Web Appendix for “Optimal Contracting with Endogenous Social Norms”
Paul Fischer and Steven Huddart
This appendix provides proofs omitted from the body of the paper.
Proof of Lemma 2:
"!
"
"2
!
!
From (8), note that ∂ 2 z/∂a2i ∂ 2 z/∂u2i > ∂ 2 z/∂ai ∂ui for
all ai ≥ 0 and ui ≥ 0, which implies that the objective is concave. Analogous to the
one-action case, therefore, the first-order conditions with respect to ai and ui of (8),
(A1)
bi h! (ai + ui ) − f ! (ai − (1 − αi )Ai − αi Sa ) ≤ 0 and
(A2)
bi h! (ai + ui ) − f ! (ui + (1 − µi )Ui + µi Su ) ≤ 0
characterize agent i’s action choices, where condition (A1) is an equality if ai > 0 and
condition (A2) is an equality if ui > 0.
The optimal choices of ai and ui implied by (A1) and (A2) must be finite because
h! (x) approaches 0 as x approaches infinity while f increases at an increasing rate.
Furthermore, ai and ui cannot both be zero because bi > 0, which implies that agent i
supplies a positive amount of at least one action. Otherwise, inequalities (A1) and (A2)
are violated because h! (x) approaches infinity as x approaches 0.
Let ai (Sa , Su ) and ui (Sa , Su ) denote the optimal action choices for agent i as
functions of the social norm parameters. Suppose ai and ui are interior for bi > 0.
Inspection of (A1) and (A2) implies that ai − (1 − αi )Ai − αi Sa = ui + (1 − µi )Ui +
µi Su at any interior optimum. By applying the implicit function rule to the first-order
conditions (A1) and (A2), (i) ai and ui can be written as implicit functions of Sa and Su
and (ii) the derivatives of ai and ui with respect to Sa and Su can be computed.
(A3)
∂ai (Sa , Su )
αi f !! [−bi h!! + f !! ]
= !!
∂Sa
f [−bi h!! + f !! ] − bi h!! f !!
αi G(i)
=
∈ [0, αi ),
1 + G(i)
1
where G(i) ≡ 1 − f !! (ai − (1 − αi )Ai − αi Sa )/bi h!! (ai + ui ) ≥ 1 because f !! > 0, h!! < 0,
and bi > 0 by assumption. In like fashion,
(A4)
(A5)
(A6)
∂ai (Sa , Su )
µi
=
∈ [0, µi /2),
∂Su
1 + G(i)
∂ui (Sa , Su )
−µi G(i)
=
∈ (−µi , 0], and
∂Su
1 + G(i)
∂ui (Sa , Su )
−αi
=
∈ (−αi /2, 0].
∂Sa
1 + G(i)
These results establish that, when ai and ui are interior, ai is increasing in Sa and Su
while ui is decreasing in Sa and Su .
We next assess how ai changes in Sa and Su on the boundaries, i.e., when ai = 0 or
ui = 0. There are four cases to consider:
Case 1: Suppose ai > 0, ui = 0, and the inequality in (A2) is strict. In this case, a
change in Sa , although it affects ai , does not affect ui because ui = 0 so
(A7)
∂ai (Sa , Su )
αi f !! (ai − Nai )
=
∈ (0, αi ).
∂Sa
−bi h!! (ai ) + f !! (ai − Nai )
Case 2: Suppose ai > 0, ui = 0, and (A2) is an equality. In this case, an increase in Sa ,
although it affects ai , does not affect ui because inequality (A2) is strict at ui = 0. A
decrease in Sa causes an increase in ui above zero because (A2) holds with equality at
ui = 0. Hence, the right-hand derivative is given by (A7) and the left-hand derivative is
given by (A3).
Case 3: Suppose ai = 0, ui > 0, and the inequality in (A1) is strict. In this case, a
change in Sa , does not affect ai because inequality (A1) is strict at ai = 0.
Case 4: Suppose ai = 0, ui > 0, and (A1) is an equality. In this case, an increase in Sa
causes an increase in ai because (A1) is an equality. A decrease in Sa does not affect ai
because ai is constrained to be non-negative.
Combined, these cases imply that for Sa above a threshold value, ai is increasing in Sa
and for Sa below that threshold, ai = 0.
2
Now consider how ai changes in Su on the boundaries. Similar reasoning to that
above implies that if ai (Sa , Su ) = 0 for some Su , then ai (Sa , X) = 0 for all X ≤ Su .
If ui = 0 for some Su , then ai (Sa , Su ) > 0 and ai (Sa , X) = ai (Sa , Su ) for all X > Su .
Thus, for Su below a threshold value, ai = 0. Above that threshold value, (A4) implies
ai in increasing in Su , until a second threshold is reached at which ui = 0. Above this
threshold, ai does not change in Su .
We next assess how ui changes in Sa and Su on the boundaries. Parallel arguments
to those above establish that ui (Sa , Su ) = 0 implies ui (Sa , X) = 0 for all X ≥ Su
and ui (Sa , Su ) > 0 implies ui (Sa , Su ) is strictly decreasing in Su . Thus, for Su below
a threshold value, ui is interior and decreasing in Su and for Su above that threshold,
ui =0. Furthermore, for Sa above a threshold, ui is zero. Below that threshold value,
(A6) implies ui is decreasing in Sa , until a second threshold is reached at which ai = 0.
Below this threshold, ui does not change in Sa .
In summary, ai is weakly increasing in Sa and Su while ui is weakly decreasing in
Sa and Su for all ai ≥ 0 and ui ≥ 0. Because ai (Sa , Su ) and ai (Sa , Su ) (although
monotone and continuous) are kinked, the derivatives below are right-hand derivatives.
For there to exist a unique equilibrium, if must be the case that there is a unique Sa
and Su that satisfy (3) and (10). Rewrite (10) as the equation
(A8)
#
ui (Sa , Su )di
Su + I #
=0
I di
and use this to characterize Su as an implicit function of Sa . The definition of Su and
the restrictions on the domain of the ui imply that Su ∈ (−∞, 0]. Because ui ≥ 0 for
all i ∈ I, it follows that the left-hand side of equation (A8) is weakly positive at Su = 0.
Furthermore, the left-hand side of (A8) (i) decreases continuously and monotonically as
3
Su decreases and (ii) approaches −∞ as Su approaches −∞ because (A6) implies that
∂
∂Su
$
#
=1 + I
>1 − µ̄
#
ui (Sa , Su )di
Su + I #
I di
∂ui (Sa ,Su )
di
∂Su
#
I di
% &&
&
&
&
Sa
>0.
It follows from these observations that for any Sa there is a unique value of Su satisfying
(A8).
Let Su (Sa ) denote the unique value of Su that satisfies (A8). Applying the implicit
function rule to (A8), we have
# ∂ui
di
∂Su (Sa )
a
= − # I ∂S∂u
.
i
∂Sa
1
+
di
∂Su
I
Now rewrite (3) as the equation
(A9)
#
ai (Sa , Su (Sa )) di
#
F ≡ Sa − I
=0
I di
and totally differentiate with respect to Sa to obtain
('
'
dF
∂ai
∂ai ∂Su
= 1−
−
di
di
dSa
∂Sa
∂Su ∂Sa
I
Ii
'
'
∂ai
∂ai ∂Su
∝
1−
−
di +
1−
∂Sa
∂Su ∂Sa
Ii
Ia
'
∂ai
∂ai ∂Su
−
di
+
1−
∂Sa
∂Su ∂Sa
Iu
'
'
∂ai
∂ai ∂Su
=
1−
−
di +
1−
∂Sa
∂Su ∂Sa
Ii
Ia
∂ai
∂ai ∂Su
−
di
∂Sa
∂Su ∂Sa
'
∂ai
di +
di
∂Sa
Iu
where Ii , Ia , and Iu are a partition of I such that j ∈ Ii iff (A1) and (A2) are equalities,
j ∈ Ia iff (A1) is an equality and (A2) is a strict inequality, and j ∈ Iu iff (A2) is an
equality but (A1) is a strict inequality. Provided the integration is over a set of positive
4
#
measure, we show that each of these integrals is positive. Obviously, Iu di > 0. From
(A7),
'
'
∂ai
1−
di ≥ (1 − ᾱ)
di
∂Sa
Ia
Ia
> 0.
Finally,
'
∂ai
∂ai ∂Su
−
di
∂Sa
∂Su ∂Sa
Ii
# ∂ui
'
'
∂ai
∂ai
∂Sa di
=1 −
di +
di # Ii ∂u
i
Ii ∂Sa
Ii ∂Su
Ii 1 + ∂Su di
$
%$
% '
'
'
'
∂ai
∂ui
∂ui
∂ai
∝ 1−
di
1+
di +
di
di
Ii ∂Sa
Ii ∂Sa
Ii ∂Sa
Ii ∂Su
$
%$
% '
'
'
'
αi G(i)
µi G(i)
αi
µi
> 1−
di
1−
di −
di
di
Ii 1 + G(i)
Ii 1 + G(i)
Ii 1 + G(i)
Ii 1 + G(i)
$
%$
%
$'
%2
'
'
G(i)
G(i)
1
> 1 − ᾱ
di
1 − µ̄
di − ᾱµ̄
di
Ii 1 + G(i)
Ii 1 + G(i)
Ii 1 + G(i)
$'
% $'
%
$'
%2
1 + G(i) − ᾱG(i)
1 + G(i) − µ̄G(i)
1
=
di
di − ᾱµ̄
di
1 + G(i)
1 + G(i)
Ii
Ii
Ii 1 + G(i)
$'
% $'
%
'
'
1
(1 − ᾱ)G(i)
1
(1 − µ̄)G(i)
=
di +
di
di +
di
Ii 1 + G(i)
Ii 1 + G(i)
Ii 1 + G(i)
Ii 1 + G(i)
$'
%2
1
− ᾱµ̄
di
Ii 1 + G(i)
$'
%2
1
di
≥ (1 − ᾱµ̄)
Ii 1 + G(i)
1−
>0,
Hence, dF/dSa > 0 and so
#
ai (Sa , Su (Sa ))di
#
Sa − Ii
=0
Ii di
must have a unique solution value for Sa because the left-hand side is (i) non-positive
at Sa = 0, (ii) strictly increasing in Sa , and (iii) approaches infinity as Sa approaches
infinity.
5
Proof of Corollary 2:
For interior a and u, (1) and (2) are equalities. The defini-
tions (3) and (10) imply P ≡ −a + Sa = 0 and Q ≡ u + Su = 0. These four equations
(i) define the equilibrium values of the endogenous variables a, u, Sa , and Su in terms
of the exogenous variables α, µ, A, U , and b and (ii) have continuous partial derivatives
with respect to each of the endogenous and exogenous variables. The corresponding Jacobian determinant is
&
&
& ∂(za , zu , P, Q) &
&
|J| = &&
∂(a, u, Sa , Su ) &
&
&
zau
∂za /∂Sa ∂za /∂Su &
& zaa
&
&
zuu
∂P/∂Sa ∂P/∂Su &
& z
= & au
&
& ∂P/∂a ∂P/∂u ∂P/∂Sa ∂P/∂Su &
&
&
∂Q/∂a ∂Q/∂u ∂Q/∂Sa ∂Q/∂Su
&
& !!
bh!!
αf !!
0 &
& bh − f !!
&
&
bh!! − f !!
0
−µf !! &
& bh!!
=&
&
0
1
0 &
& −1
&
&
0
1
0
1
= [(1 − α) (1 − µ) f !! − (2 − α − µ) bh!! ] f !!
> 0,
where subscripts on z denote partial differentiation with respect to the subscripted
variable, f !! is evaluated at either a−Na or u+Nu (and, in equilibrium, a−Na = u+Nu ),
and h!! is evaluated at a + u. Therefore, by the implicit function rule, for any change in
an exogenous variable, y,
(A10)
(A11)
da
du
∂za dSa
∂za dSu
+ zau
+
+
dy
dy
∂Sa dy
∂Su dy
da
du
∂zu dSa
∂zu dSu
zau
+ zuu
+
+
dy
dy
∂Sa dy
∂Su dy
da dSa
−
+
dy
dy
dSu
du
+
dy
dy
zaa
∂za
,
∂y
∂zu
=−
,
∂y
∂P
=
, and
∂y
∂Q
=
,
∂y
=−
where y is any one of the exogenous variables. Observe that ∂P/∂y = ∂Q/∂y =
6
∂zu /∂Sa = ∂za /∂Su = 0. Hence, (A10) and (A11) reduce to
$
%
∂za da
du
∂za
zaa +
+ zau
=−
and
∂Sa dy
dy
∂y
$
%
da
∂zu du
∂zu
zau
+ zuu −
=−
.
dy
∂Su dy
∂y
Solving this last system of equations for da/dy and du/dy yields
)
*(
da
!!
!! ∂za
!! ∂zu
= [−bh + (1 − µ)f ]
+ bh
|J| and
dy
∂y
∂y
)
*(
du
!!
!! ∂zu
!! ∂za
= [−bh + (1 − α)f ]
+ bh
|J| .
dy
∂y
∂y
The comparative-static derivatives in the statement of the corollary follow directly
from these last two expressions upon observing that 0 < −bh!! , 0 < (1 − α)f !! , and
0 < (1 − µ)f !! .
Proof of Proposition 1:
Because Nâ = (1 − α)A + α [δah + (1 − δ) al ] and
ai − Nai = ui + Nui for any agent i, it is possible to write the difference in costs between
single and separate organizations as
+
C1 − C2 =2 δf ((1 − α) (ah − A) + (1 − δ) x)
+ (1 − δ) f ((1 − α) (al − A) − δx)
− δf ((1 − α) (ah − A))
− (1 − δ) f ((1 − α) (al − A))
(A12)
+
,
δ(1 − δ)(ah − al )(α − µ)(kh − kl )
,
1−µ
where x = α(ah − al ). We first show that the term in square brackets is always non-
negative. Note that the equation (A12) equals 0 when x = 0 and that x must be nonnegative. Furthermore, the first derivative (A12) with respect to x yields
2δ (1 − δ) [f ! ((1 − α) (ah − A) + (1 − δ) x) − f ! ((1 − α) (al − A) − δx)] > 0
7
because f !! > 0 and (1 − α) (ah − A) + (1 − δ) x > (1 − α) (al − A) − δx.
Because the term in square brackets is non-negative, the costs associated with
a single organization are greater than with separate organizations if and only if
[δ(1 − δ)(ah − al )(α − µ)(kh − kl )] /(1−µ) is not too small (negative), which occurs when
(α − µ)(kh − kl ) < 0.
Proof of Proposition 2:
Differentiating the cost (16) with respect to the parameters
determining the sensitivity to the social norms yields
dC(a)
(a − A)
= −2(a − A)f ! ((1 − α)(a − A)) −
k ∝ (A − a) and
dα
1−µ
dC(a)
(a − A)(1 − α)
=k
∝ (a − A).
dµ
(1 − µ)2
Proof of Observation 1:
Let αh denote the parameter α for the agents who are
most sensitive to the social norm for the desired action and αl < αh denote the parameter α for the agents who are least sensitive to that norm. Consider first the case where
the level of desired action a exceeds the common personal norm A. Proposition 2 implies
that the αh agents are preferred. Under the contract that induces a from all αh agents
when type is observable, each of these agents attains the reservation level of utility v the
and faces a norm for the desired action Nah = αa + (1 − α)A. Assume that same contract
is offered for all agents willing to join the organization when type is not observed. The
proof for this case is completed by showing that, if all agents conjecture that only the
αh types will accept the offer, only the αh types will weakly prefer to accept the offer
and the αl types will strictly prefer to not accept the offer. An αh agent weakly prefers
to accept the offer because he anticipates attaining v. An αl type that accepts the offer
faces a lower social norm for the desired action than an αh type because αl < αh and
a > A implies Nal < Nah . As a consequence an αl type attains a level of expected utility
less than v if he joins the organization. Hence, an αl type strictly prefers to not accept
the offer. The proof for the case where a < A is analogous. Finally, the principal is indifferent between types when a = A.
8
Proof of Observation 2:
Let µh denote the parameter µ for the agents who are
most sensitive to the social norm for the undesirable action and µl < µh denote the
parameter µ for the agents who are least sensitive to that norm. Consider first the case
where the level of desired action a exceeds the common personal norm A. Proposition
2 implies that the µl agents are preferred. The proof for this case is by contradiction.
Conjecture a contract offer that induces each µl agent to accept the contract and take
action a and that induces each µh to not accept the offer. Let vl denote the level of
expected utility attained by a µl type that accepts the contract, where vl must weakly
exceed v. It must be the case that the social norm for the undesirable action faced by
any individual agent accepting the contract must be
Su = −u = U −
(a − A)(1 − α)
< U,
1 − µl
where α is the common sensitivity to the social norm for the desired action and U is the
common personal norm for the undesirable action. Because Su < U , it follows that the
norm for the undesirable action faced an agent of type µh who joins the organization
must be strictly less than that for an agent of type µl . Hence, the expected utility of an
agent of type µh who joins the organization, vh , must be strictly greater than that for an
agent of type µl , vl . The contradiction follows from the observation that a type µh agent
must strictly prefer to accept the contract and join the organization because vh > vl ≥ v.
The proof for the case where a < A is analogous.
9