class shares and economies of scope

JOURNAL OF ECONOMIC THEORY 54, 448459 (1991)

Class Shares and Economies of Scope

ENRIQUE R. ARZAC*

Graduate School of Business, Columbia University, New York, New York 10027

Received December 1, 1989: revised November 21, 1990

This paper examines the allocative role of class shares that pay dividends based upon the performance of the individual activities of multi-activity firms. The firms considered operate under economies of scope and technological uncertainty in an incomplete asset market. Investor unanimity about the choice of production plans and a constrained Pareto optimum are attained when all firms in the economy issue a class of shares for each of their activities. Partial issuance of class shares leads to a Pareto superior allocation if the ensuing changes in production plans are individually rational. Journal of Economic Literarure Classification Number: 521. b 1991 Academic Press. Inc

1. INTRODUCTION

This paper examines the allocative role of class shares issued by a firm. Class shares pay dividends based on the earnings performance of the sub- sidiary that the class represents. For example, General Motors’ class E and H stocks trade separately from the regular GM stock and their payoffs depend on the performance of GM Electronic Data Systems and GM Hughes Electronics, respectively.’

It is well known that pure conglomerate mergers or the split of a pure conglomerate does not change the allocation of resources when the conditions for value additivity are fulfilled. This may not be so in an incomplete market in which changes in tradable assets and production plans can alter the opportunity set available to investors. The market response to incom- pleteness depends on the costs and benefits of the available alternatives. Economies of scope (Panzar and Willig [ 111) may preclude the creation of

* The author gratefully acknowledges helpful comments from a referee and an associate editor of the Journal of Economic Theory, as well as financial support from the Faculty Research Fund of the Graduate School of Business of Columbia University.

’ See Murphy [lo] for a detailed analysis of General Motors’ class shares.

448 0022-0531/91 $3.00 Copyright IQ 1991 by Academic Press, Inc. All rights of reproductmn m any form reserved

CLASS SHARES AND ECONOMIES OF SCOPE 449

the number of separate firms needed in order to match investor preferences to production opportunities.’ Derivative assets, on the other hand, may fill the gap because, as Satterthwaite [ 121 has pointed out, in an incomplete market there is an incentive to create derivative assets which span observable states. Another way to allocate resources in an incomplete market is via public-good allocation mechanisms: A firm operating a bundle of activities under economies of scope can attain a Lindahl equilibrium if stockholders behave according to a public-good allocation mechanism of the sort studied by Dreze [4] and Arzac [2]. The performance of these mechanisms, however, depends on stockholders correctly revealing their preferences3 In addition, the joint stock exchange-Lindahl production equilibrium attained by internal allocation mechanisms is not necessarily a constrained Pareto optimum in the sense of Diamond [3J4

In this paper, the analysis of the role of class shares is performed within a model of multiple production activities available to the firm. The output of each activity is subject to multiplicative uncertainty such that, if each activity is incorporated as a separate firm, the stock market attains unanimity concerning production plans and value maximization is the optimizing rule for firms. A nontrivial problem arises once one allows for economies of scope since these make the creation of a firm for each activity inefficient. On the other hand, bundling several activities under a single firm eliminates spanning and renders the stock market incapable of assuring stockholder unanimity and constrained Pareto efficiency. It is in this context that class shares are shown to restore the allocation capability of the stock market.

2. THE MODEL

The model considered in this paper is that of a two-date uncertain economy with I consumers, indexed i = 1, . . . . I, J firms, indexed j = 1, . . . . J and a single physical commodity which at date 0 can be used for consumption or as input of production. The output of the firms will be available for

’ Panzar and Willig [ 111 defined economies of scope as the cost savings which result from combining two or more production activities in one firm rather than operating them separately. Under certainty, they have shown that in competitive equilibrium there must be multi-activity firms when economies of scope are prevalent.

J The mechanisms proposed by Drize and Arzac provide incentives for the correct revela- tion of preferences but both assume that stockholders behave myopically and ignore the possibility of strategic gaming over the many steps the allocation mechanism may take. Smith [ 131 discusses the experimental performance of this type of mechanism.

’ That is, the equilibrium may not belong to the set of efficient allocations attainable via a stock market. Forsythe and Suchaneck [S] have shown that internal allocation mechanisms can lead to a constrained Pareto optimum when each tirm has only one activity at its disposal.

450 ENRIQUE R. ARZAC

consumption at date 1 but it is unknown at date 0. Output uncertainty will be resolved at date 1 according to the occurrence of one of S mutually exclusive states of the world, indexed s = 1, . . . . S. A market for shares of firms operates at date 0.

A consumption plan for consumer i is a vector c’ = (cd, c;) E Ry I, where C;E R, is consumption at date 0 and 8, = (c; I) . . . . c;~) E R.1 is consumption at date 1.’ Consumer i is assumed to have a complete preference preordering 2, over R:+’ which is representable by a nondecreasing continuously differentiable function U’(8). Ub = aU’/ac; > 0; Ui = aU’/&$ = 0 V’s when cb=O; Ui.>O and U’ is strictly quasiconcave when cb>O. ri. = Uf/Ub denotes the ith consumer’s marginal rate of substitution (MRS) between c;,~ and cb. Furthermore, c denotes the set of consumption plans (cl),. f ‘= (Z-i), and f = (r’),.

A production plan for firm j is an array (x’, Y’), belonging to a production set Qi which is defined as follows: Y-’ is a K(j) x S matrix containing K( j ) date- 1 output vectors y I” = (-vi”;, . . . . v 3) E R: Each output vector corresponds to a basic activity defined by y{t = gik(XJk) @jk(s), gjk is an increasing twice continuously differentiable strictly concave function of the activity level variable xjk E R + and gjk(0) = 0. .~j = (.I?~)~. @jk(s) is a multiplicative random factor which is a function of the state of the world parameter s. Furthermore, date-0 input utilization by firm j is yi=f’(x’)~R+, where fJ is a twice continuously differentiable convex function such that afj/&yJk > 0 and f jd xk .xJk, with equality attained only when yJk = 0 for all k but one: i.e., f' = x'"' for each m when ,ylk = 0 for k#m.‘That is, xik IS the input requirement of activity jk when it operates in isolation. Y denotes the set of production plans (x’, Y-l),. Note that Qj collapses to Diamond’s [3] multiplicative uncertainty case when the firm has only a single activity at its disposal, but Qj is more general than Diamond’s characterization when the firm has access to more than one activity because uncertainty on aggregate output is not multiplicative and total input requirement is less than that needed in order to operate each activity in isolation. The present definition of Qi captures the phenomenon of economies of scope and provides an economic justification to a firm that jointly operates more than one activity (Panzar and Willig [ll]). Q’ is an instance of the production technology considered by Drize [4].

The initial endowment of consumer i consists of an amount w’> 0 of the physical commodity and shareholdings c(jo = (c(g)Jk E RS/ “” in the cj K(j f activities available in the economy, C, a$ = 1 V j, k. Final shareholdings are &= (cI~~)~~E Rx~IK(j), xi gik = 1 Vj, k. ~3, c1’, and CI denote the sets (w,),, (cr”);, and (a’);, respectively. pjk is the market price of activity k of firm j,p=(pJk),jkER?K(i’.

’ RT denotes the nonnegative orthant of the N-dimensional Euclidean space, R”.

CLASSSHARES AND ECONOMIES OF SCOPE 451

Each firm allocates the date-l output of activity k to the stock in activity k. The input cost is allocated to the kth stock according to hjk(xj), Ck hjk = yi. (One example of such a rule is hjk = .T~~J~$~~ xik Vk.) h = (hJk)jk denotes the set of cost allocation rules.

In this model, a stock-market feasible allocation is an array (c, Y, c() such that

jk

Vi

(xj, Y') E Q' Qj.

Furthermore, a constrained Pareto optimum (CPO) is a stock-market feasible allocation (c, Y, a) such that there exists no stock-market feasible allocation (c A, Y”,cr”)with~“‘~c~Viandc”‘>c’forsomei.

For the case in which each firm issues a single class of shares (i.e., when LY lo=p a;k= c(j, and pjk = p, Vk), D&e [4] has shown that this economy p’okssesleb an equilibrium consisting of an exchange equilibrium in the stock market relative to the production plans Y and a Lindahl equilibrium for each firm relative to the shareholdings tx. A Lindahl equilibrium requires that each firm maximize the present value of its production plan, using as prices the weighted average of the shareholders MRS which must be truth- fully revealed to the firm. This is so because, with a single class of shares, the stock exchange fails to equate MRS across consumers. In addition to proving existence, Dreze has shown that although every CPO is a stockholder equilibrium, there are stockholder equilibria which are not CPO.‘j

3. EQUILIBRIUM AND OPTIMALITY

Class shares are now shown to permit the stock-market economy to attain an equilibrium characterized by shareholder unanimity and constrained Pareto efficiency. That is, class shares are shown to elicit the spanning property of the firm’s individual activities without sacrificing the economies of scope attainable by their joint operation.

6 Grossman and Hart [7] introduced the concept of production social Nash optimality in order to characterize the optimality of equilibrium when the latter is not necessarily a constrained Pareto optimum.


DEFINITION 1. A stockholder equilibrium with class shares relative to the initial endowments (w, a’) is an array (c, Y, CI, p) such that

(i) (c, a, p) is an exchange equilibrium relative to the production plans Y. That is, for each individual i, c’ and a’ maximize U’(c’) subject to

and the stock market clears:

1 a:k = 1 V j, k. I

(ii) For every firm j, (xl, Yj) is a production equilibrium relative to shareholdings a. That is, (xl, Yj) satisfies

agjk pjk + hJk a& --=- axjk gjk a.# Vj, k.

Existence. Definition l(i) is trivially equivalent to Dreze’s [4, p. 143) definition of an exchange equilibrium. Thus, an exchange equilibrium with class shares exists. The requirements for a Lindahl equilibrium are also satisfied:

DEFINITION 2. For every firm j, (c, (x/ YJ), I-) is a Lindahl equilibrium relative to shareholdings a if ci” > c’ implies

Cb ̂+ ricy > c; + r’c; (2)

and (xl, Yj) maximizes

c c gJk(xjk) @jk(s) s k

on Qj.

Note that the exchange equilibrium implies the satisfaction of (2) (Dreze [4, Theorem 3.31). Furthermore, (1) maximizes (3). This follows from the fact that the optimal portfolio decision of individual i as defined in Detini-


tion l(i) implies that the aggregate p{’ = C, f I.@” is the same for all i: The optimum portfolio requires that’

or.

Denote pL” = p,‘” Vi and substitute it into (3) to obtain:

C gj’yX~k) ,@ - y.&

k

Finally, note that (1) is the condition for the maximum of (5). Therefore, we can state the following corollary of Dreze’s [4] Theorem 5.3:

THEOREM 1. Under the assumptions stated in Section 2, if short sales are bounded, there exists a stockholder equilibrium with class shares.

Remark 1. Equation (1) equates the value of the marginal product to its marginal cost. It also maximizes the market value of the firm, as can be verified by noting that (5) equals Ckpjk. Thus, the issuance of class shares reduces the Lindahl condition for a production equilibrium to market value maximization, the implementation of which does not depend on shareholders revealing their MRS to the firm.

Optimality. Consider the constrained Pareto optimal allocations attainable in the presence of class shares. Such allocations are given by the solution to the problem

max C fl;U’(cb, ct)

with respect to cb, z& and x.jk Vi, j, k, subject to

~c;+~y~-~wi<o I i I

cj-p;k@jk(s)<o Vi, s

c L:Ii_ gqxq < 0 Vi k

’ Nonnegativity constraints are assumed to be not binding throughout the paper.


where (/Ji)i are arbitrary positive weights and zfk/gjk(xjk) is the share of the activity j output received by individual i (which is the same in every state of nature as constrained in a stock market with class shares).

Note that this CPO problem consists of maximizing a strictly quasiconcave function over a convex set and, therefore, it has a unique solution. Furthermore, it is straightforward to verify that the first-order conditions for the optimum of this CPO problem are satisfied at the equilibrium conditions of Definition 1 (see Appendix). Therefore, the following properties hold:

THEOREM 2. Under the assumptions stated in Section 2:

(i) Every stockholder equilibrium with class shares is a CPO, and

(ii) every CPO such that cgi > 0 Vi is a stockholder equilibrium with class shares.

Remark 2. The results of this section hold for any arbitrary cost sharing rule h. This means that the allocative properties of class shares do not depend on the particular accounting method chosen to allocate joint costs to individual activities.

4. THE WELFARE EFFECT OF ISSUING CLASS SHARES

The issuance of an additional class of shares by a single firm does not assure the attainment of a CPO within the newly enlarged market structure if there remain linearly independent activities not spanned by class shares.8 A relevant question to consider is if the issuance of a new class of shares leads the economy to a Pareto superior (not necessarily CPO) allocation. Suppose the consumption and investment plans are at an initial stockholder equilibrium and that a firm splits each one of its shares into two new shares, each associated with a different bundle of production activities. Shareholders will reoptimize their portfolios over the now expanded choice set in an attempt to attain a higher level of utility but have the option of maintaining their holdings as they were before the split, in which case the allocation will not change. In other words, portfolio adjustments for given production plans are individually rational (i.e., no stockholder can be forced to accept a less favorable consumption plan than that attained before the share split). If the split relaxes a binding constraint, portfolio

* This is so because the stock-market feasible allocation set remains nonconvex. The source of nonconvexity is the bilinearity of the date-l constraint.


changes will move the economy to a Pareto superior allocation.’ However, the shareholders’ valuations of state contingent output will also change and, as a consequence, readjustments of the production plans have to be considered. Assume each firm readjusts its production plan to attain a Lindahl equilibrium relative to the new shareholdings. As Dreze has observed, these readjustments will be Pareto optimal but need not be individually rational as they can force agents into less preferred consumption plans.” Thus, in order to ensure that the resulting allocation is at least as good as the initial allocation for every individual, firms must undertake only those changes in their production plans which are individually rational. Once the production plans have been readjusted, individuals will reoptimize their portfolios and so on. Dreze [4, Theorem 5.31 has shown that, when individual rationality is maintained at each step, this process converges to a stockholder equilibrium. Hence we can state the following result:

THEOREM 3. An issue of class shares which relaxes a binding constraint on the consumption plan results in a Pareto superior stockholder equilibrium if .firms make only individually rational readjustments to their production plans.

Remark 3. Equation (4) holds for any activity unbundled by the issuance of class shares even when other activities remain bundled. This means that market value maximization applies to the unbundled activities. However, stock prices will not be sufficient to guide firms toward a Lindahl equilibrium if some activities remain bundled. Eliciting the investors’ MRS will still be required. This problem disappears if each firm issues a class of shares of each activity. In that case, the conditions leading to Theorem 1 and 2 are fulfilled and a constrained Pareto optimum is attained. In general, however, compensatory transfer payments among shareholders will be required in order to implement individually rational readjustments to production plans.

5. CONCLUSION

In this paper we have shown that in an uncertain economy with economies of scope, class shares permit the stock market to attain investor

9 Prior to any production adjustment, the present case satisfies Hakansson’s [9] condition for “strong global market structure dominance” of feasibility expanding changes in a pure exchange economy.

“Examples of mechanisms for production readjustment that do not maintain individual rationality include those of Green and Laffont [6] and Groves and Ledyard [S].


unanimity about the choice of production plans and that the equilibrium thus attained is a constrained Pareto optimum. These results require that a separate class of shares be issued for each activity of the firm. However, even the partial issuance of class shares will lead to a Pareto superior allocation if the ensuing changes in production plans are individually rational (which can be made so via compensatory transfers). In view of these results, we conclude that class shares can make an important contribution to economic efficiency and shareholder welfare.

Although the results of this paper assume that each activity is subject to multiplicative uncertainty, they suggest a role for class shares in the more general case in which activities are subject to nondecomposable uncertainty, i.e., when the production function is of the form gjk(x”, s): Forsythe and Suchanek [S] have shown that there are internal allocation mechanisms capable of producing constrained Pareto efficient allocations when each firm has access to a single activity subject to nondecomposable uncertainty. Dreze’s [4] examples show that this result does not carry over to the case of multiple production activities when each firm issues a single class of shares. On the other hand, the results of the present paper suggest that Forsythe and Suchanek’s theorem holds when firms issue class shares for each activity.

An important question not considered in this paper is the decision to issue class shares by the firm. It remains to be explained why class shares are not a common occurrence. Possible reasons for their limited use include: (a) the state-contingent output that they would unbundle is already readily available to investors, (b) the cost of issuing class shares and reporting income by activities is too high,” (c) managers may not welcome the scrutiny of their performance by analysts and shareholders armed with detailed reports on performance by activity, (d) investors may fear the potential transfer of wealth among classes of shares which might take place at management’s discretionI and (e) the stock exchanges’ reluctance to allow the issuance of class shares.

APPENDIX

In this appendix we show that the conditions for constrained Pareto optimality are satisfied at a shareholder equilibrium with class shares.

” Allen and Gale [l] consider the effect of issuing costs on optimal security design. I2 Murphy [lo] analyzes the possible conflicts among classes of shareholders and develops

guidelines for designing class shares which minimize these conflicts.


A.l. The Lagrangian of the CPO problem is

where 8’, cr, a:, and rjk are Lagrange multipliers. Differentiating this expression with respect to c’, +, and xjk Vi, j, k

yields the conditions

Vi, s

Vi, j, k

(AlI

t.42)

(A31

T,k dg’” dv’ --z.0 u ax-lk dx’ks

Vj, k.

In addition, the following conditions must be met,

Cc;+&$-C Iv’=0 (A51 I c;.*-;z;k@~~(s)=o, Vi,s (A61

1 ;l- gjk(Xjk) = 0, vi, k. (A71

A.2. At a stockholder equilibrium each individual maximizes its utility relative to the production plans of the firms. The first-order conditions for maximum utility of individual i result from differentiating the Lagrangian

~f,+Ca:~(p~~+hj~)-~‘-C~~~~,

% ik >

-Cs:(c~~-Caigjk(.r”)*Jk(s))

s ik


with respect to c’ and CI’, where 7~’ and 6:. are Lagrange multipliers. This yields

I+~’ (A81

Uf = bi., vi k t.49)

F !I!.$ @.ik(s) pjk::*, Qj, k.

In addition, the following conditions must be met at a stockholder equilibrium with class shares:

c; + c a;,(p,, + hjk) - ,‘d - c dI; pjk = 0 (All) jk ik

cts - c ctik g’k(.Yjk) @‘k(s) = 0, t/s, (A121 ik

~t-xf=l, Vj,k,

agjk pjk + hjk ay:, --=- a.# gjk a.dk' Vj, k. (A141

A.3. In order to verify that (A8)-(A14) satisfy (Al)-(A7), let

pk+hJk I--=Tvj, k ik and

g c(;~ = % Vi, j, k,

g

and sum (A 11) over i.

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class shares and economies of scope

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