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Relational contracts and the first-order approach
Journal of Mathematical Economics 63 (2016) 126–130
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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Relational contracts and the first-order approach Sunjoo Hwang ∗ Department of Competition Policy, Korea Development Institute, Namsejong-ro 263, Sejong City, 339-007, Republic of Korea
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Article history: Received 12 October 2015 Received in revised form 13 January 2016 Accepted 10 February 2016 Available online 18 February 2016 Keywords: Relational contracts First-order approach Local convexity of distribution function condition
abstract This paper justifies the first-order approach (FOA) to relational contract models. Optimal relational contracts pay a bonus if an agent passes an evaluation, where the cutoff point is independent of signal distribution or the agent’s cost function. Based on this independence, I find a weak FOAjustifying condition, which requires convexity of the underlying distribution-cost structure only at the cutoff point. Prominent examples (e.g., the normal or generalized error distribution with various cost functions) are consistent with this condition, but are inconsistent with existing conditions such as the Mirrlees–Rogerson condition. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Standard formal contract models cannot address many realworld incentive problems in which performance measures are non-verifiable, contracts are easily manipulated, or legal contracting is highly costly or infeasible. Relational contract models have arisen as new tools to address these complicated incentive issues.1 Relational contract models, however, have a technical issue— the need to justify the first-order approach (FOA). A principal–agent problem, whether with formal or relational contracts, is often intractable since an agent’s incentive-compatibility constraint (IC) is an abstract maximization constraint involving a set of endogenous variables (a contract). For tractability, one might use the FOA, replacing the IC with the corresponding first-order condition (FOC). However, the literature on formal contracts considers justifying the FOA a challenge due to the following reasons. First, Mirrlees (1974, 1999) shows that solutions do not exist if there are no lower bounds on payments and signals. Second, even if solutions exist, most well-known signal distributions fail to satisfy the CDFC, a standard FOA-justifying condition proposed by Mirrlees (1979) and Rogerson (1985).2 Third, the CDFC is not applicable when a signal is multi-dimensional. Therefore, the literature
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1 Levin (2003) provides a formal theoretic framework. Baker et al. (1999); Board (2011), and Cebon and Hermalin (2014) apply relational contracts to various topics such as optimality of informal authority, endogenous formation of inner circles, and efficiency of executive-pay limit, respectively. 2 See LiCalzi and Spaeter (2003) for cases consistent with the CDFC. http://dx.doi.org/10.1016/j.jmateco.2016.02.003 0304-4068/© 2016 Elsevier B.V. All rights reserved.
on formal contracts focuses on signals with lower bounds and seeks more general FOA-justifying conditions in multi-signal cases.3 To the best of my knowledge, the most general condition ever found is the CDFCL proposed by Jung and Kim (2015). This paper justifies the first-order approach (FOA) to relational contract models. I use the following two-step approach. First, I show that, if the FOC is binding, the optimal contract in a relaxed principal–agent problem in which the IC is replaced with the FOC is Levin’s bang–bang contract, which pays a fixed bonus if an agent passes an evaluation. Second, I find a FOA-justifying condition, the Local Convexity of Distribution Function Condition (LCDFC), under which the agent’s objective function is concave-in-effort given the bang–bang contract. Hence, the FOC and IC are equivalent.4 The tricky part is that if the FOC is non-binding, optimal contracts in the relaxed problem may not be the bang–bang contract. One might then seek a condition that ensures the concavity given a contract of unknown form. This should be a difficult task. However, there is a way around it: I can use the fact that for any contract that solves the relaxed principal–agent problem, there is a payoff-equivalent bang–bang contract that solves the same problem. Therefore, the LCDFC ensures the concavity whether the FOC is binding or not. Hence, it justifies the FOA.
3 Jewitt (1988) finds a set of FOA-justifying conditions when there are two independent signals. For a general multi-signal case, Sinclair-Desgagne (1994) finds two conditions (GCDFC and GSDC) that jointly justify the FOA. Conlon (2009) generalizes Jewitt and Sinclair-Desgagne’s results by suggesting a less restrictive condition (CISP). Recently, Jung and Kim (2015) find an even weaker condition (CDFCL). 4 Note that the LCDFC and CDFCL are different. L of the LCDFC stands for ‘local,’ whereas L of the CDFCL stands for ‘likelihood ratio.’
S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Note the following three positive properties of relational contract models and the LCDFC. First, Mirrlees’ non-existence result does not hold, since the self-enforcement requirement of relational contracts imposes endogenous bounds on payments. Therefore, signals without lower bounds can also be used. Second, the LCDFC is weaker than the existing conditions.5 A good thing with the bang–bang contract is that the cutoff point of pass/fail is zero (in terms of the likelihood ratio) for any distribution, cost function, or the dimension of a (vector) signal. Due to this independence, the LCDFC turns out to be convexity of the underlying distribution-cost structure only at the singular point of zero. In contrast, the CDFC and CDFCL require convexity for all points.6 Prominent examples (e.g., the normal or generalized error distribution with various cost functions) are consistent with the LCDFC, but are inconsistent with both the CDFC and CDFCL. Third, the LCDFC justifies the FOA in both single-signal and multi-signal cases in the same way. It is because the LCDFC is defined on the likelihood ratio of a signal distribution. The likelihood ratio is onedimensional even if the underlying signal is multi-dimensional. In contrast, the CDFC is a condition on the signal itself. Hence, it cannot be directly applied to multi-signal cases.7 Justifying the FOA in multi-signal cases could be more meaningful with relational contracts than with formal contracts. Since relational contracts are informal contracts, they can utilize not only objective but also subjective, non-verifiable, and internal information. Multidimensional evaluation is thus natural in relational contracting. Kvaloy and Olsen (2014) independently find a similar result in a special case.8 They consider an additive-normal case in which a one-dimensional output signal is the sum of an effort and a normally-distributed error. In this case, they show (in their Proposition 3) that the FOA is valid if the variance of the error is ‘sufficiently large.’ However, they do not provide a characterizing condition, such as the LCDFC, which determines the required degree of ‘largeness.’ When the cost function is iso-elastic, they provide (in the Remark following their Proposition 3) a characterizing condition, which is identical with the LCDFC if a constant term is adjusted. See Example 1.1 in this paper. In contrast, this paper justifies the FOA under general distributions and cost functions. For instance, Example 1.2 shows that the LCDFC can be satisfied when the error follows the generalized error distribution (which includes the normal distribution). Additionally, Example 2 shows that the LCDFC can be used with multi-signal cases. See Section 4 for more details. The organization of the paper is as follows. Section 2 presents the underlying relational contract model. Section 3 provides a FOA-justifying condition, the LCDFC. Section 4 compares the LCDFC to the existing conditions and provides consistent examples. Section 5 draws a conclusion. 2. The model The following is Levin’s standard relational contract model with a modification that an output signal is multi-dimensional. There
5 The LCDFC is weaker than the CDFCL. In single-signal cases, the LCDFC is weaker than the CDFC if the MLRP holds. For the MLRP, see Assumption 1. 6 Let F (x|a) be the distribution of a signal x given an effort a. Let κ = c (a) be the cost of the effort. Suppose that x is one-dimensional and that the MLRP holds. Then, the LCDFC holds if and only if F (x|c −1 (κ)) is convex in κ at x = xf , where xf is to be defined. In contrast, the CDFC (and CDFCL) holds if and only if F is convex at all x. 7 Jung and Kim (2015) discuss the virtue of considering conditions on the likelihood ratio rather than on the underlying signal in justifying the FOA in multisignal cases. 8 They do not focus on justifying the FOA but on characterizing optimal relational contracts when there is a team of agents. Their findings are comparable to this paper if the number of agents is one.
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are two risk-neutral players, a principal (she) and an agent (he). The discount factor is δ ∈ (0, 1). In each period t ∈ {1, 2, . . .}, they play the following stage game. The agent contributes to the output yt ∈ [0, ∞), which is nonverifiable. The principal can offer the agent a long-term informal contract, which depends on the history of a vector of non-verifiable signals, xt ∈ X ⊂ Rn . xt may or may not include yt .9 If the agent accepts the contract, he chooses an effort at ∈ [0, ∞), where the cost of the effort c (at ) is such that c (0) = 0, c ′ > 0, c ′′ > 0, limat →0 c ′ (at ) = 0, and limat →∞ c ′ (at ) = ∞. After the effort choice, the output yt and signal xt are realized according to distributions π (yt |at ) and f (xt |at ), respectively, where f (xt |at ) > 0 for all xt ∈ X and at and fa being the partial derivative. After the realization of xt , either the principal or the agent can choose whether or not to honor the informal contract. The whole principal–agent problem is a repeated game with imperfect public monitoring under a complicated structure. However, Levin (2003) shows that it can be greatly simplified to a static problem if the signal is one-dimensional, since for any non-stationary solution to the repeated problem, there exists a payoff-equivalent stationary solution. The stationarity result analogously holds in multi-signal cases. See Levin (2003) for more details. Therefore, I consider only stationary contracts without loss of generality. Time-subscripts are suppressed. Let w ∈ R be a verifiable fixed wage and b(x) ∈ R be a non-verifiable bonus schedule. Let u¯ P and u¯ A be the principal and agent’s per-period reservation payoffs, respectively. Let E[·|a] denote the conditional expectation. The stationary principal–agent problem is given by max uP ≡ E[y|a] − E[w + b(x)|a]
s.t .
a,w,b(x)
uP ≥ u¯ P
(2.1)
u ≡ E[w + b(x)|a] − c (a) ≥ u¯ A
a ∈ arg max a′
b(x) ≤
A
(2.2)
E[w + b(x)|a′ ] − c (a′ )
δ [uP − u¯ P ], 1−δ
−b(x) ≤
(IC)
δ [uA − u¯ A ] ∀x. (2.3) 1−δ
(2.1) and (2.2) are the principal and agent’s participation constraints, respectively. (IC) is the agent’s incentive-compatibility constraint, which states that an effort a can be induced if it maximizes the agent’s payoff. (2.3) ensures that informal contracts are self-enforced: the first inequality in (2.3) states that the principal’s short-term gain from reneging on a bonus payment is smaller than her long-term loss from the relationship breaking-up.10 The second inequality similarly describes that the agent has no incentive to renege on a (negative) bonus payment. Since the principal has full bargaining power (i.e., she maximizes her payoff) and there is no restriction on the fixed wage w , it can be easily shown that the agent’s participation constraint (2.2) is binding in optima.11 Thus, the binding condition of (2.2) characterizes the optimal fixed wage:
w = u¯ A + c (a) − E[b(x)|a].
(BIR)
9 The model can be easily extended in a way that x includes verifiable but costlyt to-verify signals. 10 Alternatively, one might consider different plays on off-the-equilibrium paths. However, the main result of this paper depends only on plays on the equilibrium path. 11 If the labor market for agents is competitive, the agent would have full bargaining power. However, the main results of this paper are robust to reallocation of bargaining powers.
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S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Then, uP equals yπ (y|a)dy − c (a) − u¯ A . Let s(a) denote the (per-period) total surplus:
s(a) ≡
yπ (y|a)dy − c (a).
I assume that s(a) is strictly concave and there is a unique first-best effort a∗ > 0, which maximizes s(a). Let s¯ be the sum of the two players’ reservation payoffs. Thus, (2.3) is reduced to the following self-enforcement constraint: 0 ≤ b(x) ≤ V (a) ≡ max
δ 1−δ
[s(a) − s¯], 0
∀x
max s(a) subject to (BIR), (SE) and (IC).
a,w,b(x)
(SE)
(P1)
Note that w does not play any role in the problem above: s(a), (SE) and (IC) are independent of w . Hence, I ignore w and (BIR) in the following analysis. 3. First-order approach justification The original principal–agent problem (P1) is often intractable since (IC) is an abstract maximization constraint involving a set of endogenous variables b(x). For tractability, one might use the first-order approach (FOA), replacing (IC) with the following firstorder condition (FOC). Let Cov(·|·) be the conditional covariance.13 fa (x|a)
a = c ′ (a). b(x)fa (x|a)dx = Cov b(x), f (x|a)
0 bf (x) = ∈ 0, V (af ) V ( af )
if if if
where V (a) is the value of the relationship. Note that (SE) imposes endogenous bounds on payment b(x). Relatedly, Mirrlees (1999) shows that formal contract models have no solutions if payments and signals are not bounded from below.12 This non-existence result does not hold in relational contract models. Thus, finally, the principal–agent problem is simplified to the following surplus maximization problem:
indeterminate: bf (x) is any function that satisfies (FOC) and (SE) at a∗ . (iii) If µ > 0, bf (x) is such that
(FOC)
Note that (FOC) is weaker than (IC) (if the equilibrium effort is positive). In general, the following relaxed principal–agent problem (P2) has a greater choice set than (P1). Hence, a solution to (P2) is not necessarily a solution to (P1). Thus, the FOA is not valid in general. max s(a) subject to (SE) and (FOC). a,b(x)
(P2)
I use a two-step procedure to justify the FOA. First, in Lemma 1, I characterize optimal contracts in the relaxed problem (P2) without assuming any conditions that justify the FOA. These contracts are optimal in the original problem (P1), if the agent’s objective function is concave-in-effort given these contracts. Proposition 1 provides a sufficient condition, the Local Convexity of Distribution Function Condition (LCDFC), which ensures the concavity. In other words, the LCDFC justifies the FOA. Lemma 1. Let µ denote the Lagrange multiplier of (FOC). Suppose there exists a solution (af , bf (x)) to (P2) such that af > 0. Then, (i) µ ≥ 0. (ii) If µ = 0, then af = a∗ and bf (x) is
f (x|a)
ratio fa(x|a) is zero for any density function f .
f (x|af ) fa (x|af ) f (x|af ) fa (x|af ) f (x|af )
<0 (3.1)
=0 > 0.
Proof. See Appendix. The contract bf (x) in Lemma 1 has the bang–bang shape if µ > 0: a fixed bonus is paid if and only if the likelihood ratio is positive. Note that this result is obtained independently of the Mirrlees–Rogerson condition. In contrast, Levin (2003) obtains the same result assuming the following version of the Mirrlees–Rogerson condition. Assumption 1. (a) The Monotone Likelihood Ratio Property f (x|a) (MLRP): fa(x|a) is increasing in x for all a.14 (b) Levin’s version of the
Convexity of Distribution Function Condition (CDFC): F (x|c −1 (κ)) is convex in κ for all x, where κ = c (a).15
Contract theorists typically assume the MLRP.16 However, the MLRP plays no role in the relational contract model, since it is a condition on the level of the likelihood ratio rather than the sign, which is a sufficient statistic that captures all relevant information. The following proposition shows that the LCDFC validates the FOA. The basic idea is as follows. The LCDFC is a condition that ensures that the agent’s objective function is concave-in-effort given the bang–bang contract in Lemma 1. The tricky part is that if µ = 0, a contract that solves (P2) may be not a bang–bang contract. Thus, in this case, one might seek a condition that ensures the concavity given a contract with an unknown form. This task should be difficult. However, there is a way around it: I can use the fact that for any contract that solves (P2), there is a payoff-equivalent bang–bang contract that solves (P2). Thus, the LCDFC ensures the concavity for any µ. Hence, it justifies the FOA. Proposition 1. Suppose the following condition holds.
P
fa (x|af ) f (x|af )
≤ 0 c −1 (κ) is convex in κ
(LCDFC)
where af > 0 is an equilibrium effort of (P2) and κ = c (a).17 Then, the first-order approach is valid. In other words, if there is a solution to (P2) that induces a positive effort af , it is also a solution to (P1).18
14 In the literature on contract theory, ∂ log f (x|a) is often called the likelihood ∂a ratio of f (x|a). 15 Mirrlees’ original CDFC requires F (x|a) being convex in a. Levin’s version incorporates convexity of cost function as well. 16 Although the MLRP is less restrictive than the CDFC, it is still a restrictive condition. In single-signal cases, the MLRP is stronger than the hazard rate dominance. The hazard rate dominance is stronger than the first-order stochastic dominance. The first-order stochastic dominance is stronger than the second-order stochastic dominance. In multi-signal cases, the MLRP is increasingly restrictive as the dimension n of a multi-signal x = (x1 , . . . , xn ) rises. Conlon (2009) uses the multi-dimensional MLRP. 17 For clarity, note that P fa (x|af ) ≤ 0 c −1 (κ) = f (x|c −1 (κ))dx. f (x|af )
12 If the principal pays the agent an arbitrarily low payment when the agent’s performance is arbitrarily low, the first-best outcome can be approximated, though it cannot be reached. 13 The covariance representation obtains since the expectation of the likelihood
fa (x|af )
fa (x|af ) f (x|af )
≤0
18 A necessary and sufficient condition of FOA-justification may not be attainable. As an analogue, consider a simple unconstrained optimization problem. The first-order condition leads to a global optimum if the objective function is concave. However, an ‘only-if’ result is difficult to obtain since the first-order condition generates a global optimum even if the objective function has a non-conventional form.
S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Proof. Since a = c −1 (κ), the problem of choosing a is equivalent to the problem of choosing κ . Let (af , bf (x)) be a solution to (P2) such that af > 0. Suppose af ̸= a∗ . Then, from the facts ‘µ ≥ 0’ and ‘µ = 0 implies af = a∗ ,’ it follows that µ > 0. Lemma 1 then implies that V (af )1 fa (x|af ) is an optimal contract. Generically, it is a unique f (x|af )
>0
optimal contract. Then, I can assume, without loss of generality, that bf (x) = V (af )1 fa (x|af ) . The agent’s objective function then >0
f (x|af )
equals (ignoring the constant w f ) fa (x|af )
bf (x),
fa (x|a∗ ) ∗ a f (x|a∗ )
=
fa (x|a∗ ) >0 f (x|a∗ )
bf (x)
fa (x|a )
fa (x|a∗ ) ≤0 f (x|a∗ )
b (x) f
+ ≤
f (x|a∗ )
fa (x|a∗ ) >0 f (x|a∗ )
V (af )
+
fa (x|a∗ ) ≤0 f (x|a∗ )
= V (a∗ )Cov
0
f (x|a∗ )dx
f (x|a∗ )
c ′′ (a)
f (x|a )dx ∗
c ′ ( a)
f (x|a∗ )dx f (x|a∗ )
fa (x|a∗ )
f (x|a∗ ) ∗ , a a . >0 f (x|a∗ ) f (x|a∗ )
>0 f (x|a∗ )
> 0 c −1 (κ) −κ , which is concave in κ by LCDFC.
fa (x|a0 ) a is convex in a for all q. ≤ q For each a0 , P 0 f (x|a ) For comparison, suppose that c (a) = a. Then, the LCDFC is satisfied
if and only if
P
≤ q a is convex in a f (x|af )
(4.1)
2
a . In
f
f
f
is maximized at a = a2 . Finally, the LCDFC holds if σ > a2 .21 Consider the iso-elastic cost c (a) = kam , m ≥ 2, which is more general than the quadratic cost. Accordingly, it can be shown that f
the LCDFC holds if σ > √a
2 m−1
. Relatedly, Kvaloy and Olsen (2014)
show (in the Remark following their Proposition 3) that the FOA is f
√a K0
m−1
, where K0 is approximately equal to 2.216.
only at q = 0.
β
− (|ε|)
β
error distribution is q(ε; α, β) = 2α Γ (1/β) e αβ . The variance is increasing in the scale parameter α . If the shape parameter β is equal to 2, it is the normal density. Its tail is thinner than the normal if β < 2, whereas it is fatter than the normal if β > 2. If β equals 1, it is the Laplace density. If β tends to ∞, it converges to the uniform density. The right-hand side term in (4.1) equals and
−β(a−x)β−1 αβ
β(x−a)β−1 αβ
if x ≥ a
if x < a. Suppose β > 1. The CDFC fails to hold since
20 Well-known density functions are mostly log-concave. Examples include the uniform, normal, exponential, logistic, and extreme value densities. See Bagnoli and Bergstrom (2005). 21 af depends on parameters such as σ , r and δ . If an interior solution exists, that is, af ∈ (0, a∗ ), then by Lemma 1, it can be shown that af = 2
19 See Proposition 6 of Jung and Kim (2015).
γ 2
1.2 The generalized error: The density function of the generalized
In this section, I will compare the LCDFC to the existing FOA-justifying conditions and provide prominent examples that researchers might consider useful applications. At first, consider multi-signal cases. To the best of my knowledge, the CDFCL, developed by Jung and Kim (2015), is the most general FOA-justifying condition ever discovered.19 The CDFCL is satisfied if and only if
fa (x|af )
.
a holds. To see this, note that fa(x|af ) = x− . Then, xf is equal to af σ2 2 f and the LCDFC is satisfied if σ > a(a − a) for all a. a(af − a)
valid if σ >
4. Discussion and examples
q(x − a)
f (x|af )
∗
sible and induces a∗ . Hence it is an optimal contract. Then, the agent’s objective function equals (ignoring the constant w f )
Vˆ P
q′ (x − a)
this case, − q(x−a) = xσ−2a . (4.1) holds for all x and for all a if and only if σ 2 > a(x − a) for all x and for all a, which is not true when x tends to ∞. Thus, the CDFC does not hold. In contrast, the LCDFC
Then, there is Vˆ ∈ (0, V (a )] such that c (a ) = Vˆ Cov 1 fa (x|a∗ ) , >0 f (x|a∗ ) fa (x|a∗ ) ∗ . Thus, a new bang–bang contract Vˆ 1 fa (x|a∗ ) is fea∗ | a
fa (x|a∗ ) f (x|a∗ )
>−
q′ (x−a)
1 fa (x|a∗ )
f (x|a )
q′ (x−a)
1.1 The normal error: Suppose ε ∼ N (0, σ 2 ) and c (a) =
f (x|a∗ )dx f (x|a∗ )
′
= 0.
Therefore, the CDFC holds if and only if (4.1) is satisfied for all x and for all a, whereas the LCDFC holds if and only if (4.1) is satisfied at x = xf for all a.
fa (x|a∗ )
∗
f (xf |af )
Example. 1 The univariate-additive case with the log-concave error: Suppose a random variable X = a + ε where ε ∈ R follows a distribution Q (·) with a density q(·), such that q(ε) > 0 ∀ε . Let x denote a realization of X . Then, we have F (x|a) = Q (x − a), f (x|a)
fa (x|a∗ )
fa (xf |af )
and hence fa(x|a) = − q(x−a) . Suppose q is log-concave.20 Then, the ∂2 −1 (κ)) is positive if and only if MLRP holds. Note that ∂κ 2 F (x|c
∗
F (x|a) is convex in a at x = xf ,
In contrast, the CDFC means that F (x|a) is convex in a for all x. Thus, the LCDFC is a local version of the CDFC. The following examples are consistent with the LCDFC, but are inconsistent with both the CDFC and CDFCL.
which is concave in κ if LCDFC is satisfied. Suppose instead af = a∗ . In this case, bf (x) may be different from the bang–bang contract (see Lemma 1). Then, (FOC) and (SE) imply: c ′ (a∗ ) = Cov
As the LCDFC is a local version of the CDFCL, the former is weaker. Example 1.1 shall show that an additive-normal example is consistent with the LCDFC (if the cost function is sufficiently convex), but is inconsistent with the CDFCL (for any cost function). Consider single-signal cases. Suppose that the MLRP holds. In this case, the CDFC and CDFCL are equivalent. For comparison, suppose c (a) = a. The LCDFC is then simplified to
where xf is such that
a − c (a) V (af )P > 0 f f (x|a ) f −1 fa (x|a ) f c (κ) − κ > 0 = V (a )P f (x|af )
129
Then, the LCDFC holds if σ >
δ √ . γ (δ+(1−δ) 2π)
1
γ
−
1−δ
δ
√ σ 2π .
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S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
this term tends to ∞ as x tends to ∞. In contrast, there are many cost functions for which the LCDFC holds: note that c ′′ (a) c ′ (a) β(af )β−1
maximized at a = 0. If c ′ (a) is log-convex, c ′′ (0)
β(af −a)β−1 αβ
is
is minimized at
a = 0. Thus, the LCDFC holds if c ′ (0) > . For instance, if αβ c (a) = era − 1, then the LCDFC holds if r is sufficiently large. If c (a) is iso-elastic, then similarly to Example 1.1, the LCDFC holds if α (and hence the variance) is sufficiently large. 2 The multivariate-additive case with the normal error: Suppose a random variable X = µa +ε , where X , µ, ε ∈ Rn and ε ∼ N (0, Σ ). Let x denote a realization of X . Note that fa (x|af ) f (x| ) af
= x ′ Σ − 1 µ − µ ′ Σ − 1 µ af ≤ 0 ⇔ a +
ε ′ Σ −1 µ ≤ af . µ′ Σ − 1 µ
ε ′ Σ −1 µ
Let v ≡ µ′ Σ −1 µ ∼ N (0, σv2 ), where σv2 ≡ µ′ Σ1−1 µ . Note that T ≡ a + v is a sufficient statistic for x. Consider the iso-elastic cost function, c (a) = kam , m ≥ 2. Example 1.1 analogously implies that the CDFC and CDFCL fail to hold, whereas the LCDFC holds if f σv > √a . 2 m−1
Acknowledgments I have benefited from the suggestions of three referees. Appendix Proof of Lemma 1. Note that a = af > 0 implies V (a) > 0. Suppose not. If V (a) < 0, (SE) implies that there is no solution to (P2), which is a contradiction. If V (a) = 0, (SE) implies b(x) = 0 for all x. (FOC) then implies a = 0, which is a contradiction. Let L(x) and G(x) denote the Lagrange multipliers of the first and second inequalities in (SE), respectively. The Kuhn–Tucker conditions with respect to a and b(x) are given by
δ G(x)dx 1−δ +µ b(x)faa (x|a)dx − c ′′ (a) = 0
s′ (a) 1 +
µfa (x|a) = G(x) − L(x) ∀x.
(A.1) (A.2)
I could obtain the weak FOA-justifying condition (LCDFC) since the shape of optimal relational contracts is known. In contrast, in the Holmstrom (1979) standard formal contract model, the shapes of optimal formal contracts, in general, are unknown. Therefore, one should impose stronger distributional conditions to validate the FOA in formal contract models. The relational contract model and the Innes (1990) limited liability model look similar. The limited liability model also admits a cutoff point, say q, such that incentive pay is given if the likelihood ratio is greater than q. However, the cutoff point in the relational contract model is always ‘zero’ for any signal distribution, whereas the cutoff point q in the limited liability model depends on distribution. Proposition 1 crucially relies on the fact that the cutoff point is fixed. Thus, I believe that the LCDFC does not validate the FOA in the limited liability model.
By (A.2), b(x) = V (a) if µfa (x|a) > 0, since then G(x) > L(x) = 0. (Note that it is impossible for G(x) and L(x) being both positive, since then b(x) = V (a) = 0, which is a contradiction). Similarly, b(x) = 0 if µfa (x|a) < 0. b(x) is indeterminate if µfa (x|a) = 0. Thus, the sign of µfa (x|a) determines the criterion of the bonus payment. (i) Suppose µ < 0. (A.2) implies thus that an optimal contract pays bonus likelihood ratio is negative, V (a) if the realized
5. Conclusion
Bagnoli, Mark, Bergstrom, Ted, 2005. Log-concave probability and its applications. Econom. Theory 26, 445–469. Baker, George, Gibbons, Robert, Murphy, Kevin, 1999. Informal authority in organizations. J. Law Econ. Organ. 15 (1), 56–73. Board, Simon, 2011. Relational contracts and the value of loyalty. Am. Econ. Rev. 101 (7), 3349–3367. Cebon, Peter, Hermalin, Benjamin, 2014. When less is more: The benefits of limits on executive pay. Rev. Financ. Stud. 28 (6), 1667–1700. Conlon, John, 2009. Two new conditions supporting the first-order approach to multisignal principal–agent problems. Econometrica 77 (1), 249–278. Holmstrom, Bengt, 1979. Moral hazard and observability. Bell J. Econ. 10, 74–91. Innes, Robert, 1990. Limited liability and incentive contracting with ex-ante action choices. J. Econom. Theory 52 (1), 45–67. Jewitt, Ian, 1988. Justifying the first-order approach to principal–agent problems. Econometrica 1177–1190. Jung, Jin Yong, Kim, Son Ku, 2015. Information space conditions for the first-order approach in agency problems. J. Econom. Theory 160 (2015), 243–279. Kvaloy, Ola, Olsen, Trond, 2014. Teams and tournaments in relational contracts. CESifo Group Munich CESifo Working Paper Series 4783. Levin, Jonathan, 2003. Relational incentive contracts. Am. Econ. Rev. 93 (3), 835–857. LiCalzi,, Spaeter,, 2003. Distributions for the first-order approach to principal–agent problems. Econom. Theory 21, 167–173. Mirrlees, James, 1974. Notes on welfare economics, information and uncertainty. In: Balch, M., McFadden, D., Wu, S. (Eds.), Essays in Equilibrium Behavior Under Uncertainty. Mirrlees, James, 1979. The implications of moral hazard for optimal insurance. mimeo. Mirrlees, James, 1999. The theory of moral hazard and unobservable behaviour: Part I. Rev. Econom. Stud. 66 (1), 3–21. Rogerson, William, 1985. The first-order approach to principal–agent problems. Econometrica 53 (5), 1357–1367. Sinclair-Desgagne, Bernard, 1994. The first-order approach to multi-signal principal–agent problems. Econometrica 62 (2), 459–466.
Standard formal contract models cannot address many real-world incentive problems in which performance measures are non-verifiable, contracts are easily manipulated, or legal contracting is extremely costly or infeasible. Relational contract models have arisen as new tools to address these complicated incentive issues. For tractability, the first-order approach (FOA) could be used in relational contract models. This paper finds a weak sufficient condition, the Local Convexity of Distribution Function Condition (LCDFC), which justifies the FOA for multi-signal relational contract models. Unlike existing conditions (such as the Mirrlees–Rogerson condition or the CDFCL of Jung and Kim (2015)), the LCDFC admits the normal and generalized error distributions if the agent’s cost function is sufficiently convex. This result could be obtained since the standard relational contract model proposed by Levin (2003) has a linear and stationary structure. The simple structure implies that finding a solution directly without relying on the FOA would be less difficult in the relational contract model than in formal contract models. Thus, the contribution of this paper is not in solving a difficult technical problem but rather in offering a clearer explanation of how the relational contract model differs from formal contract models (with regard to the FOA). Researchers could then apply the relational contract model to various incentive problems without much concern about the FOA.
f (x|a)
which means Cov bf (x), fa(x|a) | a ≤ 0, which is a contradiction to (FOC) and the fact that af > 0. (ii) Suppose µ > 0. Then, (3.1) characterizes optimal contracts. Suppose µ = 0. Then, (A.1) implies that s′ (a) = 0, which means a = a∗ . So any function bf (x) that satisfies (FOC) and (SE) at a∗ is optimal. References
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Journal of Mathematical Economics journal homepage: www.elsevier.com/locate/jmateco
Relational contracts and the first-order approach Sunjoo Hwang ∗ Department of Competition Policy, Korea Development Institute, Namsejong-ro 263, Sejong City, 339-007, Republic of Korea
article
info
Article history: Received 12 October 2015 Received in revised form 13 January 2016 Accepted 10 February 2016 Available online 18 February 2016 Keywords: Relational contracts First-order approach Local convexity of distribution function condition
abstract This paper justifies the first-order approach (FOA) to relational contract models. Optimal relational contracts pay a bonus if an agent passes an evaluation, where the cutoff point is independent of signal distribution or the agent’s cost function. Based on this independence, I find a weak FOAjustifying condition, which requires convexity of the underlying distribution-cost structure only at the cutoff point. Prominent examples (e.g., the normal or generalized error distribution with various cost functions) are consistent with this condition, but are inconsistent with existing conditions such as the Mirrlees–Rogerson condition. © 2016 Elsevier B.V. All rights reserved.
1. Introduction Standard formal contract models cannot address many realworld incentive problems in which performance measures are non-verifiable, contracts are easily manipulated, or legal contracting is highly costly or infeasible. Relational contract models have arisen as new tools to address these complicated incentive issues.1 Relational contract models, however, have a technical issue— the need to justify the first-order approach (FOA). A principal–agent problem, whether with formal or relational contracts, is often intractable since an agent’s incentive-compatibility constraint (IC) is an abstract maximization constraint involving a set of endogenous variables (a contract). For tractability, one might use the FOA, replacing the IC with the corresponding first-order condition (FOC). However, the literature on formal contracts considers justifying the FOA a challenge due to the following reasons. First, Mirrlees (1974, 1999) shows that solutions do not exist if there are no lower bounds on payments and signals. Second, even if solutions exist, most well-known signal distributions fail to satisfy the CDFC, a standard FOA-justifying condition proposed by Mirrlees (1979) and Rogerson (1985).2 Third, the CDFC is not applicable when a signal is multi-dimensional. Therefore, the literature
∗
Tel.: +82 44 550 4049. E-mail address: [email protected].
1 Levin (2003) provides a formal theoretic framework. Baker et al. (1999); Board (2011), and Cebon and Hermalin (2014) apply relational contracts to various topics such as optimality of informal authority, endogenous formation of inner circles, and efficiency of executive-pay limit, respectively. 2 See LiCalzi and Spaeter (2003) for cases consistent with the CDFC. http://dx.doi.org/10.1016/j.jmateco.2016.02.003 0304-4068/© 2016 Elsevier B.V. All rights reserved.
on formal contracts focuses on signals with lower bounds and seeks more general FOA-justifying conditions in multi-signal cases.3 To the best of my knowledge, the most general condition ever found is the CDFCL proposed by Jung and Kim (2015). This paper justifies the first-order approach (FOA) to relational contract models. I use the following two-step approach. First, I show that, if the FOC is binding, the optimal contract in a relaxed principal–agent problem in which the IC is replaced with the FOC is Levin’s bang–bang contract, which pays a fixed bonus if an agent passes an evaluation. Second, I find a FOA-justifying condition, the Local Convexity of Distribution Function Condition (LCDFC), under which the agent’s objective function is concave-in-effort given the bang–bang contract. Hence, the FOC and IC are equivalent.4 The tricky part is that if the FOC is non-binding, optimal contracts in the relaxed problem may not be the bang–bang contract. One might then seek a condition that ensures the concavity given a contract of unknown form. This should be a difficult task. However, there is a way around it: I can use the fact that for any contract that solves the relaxed principal–agent problem, there is a payoff-equivalent bang–bang contract that solves the same problem. Therefore, the LCDFC ensures the concavity whether the FOC is binding or not. Hence, it justifies the FOA.
3 Jewitt (1988) finds a set of FOA-justifying conditions when there are two independent signals. For a general multi-signal case, Sinclair-Desgagne (1994) finds two conditions (GCDFC and GSDC) that jointly justify the FOA. Conlon (2009) generalizes Jewitt and Sinclair-Desgagne’s results by suggesting a less restrictive condition (CISP). Recently, Jung and Kim (2015) find an even weaker condition (CDFCL). 4 Note that the LCDFC and CDFCL are different. L of the LCDFC stands for ‘local,’ whereas L of the CDFCL stands for ‘likelihood ratio.’
S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Note the following three positive properties of relational contract models and the LCDFC. First, Mirrlees’ non-existence result does not hold, since the self-enforcement requirement of relational contracts imposes endogenous bounds on payments. Therefore, signals without lower bounds can also be used. Second, the LCDFC is weaker than the existing conditions.5 A good thing with the bang–bang contract is that the cutoff point of pass/fail is zero (in terms of the likelihood ratio) for any distribution, cost function, or the dimension of a (vector) signal. Due to this independence, the LCDFC turns out to be convexity of the underlying distribution-cost structure only at the singular point of zero. In contrast, the CDFC and CDFCL require convexity for all points.6 Prominent examples (e.g., the normal or generalized error distribution with various cost functions) are consistent with the LCDFC, but are inconsistent with both the CDFC and CDFCL. Third, the LCDFC justifies the FOA in both single-signal and multi-signal cases in the same way. It is because the LCDFC is defined on the likelihood ratio of a signal distribution. The likelihood ratio is onedimensional even if the underlying signal is multi-dimensional. In contrast, the CDFC is a condition on the signal itself. Hence, it cannot be directly applied to multi-signal cases.7 Justifying the FOA in multi-signal cases could be more meaningful with relational contracts than with formal contracts. Since relational contracts are informal contracts, they can utilize not only objective but also subjective, non-verifiable, and internal information. Multidimensional evaluation is thus natural in relational contracting. Kvaloy and Olsen (2014) independently find a similar result in a special case.8 They consider an additive-normal case in which a one-dimensional output signal is the sum of an effort and a normally-distributed error. In this case, they show (in their Proposition 3) that the FOA is valid if the variance of the error is ‘sufficiently large.’ However, they do not provide a characterizing condition, such as the LCDFC, which determines the required degree of ‘largeness.’ When the cost function is iso-elastic, they provide (in the Remark following their Proposition 3) a characterizing condition, which is identical with the LCDFC if a constant term is adjusted. See Example 1.1 in this paper. In contrast, this paper justifies the FOA under general distributions and cost functions. For instance, Example 1.2 shows that the LCDFC can be satisfied when the error follows the generalized error distribution (which includes the normal distribution). Additionally, Example 2 shows that the LCDFC can be used with multi-signal cases. See Section 4 for more details. The organization of the paper is as follows. Section 2 presents the underlying relational contract model. Section 3 provides a FOA-justifying condition, the LCDFC. Section 4 compares the LCDFC to the existing conditions and provides consistent examples. Section 5 draws a conclusion. 2. The model The following is Levin’s standard relational contract model with a modification that an output signal is multi-dimensional. There
5 The LCDFC is weaker than the CDFCL. In single-signal cases, the LCDFC is weaker than the CDFC if the MLRP holds. For the MLRP, see Assumption 1. 6 Let F (x|a) be the distribution of a signal x given an effort a. Let κ = c (a) be the cost of the effort. Suppose that x is one-dimensional and that the MLRP holds. Then, the LCDFC holds if and only if F (x|c −1 (κ)) is convex in κ at x = xf , where xf is to be defined. In contrast, the CDFC (and CDFCL) holds if and only if F is convex at all x. 7 Jung and Kim (2015) discuss the virtue of considering conditions on the likelihood ratio rather than on the underlying signal in justifying the FOA in multisignal cases. 8 They do not focus on justifying the FOA but on characterizing optimal relational contracts when there is a team of agents. Their findings are comparable to this paper if the number of agents is one.
127
are two risk-neutral players, a principal (she) and an agent (he). The discount factor is δ ∈ (0, 1). In each period t ∈ {1, 2, . . .}, they play the following stage game. The agent contributes to the output yt ∈ [0, ∞), which is nonverifiable. The principal can offer the agent a long-term informal contract, which depends on the history of a vector of non-verifiable signals, xt ∈ X ⊂ Rn . xt may or may not include yt .9 If the agent accepts the contract, he chooses an effort at ∈ [0, ∞), where the cost of the effort c (at ) is such that c (0) = 0, c ′ > 0, c ′′ > 0, limat →0 c ′ (at ) = 0, and limat →∞ c ′ (at ) = ∞. After the effort choice, the output yt and signal xt are realized according to distributions π (yt |at ) and f (xt |at ), respectively, where f (xt |at ) > 0 for all xt ∈ X and at and fa being the partial derivative. After the realization of xt , either the principal or the agent can choose whether or not to honor the informal contract. The whole principal–agent problem is a repeated game with imperfect public monitoring under a complicated structure. However, Levin (2003) shows that it can be greatly simplified to a static problem if the signal is one-dimensional, since for any non-stationary solution to the repeated problem, there exists a payoff-equivalent stationary solution. The stationarity result analogously holds in multi-signal cases. See Levin (2003) for more details. Therefore, I consider only stationary contracts without loss of generality. Time-subscripts are suppressed. Let w ∈ R be a verifiable fixed wage and b(x) ∈ R be a non-verifiable bonus schedule. Let u¯ P and u¯ A be the principal and agent’s per-period reservation payoffs, respectively. Let E[·|a] denote the conditional expectation. The stationary principal–agent problem is given by max uP ≡ E[y|a] − E[w + b(x)|a]
s.t .
a,w,b(x)
uP ≥ u¯ P
(2.1)
u ≡ E[w + b(x)|a] − c (a) ≥ u¯ A
a ∈ arg max a′
b(x) ≤
A
(2.2)
E[w + b(x)|a′ ] − c (a′ )
δ [uP − u¯ P ], 1−δ
−b(x) ≤
(IC)
δ [uA − u¯ A ] ∀x. (2.3) 1−δ
(2.1) and (2.2) are the principal and agent’s participation constraints, respectively. (IC) is the agent’s incentive-compatibility constraint, which states that an effort a can be induced if it maximizes the agent’s payoff. (2.3) ensures that informal contracts are self-enforced: the first inequality in (2.3) states that the principal’s short-term gain from reneging on a bonus payment is smaller than her long-term loss from the relationship breaking-up.10 The second inequality similarly describes that the agent has no incentive to renege on a (negative) bonus payment. Since the principal has full bargaining power (i.e., she maximizes her payoff) and there is no restriction on the fixed wage w , it can be easily shown that the agent’s participation constraint (2.2) is binding in optima.11 Thus, the binding condition of (2.2) characterizes the optimal fixed wage:
w = u¯ A + c (a) − E[b(x)|a].
(BIR)
9 The model can be easily extended in a way that x includes verifiable but costlyt to-verify signals. 10 Alternatively, one might consider different plays on off-the-equilibrium paths. However, the main result of this paper depends only on plays on the equilibrium path. 11 If the labor market for agents is competitive, the agent would have full bargaining power. However, the main results of this paper are robust to reallocation of bargaining powers.
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S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Then, uP equals yπ (y|a)dy − c (a) − u¯ A . Let s(a) denote the (per-period) total surplus:
s(a) ≡
yπ (y|a)dy − c (a).
I assume that s(a) is strictly concave and there is a unique first-best effort a∗ > 0, which maximizes s(a). Let s¯ be the sum of the two players’ reservation payoffs. Thus, (2.3) is reduced to the following self-enforcement constraint: 0 ≤ b(x) ≤ V (a) ≡ max
δ 1−δ
[s(a) − s¯], 0
∀x
max s(a) subject to (BIR), (SE) and (IC).
a,w,b(x)
(SE)
(P1)
Note that w does not play any role in the problem above: s(a), (SE) and (IC) are independent of w . Hence, I ignore w and (BIR) in the following analysis. 3. First-order approach justification The original principal–agent problem (P1) is often intractable since (IC) is an abstract maximization constraint involving a set of endogenous variables b(x). For tractability, one might use the first-order approach (FOA), replacing (IC) with the following firstorder condition (FOC). Let Cov(·|·) be the conditional covariance.13 fa (x|a)
a = c ′ (a). b(x)fa (x|a)dx = Cov b(x), f (x|a)
0 bf (x) = ∈ 0, V (af ) V ( af )
if if if
where V (a) is the value of the relationship. Note that (SE) imposes endogenous bounds on payment b(x). Relatedly, Mirrlees (1999) shows that formal contract models have no solutions if payments and signals are not bounded from below.12 This non-existence result does not hold in relational contract models. Thus, finally, the principal–agent problem is simplified to the following surplus maximization problem:
indeterminate: bf (x) is any function that satisfies (FOC) and (SE) at a∗ . (iii) If µ > 0, bf (x) is such that
(FOC)
Note that (FOC) is weaker than (IC) (if the equilibrium effort is positive). In general, the following relaxed principal–agent problem (P2) has a greater choice set than (P1). Hence, a solution to (P2) is not necessarily a solution to (P1). Thus, the FOA is not valid in general. max s(a) subject to (SE) and (FOC). a,b(x)
(P2)
I use a two-step procedure to justify the FOA. First, in Lemma 1, I characterize optimal contracts in the relaxed problem (P2) without assuming any conditions that justify the FOA. These contracts are optimal in the original problem (P1), if the agent’s objective function is concave-in-effort given these contracts. Proposition 1 provides a sufficient condition, the Local Convexity of Distribution Function Condition (LCDFC), which ensures the concavity. In other words, the LCDFC justifies the FOA. Lemma 1. Let µ denote the Lagrange multiplier of (FOC). Suppose there exists a solution (af , bf (x)) to (P2) such that af > 0. Then, (i) µ ≥ 0. (ii) If µ = 0, then af = a∗ and bf (x) is
f (x|a)
ratio fa(x|a) is zero for any density function f .
f (x|af ) fa (x|af ) f (x|af ) fa (x|af ) f (x|af )
<0 (3.1)
=0 > 0.
Proof. See Appendix. The contract bf (x) in Lemma 1 has the bang–bang shape if µ > 0: a fixed bonus is paid if and only if the likelihood ratio is positive. Note that this result is obtained independently of the Mirrlees–Rogerson condition. In contrast, Levin (2003) obtains the same result assuming the following version of the Mirrlees–Rogerson condition. Assumption 1. (a) The Monotone Likelihood Ratio Property f (x|a) (MLRP): fa(x|a) is increasing in x for all a.14 (b) Levin’s version of the
Convexity of Distribution Function Condition (CDFC): F (x|c −1 (κ)) is convex in κ for all x, where κ = c (a).15
Contract theorists typically assume the MLRP.16 However, the MLRP plays no role in the relational contract model, since it is a condition on the level of the likelihood ratio rather than the sign, which is a sufficient statistic that captures all relevant information. The following proposition shows that the LCDFC validates the FOA. The basic idea is as follows. The LCDFC is a condition that ensures that the agent’s objective function is concave-in-effort given the bang–bang contract in Lemma 1. The tricky part is that if µ = 0, a contract that solves (P2) may be not a bang–bang contract. Thus, in this case, one might seek a condition that ensures the concavity given a contract with an unknown form. This task should be difficult. However, there is a way around it: I can use the fact that for any contract that solves (P2), there is a payoff-equivalent bang–bang contract that solves (P2). Thus, the LCDFC ensures the concavity for any µ. Hence, it justifies the FOA. Proposition 1. Suppose the following condition holds.
P
fa (x|af ) f (x|af )
≤ 0 c −1 (κ) is convex in κ
(LCDFC)
where af > 0 is an equilibrium effort of (P2) and κ = c (a).17 Then, the first-order approach is valid. In other words, if there is a solution to (P2) that induces a positive effort af , it is also a solution to (P1).18
14 In the literature on contract theory, ∂ log f (x|a) is often called the likelihood ∂a ratio of f (x|a). 15 Mirrlees’ original CDFC requires F (x|a) being convex in a. Levin’s version incorporates convexity of cost function as well. 16 Although the MLRP is less restrictive than the CDFC, it is still a restrictive condition. In single-signal cases, the MLRP is stronger than the hazard rate dominance. The hazard rate dominance is stronger than the first-order stochastic dominance. The first-order stochastic dominance is stronger than the second-order stochastic dominance. In multi-signal cases, the MLRP is increasingly restrictive as the dimension n of a multi-signal x = (x1 , . . . , xn ) rises. Conlon (2009) uses the multi-dimensional MLRP. 17 For clarity, note that P fa (x|af ) ≤ 0 c −1 (κ) = f (x|c −1 (κ))dx. f (x|af )
12 If the principal pays the agent an arbitrarily low payment when the agent’s performance is arbitrarily low, the first-best outcome can be approximated, though it cannot be reached. 13 The covariance representation obtains since the expectation of the likelihood
fa (x|af )
fa (x|af ) f (x|af )
≤0
18 A necessary and sufficient condition of FOA-justification may not be attainable. As an analogue, consider a simple unconstrained optimization problem. The first-order condition leads to a global optimum if the objective function is concave. However, an ‘only-if’ result is difficult to obtain since the first-order condition generates a global optimum even if the objective function has a non-conventional form.
S. Hwang / Journal of Mathematical Economics 63 (2016) 126–130
Proof. Since a = c −1 (κ), the problem of choosing a is equivalent to the problem of choosing κ . Let (af , bf (x)) be a solution to (P2) such that af > 0. Suppose af ̸= a∗ . Then, from the facts ‘µ ≥ 0’ and ‘µ = 0 implies af = a∗ ,’ it follows that µ > 0. Lemma 1 then implies that V (af )1 fa (x|af ) is an optimal contract. Generically, it is a unique f (x|af )
>0
optimal contract. Then, I can assume, without loss of generality, that bf (x) = V (af )1 fa (x|af ) . The agent’s objective function then >0
f (x|af )
equals (ignoring the constant w f ) fa (x|af )
bf (x),
fa (x|a∗ ) ∗ a f (x|a∗ )
=
fa (x|a∗ ) >0 f (x|a∗ )
bf (x)
fa (x|a )
fa (x|a∗ ) ≤0 f (x|a∗ )
b (x) f
+ ≤
f (x|a∗ )
fa (x|a∗ ) >0 f (x|a∗ )
V (af )
+
fa (x|a∗ ) ≤0 f (x|a∗ )
= V (a∗ )Cov
0
f (x|a∗ )dx
f (x|a∗ )
c ′′ (a)
f (x|a )dx ∗
c ′ ( a)
f (x|a∗ )dx f (x|a∗ )
fa (x|a∗ )
f (x|a∗ ) ∗ , a a . >0 f (x|a∗ ) f (x|a∗ )
>0 f (x|a∗ )
> 0 c −1 (κ) −κ , which is concave in κ by LCDFC.
fa (x|a0 ) a is convex in a for all q. ≤ q For each a0 , P 0 f (x|a ) For comparison, suppose that c (a) = a. Then, the LCDFC is satisfied
if and only if
P
≤ q a is convex in a f (x|af )
(4.1)
2
a . In
f
f
f
is maximized at a = a2 . Finally, the LCDFC holds if σ > a2 .21 Consider the iso-elastic cost c (a) = kam , m ≥ 2, which is more general than the quadratic cost. Accordingly, it can be shown that f
the LCDFC holds if σ > √a
2 m−1
. Relatedly, Kvaloy and Olsen (2014)
show (in the Remark following their Proposition 3) that the FOA is f
√a K0
m−1
, where K0 is approximately equal to 2.216.
only at q = 0.
β
− (|ε|)
β
error distribution is q(ε; α, β) = 2α Γ (1/β) e αβ . The variance is increasing in the scale parameter α . If the shape parameter β is equal to 2, it is the normal density. Its tail is thinner than the normal if β < 2, whereas it is fatter than the normal if β > 2. If β equals 1, it is the Laplace density. If β tends to ∞, it converges to the uniform density. The right-hand side term in (4.1) equals and
−β(a−x)β−1 αβ
β(x−a)β−1 αβ
if x ≥ a
if x < a. Suppose β > 1. The CDFC fails to hold since
20 Well-known density functions are mostly log-concave. Examples include the uniform, normal, exponential, logistic, and extreme value densities. See Bagnoli and Bergstrom (2005). 21 af depends on parameters such as σ , r and δ . If an interior solution exists, that is, af ∈ (0, a∗ ), then by Lemma 1, it can be shown that af = 2
19 See Proposition 6 of Jung and Kim (2015).
γ 2
1.2 The generalized error: The density function of the generalized
In this section, I will compare the LCDFC to the existing FOA-justifying conditions and provide prominent examples that researchers might consider useful applications. At first, consider multi-signal cases. To the best of my knowledge, the CDFCL, developed by Jung and Kim (2015), is the most general FOA-justifying condition ever discovered.19 The CDFCL is satisfied if and only if
fa (x|af )
.
a holds. To see this, note that fa(x|af ) = x− . Then, xf is equal to af σ2 2 f and the LCDFC is satisfied if σ > a(a − a) for all a. a(af − a)
valid if σ >
4. Discussion and examples
q(x − a)
f (x|af )
∗
sible and induces a∗ . Hence it is an optimal contract. Then, the agent’s objective function equals (ignoring the constant w f )
Vˆ P
q′ (x − a)
this case, − q(x−a) = xσ−2a . (4.1) holds for all x and for all a if and only if σ 2 > a(x − a) for all x and for all a, which is not true when x tends to ∞. Thus, the CDFC does not hold. In contrast, the LCDFC
Then, there is Vˆ ∈ (0, V (a )] such that c (a ) = Vˆ Cov 1 fa (x|a∗ ) , >0 f (x|a∗ ) fa (x|a∗ ) ∗ . Thus, a new bang–bang contract Vˆ 1 fa (x|a∗ ) is fea∗ | a
fa (x|a∗ ) f (x|a∗ )
>−
q′ (x−a)
1 fa (x|a∗ )
f (x|a )
q′ (x−a)
1.1 The normal error: Suppose ε ∼ N (0, σ 2 ) and c (a) =
f (x|a∗ )dx f (x|a∗ )
′
= 0.
Therefore, the CDFC holds if and only if (4.1) is satisfied for all x and for all a, whereas the LCDFC holds if and only if (4.1) is satisfied at x = xf for all a.
fa (x|a∗ )
∗
f (xf |af )
Example. 1 The univariate-additive case with the log-concave error: Suppose a random variable X = a + ε where ε ∈ R follows a distribution Q (·) with a density q(·), such that q(ε) > 0 ∀ε . Let x denote a realization of X . Then, we have F (x|a) = Q (x − a), f (x|a)
fa (x|a∗ )
fa (xf |af )
and hence fa(x|a) = − q(x−a) . Suppose q is log-concave.20 Then, the ∂2 −1 (κ)) is positive if and only if MLRP holds. Note that ∂κ 2 F (x|c
∗
F (x|a) is convex in a at x = xf ,
In contrast, the CDFC means that F (x|a) is convex in a for all x. Thus, the LCDFC is a local version of the CDFC. The following examples are consistent with the LCDFC, but are inconsistent with both the CDFC and CDFCL.
which is concave in κ if LCDFC is satisfied. Suppose instead af = a∗ . In this case, bf (x) may be different from the bang–bang contract (see Lemma 1). Then, (FOC) and (SE) imply: c ′ (a∗ ) = Cov
As the LCDFC is a local version of the CDFCL, the former is weaker. Example 1.1 shall show that an additive-normal example is consistent with the LCDFC (if the cost function is sufficiently convex), but is inconsistent with the CDFCL (for any cost function). Consider single-signal cases. Suppose that the MLRP holds. In this case, the CDFC and CDFCL are equivalent. For comparison, suppose c (a) = a. The LCDFC is then simplified to
where xf is such that
a − c (a) V (af )P > 0 f f (x|a ) f −1 fa (x|a ) f c (κ) − κ > 0 = V (a )P f (x|af )
129
Then, the LCDFC holds if σ >
δ √ . γ (δ+(1−δ) 2π)
1
γ
−
1−δ
δ
√ σ 2π .
130
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this term tends to ∞ as x tends to ∞. In contrast, there are many cost functions for which the LCDFC holds: note that c ′′ (a) c ′ (a) β(af )β−1
maximized at a = 0. If c ′ (a) is log-convex, c ′′ (0)
β(af −a)β−1 αβ
is
is minimized at
a = 0. Thus, the LCDFC holds if c ′ (0) > . For instance, if αβ c (a) = era − 1, then the LCDFC holds if r is sufficiently large. If c (a) is iso-elastic, then similarly to Example 1.1, the LCDFC holds if α (and hence the variance) is sufficiently large. 2 The multivariate-additive case with the normal error: Suppose a random variable X = µa +ε , where X , µ, ε ∈ Rn and ε ∼ N (0, Σ ). Let x denote a realization of X . Note that fa (x|af ) f (x| ) af
= x ′ Σ − 1 µ − µ ′ Σ − 1 µ af ≤ 0 ⇔ a +
ε ′ Σ −1 µ ≤ af . µ′ Σ − 1 µ
ε ′ Σ −1 µ
Let v ≡ µ′ Σ −1 µ ∼ N (0, σv2 ), where σv2 ≡ µ′ Σ1−1 µ . Note that T ≡ a + v is a sufficient statistic for x. Consider the iso-elastic cost function, c (a) = kam , m ≥ 2. Example 1.1 analogously implies that the CDFC and CDFCL fail to hold, whereas the LCDFC holds if f σv > √a . 2 m−1
Acknowledgments I have benefited from the suggestions of three referees. Appendix Proof of Lemma 1. Note that a = af > 0 implies V (a) > 0. Suppose not. If V (a) < 0, (SE) implies that there is no solution to (P2), which is a contradiction. If V (a) = 0, (SE) implies b(x) = 0 for all x. (FOC) then implies a = 0, which is a contradiction. Let L(x) and G(x) denote the Lagrange multipliers of the first and second inequalities in (SE), respectively. The Kuhn–Tucker conditions with respect to a and b(x) are given by
δ G(x)dx 1−δ +µ b(x)faa (x|a)dx − c ′′ (a) = 0
s′ (a) 1 +
µfa (x|a) = G(x) − L(x) ∀x.
(A.1) (A.2)
I could obtain the weak FOA-justifying condition (LCDFC) since the shape of optimal relational contracts is known. In contrast, in the Holmstrom (1979) standard formal contract model, the shapes of optimal formal contracts, in general, are unknown. Therefore, one should impose stronger distributional conditions to validate the FOA in formal contract models. The relational contract model and the Innes (1990) limited liability model look similar. The limited liability model also admits a cutoff point, say q, such that incentive pay is given if the likelihood ratio is greater than q. However, the cutoff point in the relational contract model is always ‘zero’ for any signal distribution, whereas the cutoff point q in the limited liability model depends on distribution. Proposition 1 crucially relies on the fact that the cutoff point is fixed. Thus, I believe that the LCDFC does not validate the FOA in the limited liability model.
By (A.2), b(x) = V (a) if µfa (x|a) > 0, since then G(x) > L(x) = 0. (Note that it is impossible for G(x) and L(x) being both positive, since then b(x) = V (a) = 0, which is a contradiction). Similarly, b(x) = 0 if µfa (x|a) < 0. b(x) is indeterminate if µfa (x|a) = 0. Thus, the sign of µfa (x|a) determines the criterion of the bonus payment. (i) Suppose µ < 0. (A.2) implies thus that an optimal contract pays bonus likelihood ratio is negative, V (a) if the realized
5. Conclusion
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Standard formal contract models cannot address many real-world incentive problems in which performance measures are non-verifiable, contracts are easily manipulated, or legal contracting is extremely costly or infeasible. Relational contract models have arisen as new tools to address these complicated incentive issues. For tractability, the first-order approach (FOA) could be used in relational contract models. This paper finds a weak sufficient condition, the Local Convexity of Distribution Function Condition (LCDFC), which justifies the FOA for multi-signal relational contract models. Unlike existing conditions (such as the Mirrlees–Rogerson condition or the CDFCL of Jung and Kim (2015)), the LCDFC admits the normal and generalized error distributions if the agent’s cost function is sufficiently convex. This result could be obtained since the standard relational contract model proposed by Levin (2003) has a linear and stationary structure. The simple structure implies that finding a solution directly without relying on the FOA would be less difficult in the relational contract model than in formal contract models. Thus, the contribution of this paper is not in solving a difficult technical problem but rather in offering a clearer explanation of how the relational contract model differs from formal contract models (with regard to the FOA). Researchers could then apply the relational contract model to various incentive problems without much concern about the FOA.
f (x|a)
which means Cov bf (x), fa(x|a) | a ≤ 0, which is a contradiction to (FOC) and the fact that af > 0. (ii) Suppose µ > 0. Then, (3.1) characterizes optimal contracts. Suppose µ = 0. Then, (A.1) implies that s′ (a) = 0, which means a = a∗ . So any function bf (x) that satisfies (FOC) and (SE) at a∗ is optimal. References