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Economics Letters 27 (1988) 105-110 North-Holland

NATURAL

TEAM SHARING AND TEAM PRODUCTIVITY

Raul V. FABELLA University

of the

*

Philippines,

Quezon City, Phdippines

3004

Received 15 January 1988

We show that when effort levels are observable, natural team sharing where one’s proportional share in total output equals one’s proportional share in total input, sustains a first-best production level only for a function unique up to a constant of proportionality.

1. Introduction Alchian and Demsetz (1972) motivated their theory of the firm by the observation that free riding in partnerships will result in the undersupply of effort levels and inefficient operation. They suggested the hiring of a principal (capitalist) who monitors agents in return for a residual claim as one way out of the problem [see also Jensen and Meckling (1976)]. Holmstrom (1982) demonstrated that where effort levels are not observable and output is shared exhaustively by team members, no sharing scheme exists that sustains a first-best production level, thus formally confirming Alchian and Demsetz. He also showed, however, that where output is not exhaustively shared, an output responsive bonus and penalty scheme exists that sustains a first-best production level. This again calls for a monitor with a residual again highlighted. Macleod (1984)

claim. The apparent following Friedman

relative inefficiency of team arrangement is (1977) counters that a dynamic supergame

version of the team game with discounting has a ‘trigger strategy’ equilibrium that sustains solutions Pareto superior to the Cournot-Nash solution. In effect the group itself acts as the monitor bound by a public commitment to shirk forever once any member shirks. Agents must have infinite lifetimes and the sustained solutions need not necessarily be ‘first-best’. There is a growing evidence that firms with some team arrangement in the form of profit sharing are highly productive relative to those without [Jones and Svejnar (1982) and Fitzroy and Kraft (1987)]. The acclaimed positive role of cooperative labor relations in Japan and Germany [Gordon (1982) and Ouchi (1981)] seem to confirm the observation. All these seem to render the team inefficiency viewpoint paradoxical. Several tracks toward reconciliation suggest themselves. One is that monitoring models of the capitalist may underestimate the shirking creativity of agents under a straight wage contract. Worker collusion to restrict output results in surplus sharing in non-pecuniary form [e.g. Frank (1984)]. Shirking creativity feeds on asymmetric information on worker effort levels between workers and monitor. Also where hiring and firing are subject to extraneous restraints, efficiency advantage of labor contracts with monitoring decreases. Another view is that non-observa* The author is grateful to Professors Ruperto Alonzo, Felipe Medalla, Dante Canlas and Cayetano Paderanga, Jr. for helpful discussions.

0165-1765/88/$3.50

0 1988, Elsevier Science Publishers B.V. (North-Holland)

R. V. Fabella / Team productioiiy

106

bility of effort levels in teams may be exaggerated in view of the incentive to monitor on the part of each team member [Fitzroy and Kraft (1987)]. In this paper, we relax Holmstrom’s non-observability assumption and explore the team productivity impact of a sharing scheme, the natural team sharing, based on observed effort levels. The problem shifts strictly from one of moral hazard to one of adverse selection. The sharing scheme introduced is designed to elicit best effort from team members. In section 2, we review the ‘cooperative program’ that generates the cooperative first-best effort levels and thus production levels. We then contrast these outcomes with the non-cooperative (Cournot-Nash) effort levels generated by the ‘individual program’. In section 3, we assume effort levels to the observable and introduce the ‘natural team sharing’ scheme. We deduce formally the Cournot-Nash first-best effort condition (nc-fbe) under natural team sharing and show that the function satisfying the natural team sharing and nc-fbe is unique up to a constant of proportionality. We start with the framework of Holmstrom (1982).

2. The cooperative versus the individual program Consider a team of n members, Following Holmstrom (1982) we endow every member i with a utility, function U, = X, - y(l;) where X, is ith share in total output and y(l;) is increasing and strictly convex on I,, the ith effort level. The production function F is concave over { I, }, i = 1, 2,. _. , n. All functions are differentiable. The cooperative program is

yy[F({I,})-

<([,>I, i=l,L...,n.

The 1” condition

gives

F’=

T(‘,

i=l,2

,...,

n,

(2)

where F’ = 6F/Sl, and y’ = dy/dl,. Eq. (2) determines the ‘cooperative first-best effort levels’, { /,* }, and thus the first-best production level F({ I,* }). Assuming members to be individually rational, each maximizes y across effort levels with the knowledge total product is shared exhaustively and that his/her share is s, F where 0 < s, < 1, is i th proportional share, and C:=is, = 1. Member i then faces the problem

my

[s,F- V(l,>ly

i=1,2

,...,

n.

If we assume with Holmstrom that effort levels are unobservable or that the sharing scheme for any reason is divorced from effort levels [say, an effort-independent social welfare function governs allocation as in Macleod (1984)], then team members take s, as given. The lo condition under this condition is F’ = I/;‘/s,,

V,=1,2

,.._, n.

(4)

This determines the ‘non-cooperative (Cournot-Nash) effort levels’, {I,““}. Since for every i, s, < 1, in it can be expected that {I?} < { f,* } and F({ 1,“‘)) c F({ f* }). A general proof is provided Holmstrom (1982). In this case, moral hazard makes for inefficient team production.

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R. V. Fabella / Team productivity

3. Observable effort levels and natural team sharing Suppose we relax non-observability sharing scheme would be to allocate Definition.

A team sharing

{.r,}={i,/ii,},

scheme

and allow effort levels to be costlessly observable. A natural output according to contribution to total effort level, X/n=,/,. {s, } is ‘natural

team sharing’

if

i,2,...,n. j=l

It is obvious that C:=,s, = 1 and output is exhausted. It is also obvious that (a/al,) Xy=,.s, = 1. The effort level f, need not be in physical units but in some, say, monetary units, so that the sum is meaningful. Thus differential human capital is allowed. The individual program is now

m?

[iir/irl,jF-

i=l,2,...,n.

V(i.)].

The 1” condition

for maximum

(5)

is

(6) Assuming F’=

that F = F’Cl,

y’,

V,=1,2

)...)

and substituting

for F in eq. (6) gives

n,

(7)

which is identical to eq. (2) above. Thus with F = F’El,, ljnc= I*. Also with this condition the second-order condition is (l,/Cl,)F” - y” < 0 is always satisfied if F is concave. We thus have the following: Proposition I. When effort levels are observable and the natural team sharing is adopted, first-best V, = 1, 2,. , , , n. levels are supplied if and only if F = F’CI,, ProoJ

We need to show only necessity.

Substituting 1.

F’CI, for F, we have 1.

F’ - L F’ + --L-F’ Cl, Cl, a contradiction.

>

Q.E.D.

Suppose

F 2 F’CI,.

effort

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R. V. Fabelia / Team productivity

We call F = F’ l,, the ‘Cournot-Nash first-best effort condition’ under natural team sharing. We now need to characterize F which allows first-best effort supply in the context of the natural team sharing. We first have the following: Proposition 2. Where effort levels are observable cmd natural team sharing is adopted, first-best levels are supplied if and only if F is homogeneous of degree 1.

effort

Proof (Sufficiency). With natural team sharing, F’Cl, = F as required for first-best effort to be supplied by Proposition 1. Let F be homogeneous of degree 1. Then F = CT=, 1, F’. Dividing both sides by F and substituting the first-best effort condition Q, we have

Thus both the Cournot-Nash

first-best

effort condition

and natural

team sharing

are satisfied.

(Necessity). We prove this by contradiction. (a) Suppose F is not a homogeneous function. Then rF f Cl, FJ, Q r > 0. This is true as well for r = 1, so assume the latter. Dividing both rides by F, we have

a contradiction. (b) Now suppose F is homogeneous of degree r f 1. Then substituting the first-best effort condition V, we have

a contradiction.

rF = Cl, FJ. Dividing

both sides by F and

Q.E.D.

Proposition 2 says that the set of functions satisfying the Cournot-Nash first-best effort condition and natural team sharing belongs to the homogeneous of degree 1 family. In fact, but for the constant proportionality, the set is a singleton. Proportion 3. (Uniqueness) Let F ((1,)) = A(E:=,l,). (a) Fsatisfies (i) F= F’C,“_,l, and (ii)XF= F({Xl,}). (b) F is unique up to a constant of proportionality. Proof We prove (a). That F satisfies (ii) is obvious. Now F’ = FJ = A, Q i, j = 1, 2,. . _, n. By (ii), F=E:,“J,F’=AC;=,l, = F’C;,,l,. We prove (b). Assume (i) and (ii). Then F = Cl, F’ = F’Cl,, for any i = 1, 2,. . . , n. This means F’ = FJ, Q i, j = 1, 2,. . . , n. The total derivative of F is dF=

n xFJd!,=F'tdl,, /=I

foranyi.

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R. V. Fobella / Team productivity

By (i) we have

dF -=_ F

cq-

j=l

k

I,

J=l

Integrating,

we have

log A being the constant

of integration.

Thus

Q.E.D.

What Proposition 3 shows is that the requirements of first-best effort and natural team sharing are together very stringent, even in the context of observable effort levels. Holmstrom’s stress on exhaustive team sharing versus Alchian and Demsetz’ stress on the need for monitoring is well placed.

In fact, in analogy

Proposition

to Holmstrom’s

result, have the following:

4. The natural team sharing fails to sustain first-best

Proof: This follows from the uniqueness is not strictly

of F({ I, }) = AU,

effort if F is strict4

satisfying

concave.

(i) and (ii) in Proposition

3. F

concave.

4. Summary We start from the Holmstrom

(1982)

result that no sharing

scheme

sustains

first-best

production

where effort levels are non-observable and output is shared exhaustively among team members. Thus a monitor is called for. This formalizes the long-standing viewpoint due originally to Alchian and Demsetz

(1972)

that free riding in partnerships

leads to undersupply

of effort

and to inefficiency.

Relaxing the non-observability condition, we introduce ‘natural team sharing’ where proportional share in total output equals the share in total input, We deduced the first-best sustaining condition, the ‘Coumot-Nash first-best effort’ (nc-fbe) and we show that only homogeneous functions qualify (Propositions 1 and 2, respectively). We then show that the production function F is unique up to a constant of proportionality (Proposition 3). This means that the natural sharing scheme fails to sustain first-best if F is strictly concave. The implication is that the output exhaustion condition is a stringtent condition to satisfy and justifies Holmstrom’s lengthy emphasis on it. The result calls into question the productive efficiency of team arrangements when F is other than the indicated one.

110

R. V Fabeila / Team productivity

References Alchian, A. and H. Dens&z, 1972, Production, information costs and economic organization, American Economic Review 62, 777-795. Fitzroy, F.R. and K. Kraft, 1987, Cooperation productivity, and profit sharing, Quarterly Journal of Economics, Feb., 23-35. Frank, R.H., 1984, Are workers paid their marginal product?, American Economic Review 74, Sept., 5499571. Friedman, J.W., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Gordon, R.J., 1982, Why US wage and unemployment behavior differs from that in Britain and Japan, Economic Journal 92, March, 13-44. Hoimstrom, B., 1982, Moral hazard in teams, Bell Journal of Economics 13, 324-340. Jensen, M.C. and W.H. Meckling, 1976, Theory of the firm: Managerial behaviour, agency costs and ownership structure. Journal of Financial Economics 3, 305-360. Jones, D.C. and J. Svejnar, 1982, Participatory and self-managed Firms (Lexington Books, Lexington, MA). Macleod, N.B., 1984, A theory of cooperative teams, Discussion paper no. 8441 (CORE, Universite Catholique de Louvain, Louvain-la-Neuve). Ouchi, W.C., 1981, Theory Z (Addison-Wesley, Reading, MA).