraint is better since it has lower efficiency loss.