21COE, University of Tokyo

MMRC Discussion Paper No. 219

MMRC
DISCUSSION PAPER SERIES

MMRC-F-219

A Monotone Comparative Statics Resulton Contract IncompletenessYasunori Watanabe
Manufacturing Management Research Center,
The University of Tokyo
March 2008

A Monotone Comparative Statics Resulton Contract Incompleteness 1
Yasunori Watanabe2
Manufacturing Management Research Center,
The University of Tokyo
March 25, 2008

1 I wish to express my gratitude, particularly to Masahiro Okuno-Fujiwara, Hirokazu Tak-izawa, and Taisuke Matsubae for their valuable advice. I also wish to thank Takahiro Fujimoto,
Tomoya Yashiro, Masao Ando, Satoshi Yoshida, and other members of the Construction Industry Architecture study group at the University of Tokyo for their helpful discussions. Of course,
I am solely responsible for any remaining errors. This work is supported, in part, by a grantfrom the Japan Federation of Housing Organizations.
2 watanabe-y@mmrc.e.u-tokyo.ac.jp

Abstract
The purpose of this paper is to generalize the incomplete contract model of Bajari andTadelis (2001) into a continuous model, and to derive the condition under which themonotone comparative statics (MCS) methods can be applied. I will show that a typeof single-crossing condition on the isoprobability curves of uncertainty is necessary andsufficient for the availability of the MCS methods when a player has a supermodular expost utility function. In this case, the greater the magnitude of uncertainty, the less his
optimal contract allows the project to proceed smoothly.

1

Introduction

Bajari and Tadelis (2001, henceforth, B-T) have developed an incomplete contract
model, and have analyzed the types of contracts in construction industries. They showedthat the difference was not based on the asymmetry between builders and owners as theprincipal-agent theory suggests, rather, it was based on the magnitude of uncertainty
they both builders and owners are confronting with.
In B-T’s model, an owner who wishes to procure a building prepares the design ofthe project in advance, and asks a builder to comply with it. Thus, the design canbe considered as a contract. The project involves certain uncertainties that are notapparent at the beginning, for example, the condition of the ground, the owner’s actual
needs, and so on. One extreme way of dealing with such uncertainties is to describe everyprescription for every possible event in the design, the builder then proceeds with the
project as suggested. However, from a practical viewpoint, it not only is very difficult toanticipate all possible events but also to describe clauses concerning them. Thus, it isnatural to include some clauses in advance, and to hold renegotiations among stakeholders
when an unforeseen event takes place. B-T analyzed the number of prescriptions thatthe owner should have in the contract in advance, that is, how specific a contract she(the owner) should prepare ex ante. However, they showed that the greater the amountof uncertainty involved, the less specific is the contract that she prefers.
B-T used a discrete model, where they defined a construction project as a combination of the number of possible states T and a probability distribution (ft )Tt=1 on them.
Moreover, they stated that “project T is more complex than project T 0 ,” if the follow0

ing two conditions are satisfied: (i) (ft )Tt=1 dominates (ft0 )Tt=1 in the sense of first-orderstochastic dominance, (ii) the states in project T are constructed by dividing some statein project T 0 . However, in their analysis, they assumed that the second condition is
automatically implied by the first condition, and treated the number of states T as anindex of complexity. Therefore, there was no discussion about the property that the1

probability distributions must satisfy. My main purpose is to derive such a property ina more general setting.
Since the work of Milgrom and Shannon (1994), the monotone comparative statics(MCS) methods have been widely researched recently. Among the recent works, seeAthey (1998) and Athey (2002) for details about the application of the MCS methods todecision-making problems under uncertainty.
This paper is organized as follows. Section 2 presents the model. The set of possible
events is assumed to be continuous, and I assume that the player’s ex post utility function
is supermodular. In section 3, I derive the condition under which the MCS methods canbe applied to the model. I will also show that the optimal contracts have a monotone
property with respect to the magnitude of uncertainty. Section 4 concludes the work. In
the appendix, I list the theorems of Topkis (1998) that are used in my proofs.
2

The Model

In this section, I propose a general model of contract incompleteness. This model isan extension of B-T’s model. The main purpose is to specify the conditions under which
the monotone comparative statics methods can be applied to the model.
A player is to implement a project with uncertainty. The uncertainty will be resolved
after the launch of the project, and an event will be brought to realization during the
project. The event affects the implementation of the project. Let t ∈ R+ be the eventthat is realized ex post. I assume that t is ex post verifiable, and is drawn from a familyof distribution functions indexed by θ ∈ Θ. Let the cumulative distribution functions be
F : T × Θ 7→ R+ . Suppose that F is atomless and is absolutely continuous in t for all θ.
The parameter θ represents the magnitude of ex ante uncertainty. The following is
an assumption about it.

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Assumption 1. For all θ 00 ≥ θ0 ,F (t, θ 00 ) ≤ F (t, θ 0 ). ∀t(1)That is, the larger θ becomes, the more uncertain it would be ex ante.
The events are ordered from the most probable event t = 0 to infinity, based on thelikelihood of its occurrence ex ante, irrespective of the parameter θ. This assumption isexpressed by allowing the density function of the events to be monotonically nonincreasing
in t.
Assumption 2. For all t00 ≥ t0 ,Ft (t00 , θ) ≤ Ft (t0 , θ),(2)where the subscripts represent the derivatives of the functions.
As the ex post events can affect the progression of the project, the player may makea list of clauses about them ex ante. I assume that there is a cost involved for including
clauses about ex post events, thus, it is impossible to have a contract that includes clauses
for all events ex ante. I will later discuss this point in detail. In this list, the player has
arrangements that describe prescriptions for each event that he has selected, moreover,he treats θ as given. For example, consider an insurance contract. The buyer has todecide which contract she should buy before any accidents take place, however, on theother hand, she knows how uncertain they would be, therefore, depending on her ex anteknowledge, she determines which clauses should or should not be included in the contract.
As I have mentioned above, it is costly to preface a contract. I represent the set of
events that are selected in the contract by S. S is assumed to be an element in thefamily of Borel set B of R+ . The cost of prefacing a contract S ∈ B is represented by
C(S) ≡ g(µ(S)) for the Lebesgue measure µ and some nondecreasing function g, whichis assumed to be twice continuously differentiable, g 0 ≥ 0 and g 00 ≥ 0.
3

When the uncertainty is resolved and some event t is brought about, if the contractcontains a clause about t, then the project would progress according to it. On the other
hand, if there is no clause about t, the stakeholders have to coordinate their actions
with regard to the implementation of the project. Therefore, the ex post difficulty of the
project depends on whether or not t ∈ S. The difficulty is represented as an ex post stateω ∈ Ω ≡ {0, 1}. As ω depends on both the ex ante contract S ∈ B and the ex post eventt ∈ R+ , I describe it as a function h : B × R+ 7→ Ω, andh(S, t) =
1 if t ∈ S

.

(3)



0 if t 6∈ S
Finally, I define the player’s ex post utility. Let X ⊆ R be a compact set. X is a set ofthe player’s ex post alternatives. The terminal utility is determined by his choice of x, theex post event t, and the ex post state ω. Let his utility function be u : X × R + × Ω 7→ R+ .
The supermodularity of u is assumed as follows.
Assumption 3. u(x, t, ω) is supermodular in (x, −t, ω).
Remark 1. This assumption implies that the order of the ex post event t not only depends on the likelihood of its occurrence but also on the player’s utility function. Although
this appears to be too restrictive, what is important in the discussions below is the supermodularity between x and ω. Thus, assuming that the utility does not depend on t does
not change the following results.
3

AnalysisFirst, I will solve the player’s problem, assuming the ex post event and state as given.
Then, taking this solution into consideration, the ex ante optimal contract will be derived.

4Given (t, ω), the player seeks an optimal x. The problem ismax u(x, t, ω)

(4)

x∈X

Let its maximized value be v(t, ω). The next lemma is derived from the results of Topkis(1998).
Lemma 1. Let (−T ) be the reverse-ordered set of R+ . Then, v(t, ω) is supermodular in(−t, ω) on (−T ) × Ω. Furthermore, there exist a greatest and a least element ofarg max u(x, t, ω),x∈X

and they are monotonically increasing in (−t, ω) on (−T ) × Ω.
Proof. Since X is a compact subset of R, there always exists a maximum value of u for all(t, ω), moreover, arg maxx∈X u(x, t, ω) is not empty. Therefore, by Theorem A.1, v(t, ω)is supermodular in (−t, ω) on (−T ) × Ω.
Moreover, by Theorem A.2, there exist a greatest element and a least element ofarg maxx∈X u(x, t, ω), and they are increasing in (−t, ω) on (−T ) × Ω.
Next, given v(t, ω) acquired above, define the player’s ex ante expected utility respective to t and ω. If he selects a set S ∈ B as an ex ante contract, then the ex ante expectedutility will become

V (S, θ) ≡Zv(t, 1)Ft (t, θ)dt +S

Zv(t, 0)Ft (t, θ)dt.
SC

He selects a contract S, which maximizes this expected utility. I can show that an interval[0, t̂], which has the origin as an end, maximizes the expected utility with respect to theLebesgue measure being constant.

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Lemma 2. For any given µ̄, let S = [0, µ̄]. ThenS ∈ arg max
V (S 0 , θ).
0

(5)

S ∈B
µ(S 0 )=µ̄

Thus, the interval S = [0, µ̄], which begins at 0, maximizes the expected payoff V under
the condition that its Lebesgue measure is equal to µ̄.
Proof. Let t0 < µ̄ < t00 and S = [0, µ̄]. Starting with S, define S 0 as follows: remove an
infinitesimal interval dt at t = t0 , and add dt at t = t00 to S. Then, µ(S) = µ(S 0 ) = µ̄,
and its associated variation dV of V is calculated as

dV = v(t00 , 1)Ft (t00 , θ)dt − v(t00 , 0)Ft (t00 , θ)dt − (v(t0 , 1)Ft (t0 , θ)dt − v(t0 , 0)Ft (t0 , θ)dt)= (v(t00 , 1) − v(t00 , 0))Ft (t00 , θ)dt − (v(t0 , 1) − v(t0 , 0))Ft (t0 , θ)dt.

This is nonpositive by Assumption 2 and Lemma 1.
Lemma 2 reveals that I can concentrate on intervals [0, t̂] as ex ante contracts. Therefore, I treat either t̂ or the probability p = Prob(t ≤ t̂) as control variables insteadof S. Given p and θ, define t̂(p, θ) as p = F (t̂(p, θ), θ). Furthermore, identifying p as
S = [0, t̂(p, θ)], the functions defined on B will be treated as functions defined on [0, 1].
Rewriting the ex ante expected utility respective to t and ω as a function of p and θ
providesV (p, θ) ≡Z t̂(p,θ)v(t, 1)Ft (t, θ)dt +0

Z ∞v(t, 0)Ft (t, θ)dt.

(6)t̂(p,θ)Before checking the property of this function, I will prove the supermodularity of t̂(p, θ).
Lemma 3. t̂(p, θ) is increasing in (p, θ) on [0, 1] × Θ. In addition, it is supermodular in(p, θ) on [0, 1] × Θ if and only if

t00 ≥ t0 =⇒ Ft (t0 , θ0 ) ≥ Ft (t00 , θ00 ),∀(t0 , θ0 ), (t00 , θ00 ) s.t. F (t0 , θ0 ) = F (t00 , θ00 )6(7)

holds.
Moreover, suppose F is twice continuously differentiable in t, differentiable in θ, andhas the cross-derivative ∂ 2 F/∂t∂θ. Then, t̂(p, θ) is supermodular in (p, θ) if and only if∂∂tFθ (t, θ)−Ft (t, θ)



≥ 0.

(8)

Proof. Since F is nondecreasing in t for all θ, the inverse function t̂ is nondecreasing inp. Furthermore, for all θ 00 ≥ θ0 and p, it follows thatp = F (t̂(p, θ 0 ), θ0 ) = F (t̂(p, θ 00 ), θ00 ) ≤ F (t̂(p, θ 00 ), θ0 ).

By Assumption 1. Since F is nondecreasing in t, I obtain t̂(p, θ 00 ) ≥ t̂(p, θ 0 ).
Suppose (7) holds for all (t0 , θ0 ) and (t00 , θ00 ) such that F (t0 , θ0 ) = F (t00 , θ00 ). F isassumed to be absolutely continuous for all θ, and has its derivative Ft . Since t̂ is theinverse function of F , there exists the derivative respective to p for all θ, and it is givenby∂ t̂(p, θ)1=.
∂p
Ft (t̂(p, θ), θ)
For all θ 00 ≥ θ0 , I have F (t̂(p, θ 00 ), θ00 ) = F (t̂(p, θ 0 ), θ0 ) = p and t̂(p, θ 00 ) ≥ t̂(p, θ 0 ). By (7),these conditions imply that

Ft (t̂(p, θ 0 ), θ0 ) ≥ Ft (t̂(p, θ 00 ), θ00 ).

Therefore,1∂ t̂(p, θ 00 ) ∂ t̂(p, θ 0 )
1
−
≥ 0,
−
=
00
00
∂p
∂p
Ft (t̂(p, θ ), θ ) Ft (t̂(p, θ 0 ), θ0 )that is, t̂ is supermodular in (p, θ). The opposite direction is straightforward and I skip
the proof.
Next, I will analyze the case in which some additional differentiability conditions hold.
7

Suppose F is twice continuously differentiable in t, differentiable in θ, and has the crossderivative Ftθ . Let p = F (t̂(p, θ), θ). Differentiating both sides of this equation by p andθ, I obtain∂ t̂(p, θ),∂p∂ t̂(p, θ)+ Fθ (t̂(p, θ), θ).
0 = Ft (t̂(p, θ), θ)∂θ

1 = Ft (t̂(p, θ), θ)

These equalities imply that1∂ t̂(p, θ)
≥ 0,
=
∂p
Ft (t̂(p, θ), θ)(9)∂ t̂(p, θ)Fθ (t̂(p, θ), θ)
≥ 0.
=−
∂θ
Ft (t̂(p, θ), θ)(10)

To prove the supermodularity of t̂, calculating the cross-derivative of t̂(p, θ) leads to thefollowing.
∂ 2 t̂(p, θ)
∂
=
∂θ∂p
∂θ




1
Ft (t̂(p, θ), θ)∂ t̂(p, θ)1
Ftt (t̂(p, θ), θ)=−+ Ftθ (t̂(p, θ), θ)∂θFt (t̂(p, θ), θ)2
1=−
Ft (t̂(p, θ), θ)2Fθ (t̂(p, θ), θ)× Ftt (t̂(p, θ), θ) −+ Fθt (t̂(p, θ), θ) .
Ft (t̂(p, θ), θ)

The expression in the curly brackets is nonpositive for any (p, θ) if and only if (8) holds
for any (t, θ). This concludes the proof.
The supermodularity of the cost function C(p, θ) = g(t̂(p, θ)) immediately follows thislemma.
I will discuss some points about the conditions in this lemma. First, to identify theequivalence between (7) and (8), it will suffice to differentiate Ft (t̂(p, θ), θ) in θ. Next, (7)8can be interpreted as follows. Suppose, given a probability p, the player has to prepare
a contract in which the probability of including a clause about events ex post is exactly
p. If he attempts to prepare this contract more specifically and include some additionalclauses, then the greater the magnitude of uncertainty, the less the increment of the exante probability. In another words, if he wishes to increase the ex ante probability bysome fixed degree, then the more greater the magnitude of uncertainty ex ante, the larger
is the number of clauses that he must include. Finally, note that (8) implies that the slopeof the isoprobability curve of F is nondecreasing in t. Therefore, this condition can be
regarded as a type of single-crossing condition about the isoprobability curve. However,they never cross each other on their domain.
Using the supermodularity of t̂ derived as above, I will prove the supermodularity ofV.
Lemma 4. If t̂ is supermodular in (p, θ) on [0, 1] × Θ, then V (p, θ) is supermodular in(−p, θ) on [0, 1] × Θ.
Proof. Differentiating V (p, θ) with p provides∂V (p, θ)∂ t̂(p, θ)= (v(t̂, 1)Ft (t̂(p, θ), θ) − v(t̂, 0)Ft (t̂(p, θ), θ))∂p∂p= (v(t̂(p, θ), 1)Ft (t̂(p, θ), θ) − v(t̂(p, θ), 0)Ft (t̂(p, θ), θ))1
Ft (t̂(p, θ), θ)= v(t̂(p, θ), 1) − v(t̂(p, θ), 0).
For any θ00 ≥ θ0 , I have
∂V (p, θ 00 ) ∂V (p, θ 0 )−∂p∂p= (v(t̂(p, θ 00 ), 1) − v(t̂(p, θ 00 ), 0)) − (v(t̂(p, θ 0 ), 1) − v(t̂(p, θ 0 ), 0)) ≤ 0.

The last inequality is derived from the fact that t̂ is increasing in θ and that v is supermodular in (−t, ω).
9I shall now prove the main theorem.
Theorem 1. Suppose either (7) or (8) is satisfied. Then, for all θ, there exist a greatest
and a least element ofarg max V (p, θ) − C(p, θ),p∈[0,1](11)

and they are monotonically nonincreasing in θ.
Proof. Note that arg maxp∈[0,1] V (p, θ) − C(p, θ) is nonempty. By Lemmas 3 and 4,
V (p, θ) − C(p, θ) is supermodular in (−p, θ). Hence, by Theorem A.2, there exist a greatest and a least element of arg maxp∈[0,1] V (p, θ) − C(p, θ), and they are monotonicallynonincreasing in θ.
4Conclusion

In this paper, I extended B-T’s model into a continuous model, and derived a condition under which the MCS methods can be applied. The intuitive interpretation of
the condition is also provided. The main result indicated that if the supermodularityof ex post utility function is assumed, then a type of single-crossing condition of theisoprobability curves of uncertainty is necessary and sufficient.

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A

Appendix

In this appendix, I list the theorems of Topkis (1998) that are used in my proofs.
The main result on the maximization of supermodular functions is the following theorem.
Theorem A.1 (Topkis (1998)). If X and T are lattices, S is a sublattice of X × T ,f : X × T 7→ R is supermodular in (x, t) on S, St is the section of S at t in T , andg(t) = supx∈St f (x, t) is finite on the projection ΠT S of S on T , then g(t) is supermodular
on ΠT S.
The next theorem assures the monotonicity of solutions.
Theorem A.2 (Topkis (1998)). Suppose that X is a nonempty lattice, T is a partiallyordered set, f : X × T 7→ R is supermodular in x on X for each t in T , and f issupermodular in (x, t) on X × T . If either(i). X is finite or(ii). X is a compact subset of Rn and f is upper semicontinuous in x on X for each t
in T ,
then each arg maxx∈X f (x, t) (being either finite or a compact subset of Rn ) is a nonempty
sublattice of X with a greatest element and a least element, and the greatest (least) elementof arg maxx∈X f (x, t) is increasing in t on T .
References
Athey, S. (1998) “Characterizing Properties of Stochastic Objective Functions.” MIT
Working Paper No.96-1R.
(2002) “Monotone Comparative Statics under Uncertainty.,” The Quarterly
Journal of Economics, Vol. 117, pp. 187–223.
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Bajari, P. and S. Tadelis (2001) “Incentives versus Transaction Costs: A Theory ofProcurement Contracts,” RAND Journal of Economics, Vol. 32, No. 3, pp. 387–407,
Autumn.
Milgrom, P. and C. Shannon (1994) “Monotone Comparative Statics,” Econometrica,
Vol. 62, No. 1, pp. 157–180.
Topkis, D.M. (1998) Supermodularity and Complementarity, Princeton, New Jersey:
Princeton University Press.

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