Many businesses organize their employees in teams. According to Lawler (2001), 72 percent of Fortune 1000 companies make use of work teams,
de-…ned as groups of employees with shared goals or objectives. A large man-agement literature has thus emerged investigating team composition, team
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
rho eta
Figure 3: Illustration for Proposition 10
compensation, team leadership, and so forth. However, this literature is mainly empirical, and the theoretical literature is conceptual rather than formal.
The economics literature on teams is, in comparison, rather small. Theory has mainly focused on how the well-known free-rider problem can be solved or mitigated, while questions related to team size and team composition has remained unanswered, or not even asked. Moreover, endogenous formation of teams, in which …rms deliberately choose to hold a team of workers ac-countable for their joint output, is not well understood.
Our paper contributes to the literature by deriving testable theoretical pre-dictions on team incentives, team size, team composition and team formation.
We’ve done so by analyzing optimal self-enforcing (relational) contracts be-tween a principal and a set of agents where only aggregate output can be observed. We have then considered how the e¢ ciency of the contract is af-fected by variations in the number of agents and in the correlations between the agents. Finally, we have compared with a situation where individual output is observable.
First, we showed that the optimal team contract entails an incentive scheme in which each agent is paid a maximal bonus for aggregate output above a threshold and a minimal bonus otherwise. We then considered optimal
team size. To the extent this is studied in the formal literature, the standard result is that more agents increases the free-rider problem and thus weakens incentives and e¤ort. In our model, this is not necessarily the case. More agents in a team have three e¤ects: First, it reduces the marginal incentive e¤ect of a given bonus, which is the standard 1/n free-rider problem. Second, it also reduces the teams’ outside option. This strengthens the relational contract and thus allows for higher-powered incentives and thus higher e¤ort.
This positive e¤ect of more agents is particularly strong if the agents’ex post bargaining power is high. Finally, it a¤ects the variance of the performance measure. For positive correlations between the agents’outputs, the variance increases, while for negative correlations the variance is reduced. The latter is bene…cial for the team because it increases the marginal incentives for each team member to provide e¤ort.
Our model thus predicts that teamwork is more robust and more e¢ cient when the team has high (ex post) bargaining power and when the team members’outputs are negatively correlated. The former implies that team-work is more e¢ cient (or prevalent) when the team is in a position to hold up values and sell their products in an alternative market. This is typically the case in human capital-intensive industries where groups of employees can potentially walk away with ideas, clients, innovations, etc
The latter - negative correlations - relates to questions concerning opti-mal team composition. In the management literature a central question is whether teams should be homogenous or heterogeneous with respect to both tasks (functional expertise, education, organizational tenure) and bio-demographic charachteresics (age, gender, ethnicity). One can conjecture that negative correlations are more associated with heterogeneous teams than homogenous teams, and also more associated with task-related diversity than with bio-demographic diversity. There is no reason to believe that e.g. men and women’s outputs are negatively correlated. However, workers with dif-ferent functional expertise may be di¤erently exposed to common shocks, or meet di¤erent sets of demands from customers or superiors. This can create negative output correlations.
Interestingly, a comprehensive meta-study by Horwitz and Horwitz (2007), investigating 35 papers on the topic, …nds no relationship between bio-demographic diversity and performance, but a strong positive relationship between team performance and task-related diversity. An explanation is that task-related diversity creates positive complementarity e¤ects. We point to an alternative explanation, namely that diversity may create negative corre-lations that reduces variance and thereby increases marginal incentives for e¤ort. The team members “must step forward when others fail”. Diver-sity and heterogeneity among team members can thus yield considerably e¢ ciency improvements.10
We have also compared with a situation where individual output is observ-able. For a parametric (normal) distribution, we have shown that the op-timal contract is an RPE (relative performance evaluation) scheme; a form of a tournament, where the conditions for an agent to obtain the (single) bonus are stricter for negatively compared to positively correlated outputs.
The e¢ ciency of the RPE contract is shown to increase with the number of agents, and to improve with higher correlation (both positive and negative).
Now, if the …rm can initially choose between organizations that allow for (a) only aggregate output or (b) individual outputs to be observed, we show that the …rm may choose (a), i.e. to organize production as a team. Thus, even if alternatives (a) and (b) are equally costly to set up initially, the team alternative may yield a higher subsequent surplus.
There are two reasons for this. One is that teams create worse outside op-tions. This is particularly the case under high ex post bargaining power.
When individual outputs are observable, high bargaining power creates quite e¢ cient spot contracts, while under team production the free-rider problem dampens the e¢ ciency of the spot contract. Hence, since worse outside op-tions strengthens the relational contract, higher bargaining power favor the team alternative.
10Hamilton et al (2003) provides one of a very few empirical studies on teams within the economics literature. They …nd that more heterogeneous teams (with respect to ability) were more productive (average ability held constant).
Second, negative correlations are even more bene…cial for the relational team contract than for the relational RPE contract. That is, although e¢ ciency in both alternatives increases with more negatively correlated outputs, the team alternative is more likely to be superior under such conditions. Hence, according to our model, team work is not only more robust and e¢ cient under high bargaining power and negatively correlated outputs. The likelihood for
…rms to deliberately choose the team alternative, even if individual output is observable, is also higher under these conditions.
APPENDIX
Proof of Lemma 1:
It remains to verify that the conditions are necessary. Given a scheme ~;~ that satis…es (IC, EA, EP) and hence implements e. Letinfy(~i(y) n1 y)
i, and let i(y) = ~i(y) i and i = ~i + 1 i. Then IC holds for . Moreover, infy( i(y) 1n y) = 0and EA holds for ( ; ), since each agent’s payo¤ is (by construction) unchanged for each realization y:
i(y)+1 ( i+E( i(y)je) c(ei)) = ~i(y)+1 (~i+E( ~i(y) e) c(ei)) Then EP also holds, since the principal’s payo¤ must be unchanged as well:
i i(y)+1 i(E(xije) E( i(y)je) i) = i~
i(y)+1 i(E(xije) E~
i(y) ~i).
Taking inf and sup in EA and EP, respectively, and adding, we get supy( i i(y) y) iinfy i(y) n1 y 1 i(W(ei) Wn(es) This shows that EC holds for , since infy i(y) 1n y = 0 for alli. This completes the proof.
Proof of Proposition 1.
The Lagrangian for the problem of maximizing total surplus ( iW(ei)
i(E(xijei) c(ei))) subject to EC and the ’modi…ed’IC constraint (2) is:
L= iW(ei) +R
1 ( iW(ei) nWs) ibi(y) (y)dy + i i R
bi(y)gl(y;l)lei(e1:::en)dy+n1 x0(ei) c0(ei) where we have used @e@
iE(bi(y)je1:::en) = R
bi(y)gl(y;l)lei(e1:::en)dy. This yields
@L
@bi(y) = igl(y;l)lei(e1:::en) (y) 0, bi(y) 0, (compl slack) If i = 0, then bi(y) = 0 for all y, hence ei = esi. For bi(y) > 0 we have
igl(y; l)lei = (y) and hence from (2)
i(c0(ei) n1 x0(ei)) = R
bi(y) (y)dy 0
Thus for ei > esi we cannot have i <0. Hence we must have all i >0.
For y > y0 we have gl(y;l) > 0 and hence (y) > 0, implying that EC is binding and at least one bonus is positive. In a symmetric solution the bonuses will thus all be equal and maximal for y > y0.
On the other hand, for y < y0 we have gl(y;l) < 0 by MLRP and hence bi(y) = 0 for all i.
Finally suppose l(e1:::en) = iei, and assume the solution is asymmetric; say that ei < ej. Let b0 = (bi+bj)=2and consider
R b0(y)gl(y;l(e1:::en))dy = 12R
bi(y)gl(y;l(e1:::en))dy+12R
bj(y)gl(y;l(e1:::en))dy
= 12c0(ei) + 12c0(ej) c0(ei+e2 j)
Hence the bonusb0(y)to each of i and j is feasible and would induce e¤ort at least ei+e2 j =e0 from each. Thus a slightly lower bonus to each is feasible and will induce e¤ort e0 from each. This yields higher value since the objective is concave. QED
Proof of Proposition 3.
To get a contradiction, suppose for givenei that FOA is not valid for arbitrar-ily large s2, and hence that, for any suchs the agent’s payo¤ is maximal for someei =ei(s)< ei. This optimum cannot be forei(s) = 0, since that would give payo¤b(1 H(ei)) + 0, wherebh(0) + c0(ei) = 0, and hence payo¤
(excluding 0) c0(eh(0)i) (1 H(ei)) = (c0(ei) )R1
ei e x
2
2s2dx, which would become arbitrarily small for s su¢ ciently large. In comparison, the corre-sponding payo¤ forei =ei isb(1 H(0))+ ei c(ei) = c0(eh(0)i) 12+ ei c(ei), which is large for s large since h(0) = p1
2 s.
Now, for ei = ei(s) > 0 being an optimum, we must have bh(ei ei) + c0(ei) = 0, where bh(0) + c0(ei) = 0 and hence bh(0)(h(eh(0)i ei) 1) = c0(ei) c0(ei), wherebh(0) =c0(ei) =const. This implies (since h(eh(0)i ei) = exp( (ei2se2i)2)) that ei = ei(s) ! ei, and hence that FOC has a solution
ei =ei(s)arbitrarily close toei forssu¢ ciently large. But this is impossible, since the slope of the FOC expression atei is bh0(0) c00(ei) = c00(ei), and hence there must be a left neighborhood where this expression is strictly positive for all s. This proves the …rst statement in the proposition.
The formulas for the agent’s payo¤s for ei = 0 and ei = ei, i.e. (c0(ei) )R1
ei e x
2
2s2dx+ 0 and c0(eh(0)i) 12+ ei c(ei) + 0, respectively, show that the former will dominate for s su¢ ciently small (since h(0) 1s) Hence FOA cannot be valid for s su¢ ciently small.
Now consider variations in . Suppose FOA is valid for 2 0;12 and hence that b(1 H(ei ei)) + ei c(ei)< b(1 H(0)) + ei c(ei)for allei < ei when bh(0) + c0(ei) = 0. Consider 0 > and let b0 < b be given by
b0h(0) + 0 c0(ei) = 0, i.e. b0h(0) + 0 =bh(0) + Then consider
b0(H(ei ei) H(0)) + 0(ei ei)
= (b0 b)(H(ei ei) H(0)) + ( 0 )(ei ei) +b(H(ei ei) H(0)) + (ei ei)
>(b0 b)(H(ei ei) H(0) h(0)(ei ei)) + (c(ei) c(ei));
where the inequality follows from FOA being valid for ( ; b). The CDF H(x) is concave for x > 0 (since then H00(x) = h0(x) < 0), hence the term multiplying (b0 b) is negative. Since b0 < b we then see that FOA is valid also for (b0; 0).
Finally consider the case of iso-elastic costs. In general, a su¢ cient condition for FOA to be valid is that the agent’s FOC has no solution for ei < ei. Due to the bell-shaped form of the normal density, this will occur if the variance s2 exceeds a critical value that yields tangency between the agent’s marginal revenue and marginal cost curves at some ei < ei This critical s2 is de…ned
by the following conditions:
bh(0) + c0(ei) = 0 =bh(ei ei) + c0(ei) (23)
bh0(ei ei) =c00(ei) (24) We have h(x) = (xs)1s, where (z) = e z2=2=p
2 is the standard normal density. For iso-elastic costs the above conditions (23 - 24) thus take the form
b
s (0) + km(ei)m 1 = 0 = bs (eisei) + km(ei)m 1 and
b
s (eisei)eisei1s =km(m 1)(ei)m 2 Letting now
B = sbm (0); 0 = sm 1, d = esi, d= esi; the above conditions, re‡ecting (23 - 24) can be written as
B + 0 km(d )m 1 = 0 =Bexp( (d 2d)2) + 0 kmdm 1 and Bexp( (d 2d)2)(d d) = km(m 1)dm 2
For = 0 this yields
exp((d 2d)2) = (dd)m 1 and (d d) = (m 1)d 1 Letting x= dd this yields
d2(x 1)2=2 = (m 1) lnx and (x 1)d2 =m 1
and hence x is the solution to x 1 = 2 lnx, i.e. x 3:512 9. So we have (d )2 = d2x2 = mx 11x2, and hence esi = d = K0p
m 1, where K0 = q x2
x 1 2:216. This proves the …rst assertion for iso-elastic costs.
To verify the second assertion, note that for m = 2 the conditions (23 - 24) de…ning the critical s take the form
B + 0 2kd = 0 =Bexp( (d 2d)2) + 0 2kd and Bexp( (d 2d)2)(d d) = 2k
This yields
2k(d d) = B 1 exp( (d 2d)2) = 2kexp((d 2d)2)(d d) 1 1 exp( (d 2d)2) and hence
(d d)2 = exp((d 2d)2) 1 :
So (d d)2 =z where ln(z+ 1) =z=2, i.e. z 2:513. This yields B + 0 2kd = 0 and Bexp( z2)p
z = 2k where 0 = s and d = esi. Hence we have
ei =2k
s =d 2k0 = 2kB = exp(z2)=p
z 2:216 This completes the proof.
Proof of Proposition 4
We have H(x;s) = (xs), and h(x;s) = (xs)1s where () is the N(0,1) CDF and ()its density. The relations (8 - 10a) can then be written as
b(1 (
s )) c(e ) b(1 (e e0
s )) c(e0) (25) b (
s )1
s c0(e ) = 0 =b (e e0
s )1
s c0(e0) (26)
b 1 W(e ) (27)
Note …rst that for the minimals=sc for which the FOA is valid, all relations hold with equality, and = 0. Denote the associated e¤ort and bonus by e =ec and b =bc, respectively.
We show below that for any s < sc, the optimal threshold scheme of this type has all relations (25 - 27) binding, and implements some e 2(ec; eu):
Given the latter property, it is straightforward to see thate !eu ass !0.
For suppose that (at least along a subsequence)e !el < eu ass !0. Note that we then must have s ! 1 as s ! 0. For if not, then b ! 0 by FOC for e in (26), which implies a negative payo¤ at e . For the same reason we must also have e es0 ! 1. Then we must havee0 !e0l = 0 as s!0, for otherwise the payo¤ at e0 would converge to c(e0l)<0. This is impossible, since the payo¤ ate0 exceeds that ate= 0, and hence must be non-negative.
Taking limits in the …rst relation (25) with equality, we then get limb 1 c(el) = 0, and hence from the last equation (for b) that c(el) = 1 W(el).
This cannot hold for el < eu, hence we must haveel =eu.
It remains to prove the claim that for any s < sc, the optimal threshold scheme of this type has all relations (25 - 27) binding, and implements some e 2(ec; eu):
We …rst show that for any s < sc, e¤ort e = ec can be implemented with b = bc, and a suitable choice of . Indeed, …x e = ec and b = bc, and let (s) and e0(s) be de…ned by the FOCs (26) for e and e0, respectively. For s =sc we have = 0 and all relations hold with equality. We can show that the payo¤ di¤erence will increase ass decreases; this follows from (as shown below) dds <0, where is the payo¤ di¤erence;
=b( (e e0
s ) (
s )) (c(e ) c(e0)); (28) and = (s) and e0 =e0(s).
This shows that for any s less than the critical level sc, it is feasible to implement the e¤ort e = ec associated with sc, and that this can be done by adjusting the threshold downwards, keeping the bonus …xed at b = bc (the FOA bonus associated with sc). Moreover, the agent’s payo¤ at e is then strictly larger ( >0) than his payo¤ at the other local maximum e0. (And the payo¤ at e0 is positive, since it exceeds the payo¤ at e = 0.) For
…xed s < sc this leaves room for increasing the bonus and/or the threshold (reducing ), thereby increasing the implemented e¤ort.
Note that by increasing b, keeping …xed, e¤ort e will increase (by FOC), and the payo¤ di¤erence will also increase, since ddb = (e es0 ) ( s )>0 (by the envelope property). Hence b and e can be increased if the EC constraint b 1 W(e ) is not binding.
By reducing , keeping b …xed, e¤ort e will increase (by FOC), and the payo¤ di¤erence will decrease, since dd = (c0(e ) c0(e0))1s > 0. Moreover, this relaxes the EC constraint b 1 W(e ), and is hence feasible. Thus can be reduced and e¤ort e increased if the payo¤ constraint 0 is not binding.
These arguments show that, for givens < sc, the optimal (feasible) threshold bonus has the EC constraint as well as the payo¤ constraint binding. Thus all the relations (25 - 27) will be binding, and the scheme implements an e¤ort e > ec. This veri…es the claim.
It remains to prove dds <0, where is given by (28), = (s)ande0 =e0(s) are given by the FOCs in (26), and b and e are kept …xed (e =ec; b =bc).
To this end, note that we have, for the payo¤ at e0:
d
ds b(1 (e es0 )) c(e0) = b (e es0 )dsd(e es0 ) c0(e0)deds0
= b (e es0 )s12
d(e0+ )
ds s (e e0 ) c0(e0)deds0
=c0(e0)1s d(eds0+ )s+ (e e0 ) c0(e0)deds0
=c0(e0) dds +e es0 Similarly, for the payo¤ at e :
d
ds b(1 ( s )) c(e ) = b ( s )dsd( s )
=b ( s )s12(sdds ) =c0(e )(dds s) Hence
d
ds =c0(e )(dds s) c0(e0) dds + e es0
= (c0(e ) c0(e0))(dds s) c0(e0)e se0
From the FOCs (26) and the fact that 0(z) = z (z)we obtain
Letting h(x) = ff(x;e)e(x;e) here denote the likelihood ratio, we have, integrating by parts hence we see that s(n) is increasing in n.
This implies that c0(e)s(n;e)x0(ei) shifts down with n, and hence that e¤ort per agent (e) increases.
Proof of Proposition 6
Consider the problem of maximizing total surplus ( iW(ei) = i(E(xijei) c(ei))) subject to (14) and the ’modi…ed’ IC constraint (15). Letting x = (x1:::xn) and e= (e1:::en), the Lagrangian for the problem is
L= iW(ei) +R
1 ( iW(ei) nWs) ibi(y) (x) + i i R
bi(x)fei(x;e) + n1 x0(ei) c0(ei) ; where we have used @e@
iE(bi(x)je) = R
bi(x)fei(x;e) (and the integrals are multiple integrals over vector x). This yields
@L
@bi(x) = ifei(x;e) (x) 0, bi(x) 0, (compl slack) If two agents are paid a positive bonus, then ifei(x;e) = (x) = jfej(x;e), so their weighted likelihood ratios must be equal; iff(x;e)ei(x;e) = jffej(x;e)(x;e). But this can only occur for a set of measure zero, hence at most one agent is paid a bonus (almost surely).
If fei(x;e) < 0 then bi(x) = 0. If fei(x;e) > 0 then (x) > 0, and agent i is paid the bonus (bi(x) > 0) if and only if he has the largest weighted likelihood ratio. Also, the bonus is maximal since EC is binding.
In a symmetric solution the weights (multipliers) i will be equal, and hence the agent with the largest likelihood ratio will get the bonus, provided this ratio is positive.
Now consider variables with identical variances and identical correlations (corr(xi; xj) = alli6=j.). The multinormal density has the form
Cexp( 12(x e)0 1(x e))
where is the covariance matrix. Under our assumptions we have =s2R, where the correlation matrix R can be written as
R = (1 )I+ J;
where each element of J isJik = 1. Note that J2 =nJ. We will show below that
R 1 = 1
(1 + (n 1) )(1 )Q, where Q= (1 + (n 1) )I J (29) Note that the matrixQ has elements(1 + (n 2) ) on the diagonal, and o¤ the diagonal.
From the formula for R 1 and the de…nitions of k1; k2 in the text it follows that the quadratic form in the multinormal density can be written
1
2(x e)0 1(x e) = 12(k1 izi2+k2 i6=jzizj); zi =xi ei Di¤erentiation of the density wrt ei then yields the formula (18) for the likelihood ratio in the text
From the formula (18) it follows that agent i0s likelihood ratio is positive i¤
the inequality in (19) holds. We now verify the last equality in (19), i.e. the validity of the expression forE(xijx i). To this end note that for the normal distribution the conditional expectation of, say x1 can be written
E(x1jx 1) = E(x1) + 12 221(x 1 Ex 1);
where 12 = s2( ; :::; ) is the (n 1) dimensional vecor of covariances cov(x1; xj), j > 1, and 22 is the covariance matrix for x 1 = (x2:::xn)0. It follows from (29) that s2 221 has the same form as R 1, with n replaced by n 1. Hence 12 1
22 = ( ::: )Rn11, and each element of this (1 n) matrix is, from (29):
(( ::: )Rn11)i = (1+(n 2) )(1 )((1 + (n 3) ) (n 2) ) = (1+(n 2) ) This veri…es the last equality in (19).
It remains to verify the formula (29) forR 1. To this end consider RQ = ((1 )I+ J)((1 + (n 1) )I J)
= (1 )(1 + (n 1) )I+ (1 + (n 1) ) J (1 ) J 2nJ
= (1 )(1 + (n 1) )I+ 0J This proves the formula for R 1.
Finally we check positive de…niteness, and verify that Ris positive de…nit i¤
1 + (n 1) >0. To verify this we will show that the determinant of R is
n= (1 )n 1((n 1) + 1)
To see this, note that the element qij in the inverse matrix Q =R 1 equals Cji= n, where Cji is the cofactor of element ji in matrix R. So for element nn we have qnn = Cnn= n = n 1= n, hence (1+(n1+(n1) )(12) ) = n 1= n. The formula for n then follows by induction.
Proof of Proposition 7
We will show that for e1 =e2 the marginal gain from e¤ort is R
Bfe1(xje1e2) = 1s (e1se1; )
where (a; ) is the function de…ned in (22) in the text. The agent’s FOC then takes the form bs (0; ) =2 c0(e1) = 0, which is precisely the formula (21) stated in the proposition.
The normal density depends on (vector) x via a quadratic form in x e„
hence it satis…es fei(x;e)dx= fxi(x;e). Taking account of the de…nition of the set B of outcomes (the set where agent 1 is paid a bonus) in (20) , we thus have
R
Bfe1(xje1e2) = Re2
1dx2R1
e1+ (x2 e2)dx1+R1
e2 dx2R1
x2 dx1 fx1(xje1e2)
= Re2
1dx2[f(xje1e2)]xx1=1
1=e1+ (x2 e2)+R1
e2 dx2[f(xje1e2)]xx1=1
1=x2
where
f(xje1e2) =kexp (x1 e1)2+(x2 2(1e2)2 22 (x)s2 1 e1)(x2 e2)
=kexp ((x1 e1) (x22(1e2))22+(1)s2 2)(x2 e2)2 , k = 1
2 s2p
1 2
This implies
[f(xje1e2)]xx1=1
Hence we have
Given that the marginal gain to e¤ort for agent 1 in the modi…ed tournament scheme can be written as
2 , this in turn implies
a(0; )
This can be rearranged to yield the formula stated in the proposition.
We will now show that for s su¢ ciently large, the FOC has a single solution (e1 =e1), which then must be a maximum, since the local SOC holds strictly for s large. To get a contradiction, suppose that, for every s0 > 0 there is s > s0 such that FOC has a solution e1 =e1(s)6=e1, i.e. such that
b
s (e1se1; ) + c0(e1) = 0 = bs (0; ) + c0(e1);
implying
(e1se1; )= (0; ) = cc00(e(e1)
1)
Then letting s! 1 (if necessary along a subsequence) we see that e1(s)! e1. Hencea(s) = e1(s)s!e1 !0, and the last equation above yields
1) . This yields a contradiction and thus completes the proof.
Proof of Proposition 9
It follows from (5) that the critical discount factor to implement …rst best e¤ort eF B is for a team with n independent agents given by
c0(eF B) 1n snM = F B
Consider now quadratic costs: c(e) = k2e2, with associated surplus per agent:
W(e) = e k2e2
First-best e¤ort and surplus is then
: 1 = keF B; WF B = 1k k2(1k)2 = 2k1
Spot e¤ort is given by =n) =c0(es) = kes; and surplus is then:
W(es) = nk k2(nk)2 = 2kn12 (2n ) Hence the critical discount rate is here given by
F B
d
Hence, if 3 2, i.e. 23, then the critical discount rate F B is increasing for n 2, meaning that teams with n >2 do worse than teams with n = 2 with respect to achieving FB.
The critical discount rate F B is always increasing for n > 3 (since < 1), hence teams withn >3will always do worse than teams withn= 3regarding achieving FB.
Hence we have (for quadratic costs): wrt achieving FB, the optimal team size is n= 2 if < 0 = 0:805 42, and n= 3 if > 0.
This proves the …rst part of the proposition. The second part (comparison with the RPE tournament) follows from the proof for Proposition10 below.
Proof of Proposition 10
For normal distributions, we have that e¤ort in the modi…ed tournament is for given bonus given by FOC, see (21):
bp1
Consider next the relational Team contract.
The maximal e¤ort per agent that can be sustained in the team is given by (5), where now M =p
2 and s22 = 2v(1 + ), and hence (5) is c0(e) n1 p
2 p
2v(1 + ) =b= 1 (W(e) Ws(n))
Compare now critical 0s to implement FB. They are given by the following conditions, for the tournament and the team, respectively
1 = W(ec0(eF BF B)) Ws Hence the tournament has highest critical F B i¤
1< W(eF B1 ) Ws
where the last equality follows from straightforward algebraic calulations.
Consider then the critical combination 1221 p2p1p +1 = 1, ie
= 2(p +1+p +1 p2)
2p +1+p +1 p 2
The tournament has highest critcal F B above the curve de…ned by ( ).
Di¤erentiating, we obtain
The increasing curve shows that the tournament gets relatively better vis-a-vis the team when increases.
References
[1] Alchian, Armen A., and Demsetz, Harold. 1972. “Production, Informa-tion Costs, and Economic OrganizaInforma-tion.”American Economic Review 62 : 777–95.
[2] Baker, George, Robert Gibbons and Kevin J. Murphy. 2002. Relational contracts and the theory of the …rm. Quarterly Journal of Economics 117: 39-94.
[3] Baldenius, Tim and Jonathan Glover. 2010. Relational Contracts With and Between Agents. Working paper.
[4] Bar-Isaac, H. 2007. Something to prove: Reputation in teams.”RAND Journal of Economics, 38: 495–511.
[5] Bewley, Truman. 1999.Why Wages Don’t Fall During a Recession. Cam-bridge, MA: Harvard University Press.
[6] Bloom, Nicholas, and John Van Reenen. 2010. Why do management practices di¤er across …rms and countries?" Journal of Economic Per-spectives, 24: 203-24.
[7] Bull, Clive. 1987. The existence of self-enforcing implicit contracts.
Quarterly Journal of Economics 102: 147-59.
Quarterly Journal of Economics 102: 147-59.