on optimal operation of a jointly owned enterprise

Journal of Mathematical Economics 10 (1982) 269-278. North-Holland Publishing Company

ON OPTIMAL OPERATION OF A JOINTLY OWNED ENTERPRISE*

David GALE University of California, Berkeley, CA 94720, USA

Hilton MACHADO** University of Brasilia, 70.000 Brasilia, D.F., Brazil

Received November 1981, accepted January 1982

An enterprise is owned jointly by m agents, the ith agent’s share being Bi>O where xi Oi= 1. The enterprise is able to produce some non-negative n-vector x of goods where x lies in some convex production set X. An operation consists of choosing a vector from X and distributing it among the agents. The problem is to find an operations such that the value of the ith agent’s bundle measured in a given price system is proportional to 0, and such that the operation is (Pareto) optimal with respect to the agents’ preferences. It is shown under standard assumptions that operations which are both optimal and proportional always exist. It is also shown that these operations are unique if (a) X is given by a separable production function, and (b) when X represents production of a single good over n time periods.

1. Introduction

An enterprise such as a farm or a firm is owned jointly by m agents, the share of agent i being 8,, where c Bi= 1. The enterprise is able to provide various amounts of n goods, thus, a non-negative n-vector from some production possibility set X (non-trivial) in R’!+. Each agent has a preference ordering over R:. The problem is then to decide which vector x in X should be produced and how this vector should be distributed among the owners of the enterprise. An obvious requirement for any such scheme is that it should be (Pareto) optimal with respect to the owners’ preferences. A second requirement is that the distribution should in some way reflect the shares of the different owners. In order to formulate the latter we will assume that there is some exogenously given set of prices for the various goods. A distribution will then be called proportional if the values at these prices of the goods-vector distributed to each owner is proportional to his share of the

*Research partially supported by the National Science Foundation, Grant SES-7805196 and the Office of Naval Research under Contract NOOO14-76-C-0134 with the University of California.

**Visiting Research Associate at the Operations Research Center, University of California, Berkeley, with the support of Conselho National de Pesquisas, Brazil.

030&4068/82/00O&OOOOcHNH>o/$o2.75 0 1982 North-Holland

210 D. Gale and H. Machado, A jointly owned enterprise

enterprise.. Without loss of generality we may assume that all prices are equal

to one (simply define the unit of each good appropriately), so in a proportional distribution the total amount of goods received by each agent is proportional to his share. A feasible operation of this model consists of a choice of a vector x = (xi,. . ., x,) from X and a distribution x=(x1,. . ., x”l) in R’;” such that Ci xi = X, xi E R;.

The purpose of this paper is to show that, under the usual assumptions of closedness and convexity of production and preferences, there always exists

an optimal, proportional operation. The proof is fairly standard and is related to a much more detailed study of Balasko (1979) which has been

generalized by Keiding (198 1). However, these do not include production. For the sake of completeness we include our own short proof. Our main concern here is with uniqueness. For general production sets and utility functions there can be more than one optimal proportional operation. We show here, however, that uniqueness does hold in two special cases. First when all utility and production functions are additively separable. This generalizes a result of

Gale and Sobel (1979). The second case is that of a standard one-sector neo- classical production model operated over n time periods. The II goods are

then the outputs of each time period, and the decision problem is to

determine for each period how much output should be reinvested and how the remaining output should be distributed among the owners of the

enterprise. We remark that this paper is a sequel to Gale and Sobel (1979, 1982) and

Sobel (1981) concerned with the case in which the enterprise produced only one good, the amount of which was a random variable over which the owners had no control. We here eliminate the stochastic element but allow the owners to determine the output vector as well as its distribution. The

interest in the present result like that of its predecessors lies in the uniqueness theorem. We have here an instance, as with the Shapley value and the Nash bargaining problem, where a natural bargaining-type problem has only one solution satisfying certain natural requirements.

The condition of separable utility is, of course, a strong one. It is natural, however, for the second case which was the original motivation for this study. Here the enterprise is to operate over n time periods and the goods can be taken to be the profits of each period. These can be controlled by the owners who must decide how much prolit to distribute among themselves in each period and how much to reinvest for the next period, input and output being related via given production functions f,. Under the usual assumption that each owner’s utility of income is additive over time, we have an example which satisfies the separability condition above.

An interesting but as yet unsettled question would involve incorporating a stochastic element in the dynamic model of the previous paragraph. Suppose the return on investment depends on a random element as well as the

D. Gale and H. Machado, A jointly owned enterprise 211

amount invested. What are the appropriate investment and distribution strategies for obtaining analogues of our uniqueness results above? The

existence problem for the stochastic version of this model is settled under fairly general hypotheses in Machado (in preparation).

2. Existence

We assume each agent has a strictly increasing, strictly convex, closed preference ordering on R; which can therefore be represented by (quasi- concave) continuous utility functions. The production set is compact and convex in R;.

To prove existence of optimal proportional operations, let Xc R"" be the set of all distributions x=(x1,. . ., A?‘) of the model where xi =(xil, . . ., Xin) is

the vector distributed to agent i. Choose utility functions pi such that pi =0 and define ,u:X+Ry by p(x)=(~~(x’),. ..,p,(x”)). If X is the set of all

optimal operations, it is well known that 8 =r_l(_%) is homeomorphic to the unit simplex CmP l [see e.g. Arrow-Hahn (1971, pp. ill-112)]. Further, p restricted to 8 is one-to-one because of the strict convexity of preferences so /.-I is a homeomorphism from 0 to X. Define the map E: X-{O}-+CmP1 by E(x)=(e.x’,..., e. xm)/(e. x), where e is the n-vector all of whose coordinates are one and x=Xx’. Let +=Eop-l, then 4 is well defined because 0$X and continuous from I? to Cm-‘. Further, for any subset S of { 1,. . .,m} let

OS= {UE 0; u,=O for in S}, the image of the corresponding face of the unit simplex. By monotonicity, pi(xi) = 0 implies xi =O, hence e. xi =0 so that 4

maps every face of l-? onto the corresponding face of Cm- ‘. The standard homotopy result then implies that 4 is surjective. Thus, for some optimal operation x and some positive number 1, we have e .xi = idi for all i, as desired.

3. The case of separable production and utility

We shall assume from here on that each agent’s preferences are given by a separable utility function. Thus

where each pij has a positive decreasing derivative. In this section we also

assume that the production set X is separable meaning

x= x=(x1,..., i

xn)2°; Cfj(xj)S l 5

j I where the fj have positive increasing derivatives.


Separable production sets arise from situations in which there is a single scarce resource, say one unit of labor, which can be allocated to producing the various consumption goods. Thus, the output of good j is given by g,(Z) where gj is concave, smooth and strictly increasing. Assuming free disposal and letting fj=gJ: I, we see that a vector of outputs (x,, . . .,x,) is producible if and only if cjfJ(xj)s 1. We also remark that any production set with only two goods is separable. Namely if X = {( x1,x,); x2 5 f(q)} where f is concave and non-decreasing, then X = {( x1,x,); -$(x1)+(1-x2)51}, so X is of the desired form. Another obvious case is that of a simplicial production set X={xER:; c. x 5 l}, c a positive vector.

We first prove the following general lemma:

Lemma 1. Zf X=(X’ , , . ., 2”‘) is an optimal operation where xi # 0 for all i, then there exists a weight vector a =(c(~, . . ., a,,,)> 0 such that

ai&{Tij) 5 f >(Tj) for all i 5 m, j 5 n,

whenever Xij > 0. (1)

ProoJ: By the Fundamental Theorem of Welfare Economics there exists a non-negative n-vector q such that

X maximizes q*x for XEX, (2)

_9 maximizes ,L&‘) subject to q. xi 5 q. A?. (3)

For our special case (2) means that X maximizes q. x subject to 1 f;(Xj) 5 1, so, by the Kuhn-Tucker Theorem, there exists ,I>0 such that

=Af$T) for Xj> 0. (4)

Since (2) and (3) are unchanged when q is multiplied by a positive number, we may take 1= 1 in (4).

Next, applying the Kuhn-Tucker Theorem to (3) implies that there exists yi > 0 such that

=Yiqj for Xij>O. (5)


If we now define cli = l/y, and note that Xij> 0 implies Xj>O, then

combining (4) and (5) gives (1). 0

From now on we will assume n and y to be optimal operations and a and /? the corresponding weight vectors.

Lemma 2 (Sobel). If CQ//?~ 2 a,J&, then yij > xij implies ykj 2 xkj for all j.

Proof: We argue by contradiction. Suppose xkj > ykj, then

from Lemma 1. So

but using the fact that the p’s are concave contradicts ai/8i~uk/Bk. 0

Lemma 3. If yij > Xii and yj> xj, then pi > ai.

Proof: From Lemma 1,

BiPiJcVij) =_f;(Yj) 2T.filxj)

2 c(iPiJ<xij)

> aiP&ij)

(convexity Of fj)

(Lemma 1 again)

(by concavity of pij).

Since pLij is positive, the result follows. 13

Lemma 4. If operations x and y are not the same, there is an agent k such that xk f yk and elk > &,

Proof: If x#y, then we may assume x f y since if x =y then we are in the special case treated in Gale and Sobel (1979) and hence x=y. It follows that Xj>yj for some j, for if not we would have x sy and x would not be efficient. But then xkj>ykj for some k, so from Lemma 3, txk>fik 0

We now prove our main result:


Theorem 1. There is only one optimal proportional operation.

Proof Assume x and y are two different optimal proportional operations, z and /? the associated weights, ASS the corresponding proportionality factors (for all agents, e.x’=M,, e.y’=%I,).

Now order the agents so that ~i/Pi is non-increasing and let k be the smallest index such that xk#yk. Then by Lemma 4 and the ordering of agents, we have ak >Pk. We claim that ykj5xkj for all j, i.e., yksxk. This would contradict the hypothesis 156. To prove the claim, assume that ykj>xkj for some j. Then by Lemma 2 yij~xij for i > k, and by definition of

k, yij=xij for i< k. Hence yj>xj, which by Lemma 3 would imply Bk > ak, a contradiction. 0

4. The multi-period production model

Here a single good is produced in each of T consecutive periods, and agent i owns f!Ii shares of the enterprise, 61i a (positive) fraction of the initial input x0= 1. The output associated with input x,_~ at the beginning of

period t is yt=f;(x,_ 1). The production functions f, have positive non- increasing derivatives and f,(O) = 0.

The output of period t is split into individual consumption tit by the different agents i and savings x,, input for the next period. Now pi,(c)

measures the utility to agent i of consumption or income c in period t and it is assumed that pit has a positive strictly decreasing derivative on R+. We are also given a price sequence p = (pi,. . ., pT)>O, where pt stands for the unit

price of the commodity in period t, but appropriate scaling of production

and utility functions allows for the usual simplification, pt= 1 for all t, which

is assumed here. A feasible operation is now a matrix s =(cl,. . ., cm, x) where ci =(cil,. . ., ciT)

and x=(x1,..., xT) satisfy the following conditions:

Cit 2 O, x,20 for all i,t, xT=O, (6)

Ccit+x,=f,(x,-J for all tzl. L

(7)

For short, we will write s =(c, x) where c is the m x T consumption matrix (c,J and x is the savings-vector; we occasionally refer simply to c rather than to s.

As before, agent i judges a given operation or scheme s on the basis of two quantities,

U,(S) = C pit(cit) and Y(S) = e. ci, t

and the problem is to find in the set S of all feasible operations one which is Pareto optimal and proportional, i.e., the m-vector U(s) is a maximal element


in the partial ordering of R”, and V(s) is a positive multiple of the share-

vector 8. Here we use a different notation, better adapted to the model. It is easy to

verify that this is a particular case of the general problem described in the Introduction. In particular, optimality of a scheme S is equivalent to the

following property: For some CI E Cm-l,

S maximizes the function ~1. U(s) over the set S. (8)

Furthermore, the existence of optimal proportional schemes follows from our

result in section 2 by taking for production set X = {e . c; (c, x) E S}. To prove uniqueness we first establish some necessary conditions for

optimality. Roughly they say that an agent may decide to sacrifice part of his

present income for the sake of some agent’s present, past or future consumption but that no increase of the critical value CI. U(s) will result from

this.

Lemma 5. Let (c,x) be an optimal operation associated with the vector CI in Cm-‘. If for some agent i and period t, cit>O, then the following ‘backward inequality’,

cci/-4t(cit) fi fXxv - 1) 2 ajkjs(cjs), o=s+ 1

holds for every agent j and every period ss t (by convention the empty product has value one). On the other hand, if cjS > 0, j and s as before, then we have the Tforward inequality’,

Proof: We prove (9) when s<t, the other cases are simpler. Consider the operation (c’,x’) obtained from (c,x) as follows:

c;~ = cjs + A,, A,>O,

x:=x,- A,,

x:+~=xs+t-As+13 A,+ I =f,+ 1(x,) -_f- lb:)> . . .

Xt-l=Xt-l --At-l, At-,=f,-1(x,-,)-f;-,(x~-,),

c;, = tit - A,, 4=_Lb-l)--f,(x;-l~~ (11)

(all other entries as before).


By definition, s’=(c’, x’) satisfies the feasibility condition (7). As tit> 0 we have ft(x, _ J > 0, so x, _ 1 > 0 and f, r(x, _ J > 0. Recursively, x, > 0 for all v in the interval s5 us t- 1. Continuity and monotonicity of the production functions now guarantee that, for small values of A,, we have XL 20 and cl,20 so that condition (6) also holds and S’E S. By optimality of s, we have

0 2 tl . (U(d) - U(s))

= oLJ&js(c&) - Pjs(cjs)) + Cli(Pit(CIt) - PdciJ)

= ~.$Pjs(Cjs; Ask’s - G/Jit(CiG - A&f,,

where we use the standard notation

6F(x; z) = (F(x + z) - F(x))/z.

On the other hand, with the same 6 notation,

4=&Xx,-,; -A,-&L1,

A s+~=K++(xs; -AJA,,

so that, by multiplication,

(12)

(13)

(14)

Substituting this in (12), dividing by A,>0 and passing to the limit as A,-+0 (all A,-+O, necessarily) we get (9), as wanted. 0

Lemma 6. Let (c, x) and (d, y) be optimal operations, CI >O and B >O the associated vectors in Cm- ‘, 15 i, j 5 m and 15 s 5 t 5 T. If we have

and

then

(Cit-ditXcjs-djs)<O,

(ci,-dJ(x,-y,)zO for al2 ssvst-1,

(tit -d3(4Bi - aj/aj) > 0.

(15)

(16)

(17)

Proof Because of the symmetry we may as well assume that cit>dif, cjS < djS, x, 2 y, for all the V’S and prove that ~i/Pi > aj//Ij. AS ci, > 0 and d, > 0,


Since utilities were assumed to be strictly concave, (18) gives

clif4t(dit) fi fXYu- 1) ’ OLj&tdjs). s+1

(18)

(19)

(20)

Multiplying now (19) and (20) together, we get NJj?i>aj/pj as wanted. 0

The proof of uniqueness relies essentially on the impossibility of certain

specific patterns in the order-relationship of two schemes, an idea introduced in Sobel (1981). From now on we assume that (c,x) and (d, y) are two given optimal proportional schemes with, say, V(c)=A8 and V(d)=@ where Alr>O, and c( and j3 (necessarily positive) are the corresponding vectors in C”~‘. Furthermore, we assume that the agents are so arranged to make the quotient xJ/Ii an increasing function of i. One immediate consequence of this

set-up is that, if cited, and cjs <djs for i 5j and some pair t, s, then none of the following may occur:

t5.s and x,sy, for tsoss-1, (21)

tzs and x,zy, for ssvgt-1. (22)

(The proof is a straight forward application of Lemma 5.)

Fig. 1 indicates sketchly the five impossibility situations. It is worth

noticing the central role played by Lemma 4 whose ‘give-away’ technique goes back to Gale and Sobel (1979). Case (c) is explicit and basic in Sobel

(1981).

c.d. relation

x, y relation

(4 (b)

(4

Fig. 1

(4


Proof of uniqueness

Consider the smallest i such that Cit> di, for some t (if none exists, then c =d as remarked previously). Observe first that cjsgdjs for all js i and s so that Mjg ~0,. Hence by the hypothesis on 2 and q, we have equality and cjs=dj, for all such pairs.

Assume first that x(x,- J SJ(y,_ J. The impossibility case (c) described above applies to guarantee that Cj,~djr for all j. Hence ~jcjt>~jdjt and x, < y,. Cases (a) and (b) now apply to force cjt + 1 2 dj, + 1 for all j 2 i, hence for all i, and zjCjt+11Cjdjt+1* As x, < Y, we have f,+ &) -4 + l(~,) so that, as

before, x f+ 1 < yt+ 1. Recursion leads now to the conclusion that in the !ast period T, ciT 2 d,, for all i, and fT(xT_ 1) <fT(yT- &, a contradiction since there are no savings in period T (q=yT=O).

If we assume instead that fi(~~_~)>ft(y~_ J, we can use impossibility cases (c) and (d) in a backward step-by-step procedure similar to the one above to conclude that, in period 1, Gil zdil for all agents i and x1 >yl, a contradiction since the initial output is fixed (fi(l)).

References

Arrow, K.J. and F.H. Hahn, 1971, General competitive analysis (Holden-Day, San Francisco, CA).

Balasko, Y., 1979, Budget constrained Pareto eficient allocations, Journal of Economic Theory 24359-379.

Gale, D. and J. Sobel, 1979, Fair division of a random harvest, in: General equilibrium, growth and trade (Academic Press, New York) 193-198.

Gale, D. and J. Sobel, 1982, Optimal distribution of output from a jointly owned resources, Journal of Mathematical Economics 9, 51-60.

Keiding, H., 1981, Existence of budget constrained Pareto ef%zient allocations, Journal of Economic Theory 24, 147-159.

Sobel, J., 1981, Proportional distribution schemes, Journal of Mathematical Economics 8, 147~ 159.

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