Second-Best Contract in Period Two and Firing Decision

Consider the contracting problem in the second period. For period two a non-commitment contract is infeasible. After period two the employment of the CEO at the firm ends and the firm will not pay compensation, when it is not enforceable in court. The CEO anticipates this behavior of the firm and only accepts a contract that is completely enforceable in period two.¹³ In this scenario the firm and the CEO sign a contract that includes the base salary, a, and the participation rate, b, of the CEOs compensation.

If the firm discontinues employment of the CEO after period one, the probability for a good match between the new CEO and the firm is again q. When the CEO stays, the probability of the firm to employ a good match isp(eix) instead ofq. The CEO’s expectation to be a good match in period two is pi, with i ∈ {H, L}, and depends on the match quality in the first period. This incorporates the possibility that the match quality is identical in both periods and that the match quality changes, for example due to an external shock.¹⁴ The optimization problem for a new and an incumbent CEO only differ in the probabilities for a good and a bad match. The optimization problem below derives the optimal contract for an incumbent CEO. Simply replacingp(eix) andpi withqin the optimal second period contract gives the optimal contract for a new CEO.

Considering the uncertainty about the match quality, represented by the beliefs of the firm and the CEO, the firm’s optimization problem for the second period is

maxa,b E_P^{F C} = p(eix)[(1−b)µHeH] + (1−p(eix))[(1−b)µLeL]−a, (3.9) s.t. EU_A^{F C} = pi[bµHeH +a−e²_H

2 ] + (1−pi)[bµLeL+a− e²_L

2 ]≥U, (3.10) bµiei+a− e²_i

2 ≥U,∀i∈ {H, L}, (3.11) and e^{F C}_i = bµi,∀i∈ {H, L}. (3.12)

13A contract with discretionary variable compensation is infeasible, because the CEO anticipates that the firm refuses to pay and consequently exerts zero effort under such a contract.

14If the match quality is the same in both periods and the CEO is a good match in the first period,p_H= 1 andp_L= 0. For a bad match in the first period,p_H= 0 andp_L= 1. If the match quality may change from period one to period twop_i∈(0,1) reflects the beliefs for the two match qualities.

The net profit of the firm (3.9) reflects the uncertainty about the match quality at the time of contracting. In the second period the firm infers the probability of a good match for an incumbent CEO, p(eix), from the realization of the firm profit in period one, eix. The firm’s expected payoff again consists of expected residual firm profits minus the base salary. The first part of (3.9) is the payoff from a good match, the second part is the payoff from a bad match, both multiplied with the probability of the respective match quality, p(eix) and 1−p(eix). (3.10) represents the individual rationality constraint of the CEO at the time of contracting. The expected utility of the CEO is the expected payoff from compensation minus effort costs, weighted with the probabilities of a good and a bad match. The probability of the CEO for the match quality in the second period, pi, depends on the match quality she observes in the first period,i∈ {H, L}. Equation (3.11) is the minimum utility constraint. It ensures that the first-best solution is unfeasible and states that the CEO receives the reservation utility independent of the match quality. As the firm does not know pi, it could try a signaling mechanism to find out pi. I do not consider such a mechanism. Asbµiei+a−^e₂²ⁱ is lower for L, given (3.12), restrictions (3.10) and (3.11) can be replaced by restriction (3.10) for i =L only. This implies that pi is irrelevant for the optimization problem (3.9) to (3.12). Condition (3.12) represents the optimal effort choice of the CEO, which balances the marginal costs and the marginal benefit of effort as in first-best.

The principal has to solve the problem (3.9) to (3.12) given that he offers the same contract to both types of CEOs. The following proposition summarizes the second-best contracts for an incumbent CEO and for a new CEO for period two with full commitment.

Proposition 2 The optimal second-best contract with full commitment in the sec-ond period gives the an incumbent CEO a shareb^{F C} = _E[µ2]+p(e^E[µi²x)∆(µ^{] 2}), with∆(µ²) = µ²_H −µ²_L, of firm profit and a base salary of a^{F C} =U −^{(bF C}₂^µL)2. To receive the full commitment contract for a new CEO replace p(eix) with q.

Proof: See appendix.

Optimal variable compensation of the CEO under full commitment, b^{F C}, is smaller in second-best than variable compensation in first-best. The numerator of b^{F C} is smaller than the denominator as long as the two match types differ (∆(µ²) > 0).

This implies b^{F C} < 1 and therefore the participation rate as well as the optimal

effort of the two match types under the full commitment contract in period two are lower than in first-best.

In the second-best world the firm cannot offer type-specific contracts. Satisfying the utility constraint (3.11) with equality for both match types is infeasible when both match types receive the same base salary. The expected utility of a good match CEO is larger than her reservation utility, whereas the bad match only receives the reservation utility. This gives the good match CEO a rent. The firm faces the problem that providing stronger incentives increases the rent of the good match CEO. The optimal contract balances the costs for the rent of the good CEO and the benefits from providing incentives. Consequently, the optimal second-best variable compensationb^{F C} declines when the difference in expected productivity of the two types of CEOs ∆(µ²) rises. The rent of the good CEO increases, when the difference between the two match types increases. This makes providing incentives more costly for the firm and decreases the optimal b^{F C}.

It remains to characterize the firing decision of the firm. The firm fires the old CEO, if the expected second period profit of a new CEO is larger than the expected second period profit from the incumbent CEO plus the non-committed compensation in the first period

E_P^{F C}(q, a^{F C}_q , b^{F C}_q )> E_P^{F C}(p(eix), a^{F C}_p(eix), b^{F C}_p(eix)) +w^{N C},

withEP as given in equation (3.9). w^{N C} is the non-committed compensation in the first period, which is equal toa^{V C} orb^{P C}eix, when the firm does not commit to the base salary or the bonus, respectively.¹⁵ The firm offers the CEO the optimal second period-contract{a^{F C}, b^{F C}}, which depends on the probability of a good match q or p(eix), respectively. This condition implies that the firm fires the CEO when the realized first period firm profit suggests that the probability of a good match is larger for a new CEO than for an incumbent CEO,q > p(eix) and the firm chooses the full commitment contract in the first period. This is equivalent to saying that for all forms of commitment there exists some threshold for the firm profit such that the firm fires the CEO when realized profit is lower and keeps the CEO when realized profit is higher. The CEO knows this and takes the firing decision as given.

For the firm the firing decision is essentially an investment problem. It has to decide whether or not to invest the non-committed compensation to retain the

15The values for a^{V C} andb^{P C} are derived in the following sections.

CEO. The potential reward is the excess profit an incumbent CEO yields compared to a new CEO. The firm invests the non-committed compensation, if the difference between the payoff of the incumbent and the new CEO is larger than the non-committed compensation.¹⁶