9. Incomplete Contract and Corporate Control Rights

In this section, we will discuss Aghion and Bolton (1992), which explicitly talks about corporate control rights, and how this right affects efficiency.

Timing: there are three periods.

• \(t=0\):
• Entrepreneur \(E\) has a project requiring investment of \(I\).
• \(E\) has no wealth, so he raises money of \(I\) from a creditor \(C\).
• \(E\) and \(C\) sign a contract specifying corporate control at \(t=1\).
• \(t=1\):
• The party specified in the contract controls the firm, and chooses an action \(a\in\mathcal A\) where \(\mathcal A\) is some action space.
• \(t=2\):
• The project return \(X(a)\) is realized to the public, and the private benefit \(B(a)\) is private information only revealed to \(E\).
• \(C\) takes away all of \(X(a)\), and \(E\) takes away all of \(B(a)\).
  • Assume that investment is efficient, i.e. \(\max_a\{X(a)+B(a)\}>I\).

The first-best solution should maximize the total surplus by choosing the most efficient \(a^\star\), i.e.

$$a^\star\in\arg\max\{X(a)+B(a)\}$$

No-negotiation baseline: \(E\) chooses \(a^E\) optimally for himself to maximize only \(B(a)\), i.e.

$$a^E\in\arg\max B(a)$$

This implies

$$X(a^\star)\ge X(a^E)\quad\text{and}\quad B(a^\star)\le B(a^E)$$

which is true because:

• \(E\) can always at least choose \(a^\star\), so \(a^E\) cannot be worse than \(a^\star\) to \(E\), i.e. \(B(a^E)\ge B(a^\star)\);
• \(a^E\) is away from the optimal action \(a^\star\), so the total surplus becomes smaller, i.e.

$$X(a^\star)+B(a^\star)\ge X(a^E)+B(a^E)$$

• then, it must be that \(C\) is losing away from \(X(a^\star)\) (i.e. \(X(a^E)\le X(a^\star)\)).

This introduces two types of inefficiencies:

• Ex-ante inefficiency: it is possible that \(X(a^E)under-investment problem.
• Ex-post inefficiency: \(E\) chooses an inefficient action \(a^E\) instead of the efficient \(a^\star\).

Suppose \(E\) has all the bargaining power, then:

• \(C\) would like to bribe \(E\) to choose \(a^\star\) by compensating him up to

$$X(a^\star)-X(a^E)$$

• In the end, \(C\) always has a payoff of \(X(a^E)\) (\(E\), with all the bargaining power, grabs the entire gain from the bribe).
• This resolves the ex-post inefficiency since the action is now optimally chosen as \(a^\star\).

No-negotiation baseline: \(C\) chooses \(a^C\) optimally for himself to maximize only \(X(a)\), i.e.

$$a^C\in\arg\max X(a)$$

This implies

$$X(a^\star)\le X(a^C)\quad\text{and}\quad B(a^\star)\ge B(a^C)$$

which is true because:

• \(C\) can always at least choose \(a^\star\), so \(a^C\) cannot be worse than \(a^\star\) to \(C\);
• \(a^C\) is away from the optimal action \(a^\star\), so the total surplus becomes smaller, i.e.

$$X(a^\star)+B(a^\star)\ge X(a^C)+B(a^C)$$

• then, it must be that \(E\) is the one losing (i.e. \(B(a^C)\le B(a^\star)\)).

This also introduces two types of inefficiencies:

• Ex-ante inefficiency: it is possible that \(X(a^C)
• Ex-post inefficiency: \(C\) chooses an inefficient action \(a^C\) instead of the efficient \(a^\star\).

Suppose \(E\) has all the bargaining power:

• \(E\) would like to bribe \(C\) to choose \(a^\star\) by compensating him up to

$$B(a^\star)-B(a^C)$$

• However, \(E\) has no cash, so he cannot bribe \(C\) at all.
• So, there is no way for them to ex-post renegotiate. Therefore, the ex-post inefficiency could not be solved.

Recall from Case 1 and Case 2:

• When \(E\) controls the firm: (after ex-post bribery) the efficient \(a^\star\) is chosen; \(C\) gets \(X(a^E)\) and \(E\) gets \(B(a^E)\) (baseline private benefit).
• When \(C\) controls the firm: the inefficient \(a^C\) is chosen; \(C\) gets \(X(a^C)\) and \(E\) gets \(B(a^C)\).

Given that \(E\) has all the bargaining power, \(C\) always breaks even. So, higher total surplus implies higher individual surplus to \(E\) (\(E\) is the residual claimant on the total surplus net of \(C\)'s input).

• Since it attains efficiency when \(E\) controls the firm, the more efficient outcome should be that \(E\) controls the firm as much as possible.
• Assume that we flip a coin, which yields probability \(p\) of \(E\) controlling the firm, and probability \(1-p\) of \(C\) controlling the firm.
• So, for efficiency purposes, we need to make \(p\) as high as possible such that \(C\) breaks even, i.e.

$$I=pX(a^E)+(1-p)X(a^C) \tag{9.1}$$

• In conclusion, the optimal choices are:
• When projects are profitable, i.e. the worst payoff to \(C\) satisfies \(X(a^E)\ge I\), \(E\) should have the control (\(p=1\)).
• When projects are bad, i.e. the best payoff to \(C\) satisfies \(X(a^C)=I\), to make the creditor have an incentive to invest, \(C\) has to have the firm control (\(p=0\)).
• When projects are not good not bad, i.e. \(X(a^E)

Suppose that the payoff \(X\) is also a function of a verifiable random state variable \(s\) in addition to action \(a\), i.e. \(X(a,s)\).

• We still let \(E\)'s private benefit \(B(a)\) be only a function of action \(a\).
• \(s\) could be interpreted as earnings, which is not the cash \(X\) itself, but a signal closely related to cash.

Assume that \(X(a,s)\) is determined by \(a\) and \(s\) as follows:

$$X(a,s)=f(s)\cdot a+b(s) \tag{9.2}$$

where \(f(\cdot)\) and \(b(\cdot)\) are some continuous functions such that \(f(s)>0\) and \(f'(s)<0\) for \(\forall s\).

• the assumption \(f(s)>0\) ensures that action always has a positive effect on \(X\);
• the assumption \(f'(s)<0\) means that \(a\) becomes less important when \(s\) is high enough.

Still keep the assumption that

$$X(a^E,s)

which means:

• at \(C\)'s optimal action, investment is attractive to \(C\) for any state \(s\);
• at \(E\)'s optimal action, investment is not attractive to \(C\) for any state \(s\).

The state-contingent control regime requires that \(E\) controls the firm for \(s\ge T_s\), and \(C\) controls for \(s

• If \(T_s\) is very high, \(C\) would always control the firm. Then, by (9.3), \(C\) would always strictly want to invest.
• If \(T_s\) is always low, \(E\) would always control the firm. Then, by (9.3), \(C\) would not want to invest.
• So, from (9.2), since \(X(a,s)\) is a continuous function of \(a\) and \(s\) changes continuously, we can see there must exist some threshold \(\bar T_s\) such that \(C\) breaks even.

Recall the fact that \(C\) would take over control rights at low state \(s\). The assumption \(f'(s)<0\) simply means \(C\) would have control when the action is very important (low \(s\Rightarrow\) high \(f(s)\)). This is good because \(C\) would put in a more efficient action since the action becomes more important. So, such threshold control switching could reduce the efficiency loss from \(C\)'s control.

Drawbacks of Aghion and Bolton (1992):

• It is very hard to justify the assumption \(f'(s)<0\);
• Aghion and Bolton (1992) does not link control switching with debt-repayment default;
• This intuition is wrong:
• Aghion and Bolton (1992) implies that \(E\) always gives the efficient action and that it is \(C\) who introduces inefficiency;
• however, in reality, this is always flipped: \(E\) always introduces inefficiency.