Shapley value and disadvantageous monopolies

JOURNAL

OF ECONOMIC

Shapley

THEORY

Value

16,

513-511

(1977)

and Disadvantageous

At least one monopoly in a two-sector model must be advantageous, according to the Shapley value. If the elasticity of substitution is a constant at least equal to one, then each monopoly has an advantage if the other sector is not monopolized. For elasticity of substitution fessthan one, monopoly may be disadvantageous, depending on the distribution parameter.

1.

INTRODUCTION

In his discussion of the phenomenon of core and disadvantageous monopoly, Aumann [1] remarked, “It is known that the Shapley value is signi~~a~t in some economic contexts, and it may well turn out to be significant in accounting for monopoly power as well.” His implication seeme the Shapley value will always impute an advantage to monopoly over its competitive counterpart. In the two-sector examples of [4] and [6] this indeed is true. This paper finds conditions for monopoly on either side of a twosector model to be advantageous, and then exhibits a wide class of disadvantageous monopolies. Apparently the Shapley vaiue, like the core, appears to have a limited success in accounting for monopoly power. The paper is organized as follows, The next section sets out a two-sector model: with the feature that either sector may form a monopoly. The production interpretation of [2] is retained, and the Shapley values according to various states of competition are computed. Section 3 contains the results. At least one monopoly must be advantageous, if the other sector is not monopolized. Sufficient conditions are given for a particular monopoly to be advantageous. For the case of CES production functions, both mono~o~e3 are advantageous if the elasticity of substitution is at least one, but a disadvantageous monopoly is possible when the elasticity of substitution is less than one. The last section briefly discusses the results,

2.

MODEL

AND

ASSLMYWMS

The economy consists of two types of agents, of measure one each, two productive resources, and one output. Formally, the agents of the economy are represented by the measure space (T, r;i, L), where ;r = [O, 21, 17 is the 513 Copyright Ail rights

0 1977 by Academic Press, Inc. of reproduction in any form reserved.

ISSX

00224531

514

ROY

GARDNER

class of admissible coalitions, and L is Lebesgue measure. Every admissible coalition is measurable, but what coalitions are admitted depends on the state of competition, There are two productive inputs, x, and xz , and one output y. The production function F(x, , XJ exhibits constant returns to scale, has F(0, 0) = 0, and is available to any admissible coalition. An infinitesimal agent dt of type 1, belonging to Tl = [0, l] has the endowment X,(dt) = 1, X,(dt) = 0. An infinitesimal agent dt of type 2, belonging to T, = (I,21 has the endoment iCl(dt) = 0, X,(dt) = 1. The utility of any agent is given by y(dt), the amount of output allocated to him. Thus, the economic situation corresponds to a cooperative game with transferable utility. Different states of competition are represented in the model by different classes of admissible coalitions. Perfect competition prevails when 17 = 2, the class of Bore1 subsets of [0,2]. Let &S) denote the measure of agents of type i in coalition S. Then under perfect competition, the vector measure is. nonatomic. Apart from perfect competition, there I”(S) = (Pm Pm may arise two cases of imperfect competition. A type-l monopoly exists when 17 = & , where Z; = {S E .J? S 1 T, or &S) = 01. A type-2 monopoly exists when 17 = Z2 , where ,Zz = {S E 2‘: S 1 T, or &S) = 01. In either of these cases, the vector measure p is atomic. Let u be the characteristic function, whatever the state of competition. It is clear that

when pz(S) > 0. Further, v(S) = 0 when p = (0, 0). It will be assumed that f(O) 2 0,

f ‘(t> > 0,

f”(t) < 0,

hi

tf(l/t)

= 0.

(2)

Denote by y(S; .ZT) the Shapley value of coalition S in the state of competition If. In perfect competition, y(S; n) is given by the diagonal formma [3]. Thus, dT,;

z) = f’(l),

TV,;

z) = f(l)

(3) -f’(l).

These represent both the axiomatic and asymptotic values, as well as the competitive equilibrium imputations. For a situation of imperfect competition as formalized by Z1 and ,& , Hart [5] has shown the existence of many axiomatic values, but in these

SHAPLEY

VALUE

AND

MONOPOLY

cases the asymptotic value is still unique. The required v&es

515

are given by

Formula (4) expresses the limit to the process of spreading the monopoiy uniformly over the unmonopolized sector. By the Pareto efficiency of the asymptotic value, (4) implies that the remainder of the output J(l) is distributed over the unmonopolized sector. Pareto efficiency implies that monopoly does not gain its power by sabotaging the price system, but rather that it distorts only the competitive distribution of the product, preserving the optimal allocation of resources.

3.

MQNOPOLY

VS

COMPETITION

This section considers whether monopoly has an advantage over competitive behavior in terms of the Shapley value. A type-l monopoly is adva~tageous if and only if

From (3) and (4), this implies

A type-2 monopoly

is advantageous if and only if

that is,

A basic result on monopoly

advantage is the following:

PROI?OSITION 1. At least one monopoly is advantageous.

Proo$ Suppose that both monopolies follows from (5) and (6) that

are disadvantageous.

Then it

516

ROY GARDNER

From (2), the integrand equals 2j(l) at t = 1 and at least 0 at t = 0. Thus, to exhibit the desired contradiction it suffices to show that the integrand is a concave, increasing function of t. The first derivative, f’(t) +f(l/t) (l/t)f’(l/t), is greater than zero by the strict concavity off; while the second, J”(t) + t-“f”(l/t) is negative, again by strict concavity off. Given that at least one monopoly is advantageous, one asks for conditions under which a given monopoly is advantageous. PROPOSITION 2. If the competitive imputation to factor 2 is greater than that to factor 1, type-l monopoly is advantageous.

Proof.

Integrating

(5) by parts and rearranging, f(l)

-f’(l)

>f’(l>

- ~h/Q

dt.

0

The integral on the right-hand side is positive, so if f(1) -f’(l) > f’(l), type-l monopoly is advantageous. But in light of (4), this last inequality is the content of the hypothesis. PROPOSITION 3. If the competitive imputation to factor 1 is an increasing function of the input l/input 2 ratio, then type-2 monopoly is advantageous.

PuooJ: Integrating

(6) by parts and rearranging,

.rlf(t)
Consider the set of economies (t: 0 < t 6 I} where t represents the input l/ input 2 ratio. Then f(t) represents the competitive imputation to factor 1, as well as being the above integrand. The limit of the integrand as t --j 1 is f’(l), while the integrand increases by hypothesis. Somewhat sharper results can be obtained by restricting the production function to have a constant elasticity of substitution, in which case F(x, , x2) = [8(x,)-B i (1 - Q(x&~]-‘/~, where the elasticity of substitution,

withpa-1,0<8<1, U, equals I/(/? + 1).

(7)

Clearly f = [8(x)-B + 1 - 8]--IIB satisfies (2). Applying Proposition 2, 6 < Q is sufficient for advantageous type-l monopoly; from Proposition 3, /3 < 0 is sufficient for advantageous type-2 monopoly. Indeed, it is easy to show PROPOSIITON 4. If CJ3 I, then for every 8, 0 < 6 -=c1, both monopolies are advantageous.

It is also easy to prove that

SHAPLEY

R~OPOSITIOIG’

5.

VALUE

AND

MONOPOLY

5119

If CT< 1, then there exist disudva~ta~~~us

monopoiies of

each type. 4.

CONCLUSION

What is most striking about examples of disadvantageous monopoly is t utter lack of pathology in the underlying economic situation. ask for better behaved production functions than CES; yet, for ~~bst~tut~o~ possibilities sufficiently inelastic, there arise disadvantageous monopolies. That at least one side of the market can be advantageously monopolize gives the Shapley value some success in accounting for monopoly po However, this is only half of what the received theory of rno~o~o~y inte

REFERENCES

1. R. J. AUMANN, Disadvantageous monopolies, J. Econ. Theory 6 (19731, l-11. 2. R. J. AUMANN, Values of markets with a continuum of traders, Econome~nka 43 (1975), 61 l-646. 3. R. J. AUMANN AND L. S. SHAPLEY, “Values of Non-Atomic Games,” Princeton Univ. Press, Princeton, NJ., 1974. 4. P. CHAMPSAUR, Cooperation versus competition, J. &on. Theory 11 (1975), 394417. 5, S. HART, Values of mixed games, Int. J. Game Theory 2 (1973), 69-85. 6. L. S. SHAPLEV AND M. SHIJBIK, Ownership and the production, function, Quarf. J. Econ. 81 (1967), 88-111. RECEIVED:

October 19, 1976;

REVISED:

July 18, 1977 Department of Economk~ Iowa state university Ames, Iowa 5001 i