Externalities, Welfare, and the Theory of Games Author(S): Otto A

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Externalities, Welfare, and the Theory of Games Author(s): Otto A. Davis and Andrew Whinston Reviewed work(s): Source: Journal of Political Economy, Vol. 70, No. 3 (Jun., 1962), pp. 241-262 Published by: The University of Chicago Press Stable URL: http://www.jstor.org/stable/1828857 . Accessed: 07/08/2012 13:21

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http://www.jstor.org EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES'

OTTO A. DAVIS AND ANDREW WHINSTON Carnegie Institute of Technology and Yale University

I. INTRODUCTION at the conceptual level, since problems of T HAS traditionally been argued that, uncertainty and the non-existence of if firms create external economies equilibrium arise. Finally, we note that and diseconomies, the proper role of this latter possibility poses some diffi- a welfare-maximizing government is to cult problems for policy-makers, and we constrain the behavior of firms by ar- attempt to outline and explore briefly ranging rates of taxes and subsidies in alternative policy approaches. order to equate private with social bene- The analytic approach which we shall fit. We attempt to establish both the employ involves the consideration of two conditions under which this classical firms in a competitive industry. The policy prescription might work and is traditional or classical approach, on the needed, and those under which it cannot other hand, often involves an analysis of be expected to work. externality between competitive indus- First, we argue that motivation exists tries. We choose to depart from this for firms themselves to try to eliminate traditional approach for several reasons. externalities in production through merg- First, the firm is an entity which fits er. Second, we attempt to show that more easily into the framework of our technological externalities can be divided analysis. Second, and more fundamental, neatly into two cases, which we label it is individual decision units-firms- "separable" and "non-separable," re- which react to externalities so that it spectively. Third, if merger has not seems more "natural" to conduct the eliminated the externalities, we argue analysis at that level.2 Furthermore, con- that the classical scheme of per unit centration upon the industry (as op- taxes and subsidies can be clearly suc- posed to the firm) requires a certain cessful in equating private with social amount of aggregation which tends to benefit only in the separable cases. mask some of the more important and interesting points at issue. This aggrega- Fourth, if the externality is non-sepa- tion is especially misleading with respect rable, we argue that it is not clear that to public policy regulation, where the the classical prescription can work even problem is made to appear much more 1 This paper was written as part of the project simple than it actually is. Finally, "The Planning and Control of Industrial Opera- tions" under a grant from the Office of Naval Re- utilization of the firm as the basic search and the Bureau of Ships at the Graduate analytic unit gives a level of generality School of Industrial Administration, Carnegie Institute of Technology. The authors would like to 2 Interestingly enough, J. de V. Graaff also con- express their appreciation to Professors W. W. siders that externalities are a phenomenon which Cooper and J. F. Muth, both of Carnegie Institute relates to the firm rather than the industry, and, of Technology, Dr. R. R. Nelson, Council of Eco- furthermore, he seems to think this point quite im- nomic Advisers, and Professor James M. Buchanan, portant (see his Theoretical Welfare Economics University of Virginia, for their very helpful com- [Cambridge: Cambridge University Press, 1957], ments and criticisms. p. 19). 241 242 OTTO A. DAVIS AND ANDREW WHINSTON which is greater than can be obtained where the subscripts refer to the respec- through the traditional approach. The tive firms, C represents cost, and q indi- reason for this is that no two firms in an cates the output level.4 If each firm industry may be affected identically by maximizes profit, we have the relation- an externality, or some firms in an indus- ships try may impose (different "amounts" of) p = - , and p a 2 (2) an externality upon other production firms in the units, while the remaining where p represents price . Each firm must industry may not create externalities. maximize its profit with respect to the And it should be emphasized at the variable under its control, although the outset that our concern with firms level of its profit depends by assumption within the same industry is only a device upon the output level of the other firm. to simplify the analysis. A more elabo- It is well known that the welfare as- rate use of subscripts would allow the sociated with the production of the firms under consideration to be in differ- commodity can be measured by the dif- ent industries. ference between social benefit and social Yet another result of our approach cost, and that in a competitive market will be a demonstration that externality the social benefit can be measured by the problems involve many aspects of duop- firms' total revenue, P(qi + q2), while oly problems. This will be particularly social costs can be measured by the striking in the case which we consider- firms' total costs, Ci(qi, q2) + C2(ql, q2). reciprocal externality between two firms It follows that, in order to maximize wel- -but, peculiarly enough, these duopoly- fare, the joint profits of the firms must like problems remain even if the number be maximized. In other words, using P of firms under consideration is expanded to represent profits, let to n. We shall not attempt such an ex- pansion here, however, since all relevant P =P1+P2 = p( q1+ q2) () aspects of the problem seem to be con- -Ci(q1, q2) -C2(q,, q2) tained in the two-firm case, so that a sufficient level of generality can be represent the total profits of these two achieved without resort to additional firms as indicated by the relevant sub- complications. scripts. A necessary condition for maximization under the indicated assump- II. MOTIVATION AND MERGER tions is ap Consider two firms in a purely compet- _ aCl_ aC2 0 itive industry which are related through their cost functions (external economies lop P 4C1=C0_ C2 and diseconomies on the production a q2 a q2 a q2 side).3 Assume that the cost functions are 4 It should be emphasized that we consider only C1 = C1( q1, q2) technological externalities in this paper. We are not concerned with possible problems associated with q2), C2=C2(q1, pecuniary externalities. The usual convexity condi- are assumed whenever appropriate. 3 The analysis in this paper is conducted within tions the context of a competitive industry, although some I Although we assume the two firms to be in the of our results are applicable even if the market struc- same industry, this assumption is not necessary ture is not competitive. We have made the competi- either here or in the remainder of the paper. All tive assumptions simply because of a desire to make that is required for the general case is to assume two welfare statements which require such a framework. prices, pAand P2, instead of the single price p. EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 243 and a sufficient condition is case of a negatively sloped supply curve, suggested the possible use of taxes and a a 2P2< ? 2P2 < ? subsidies as one way to handle this type )(5) of difficulty. Meade has effected a mod- 82p 82p /t 82p 2 ern statement of this classical solution.7 q2 q~ 4)0q13q2 In particular, Meade argues that a tax- subsidy solution is sufficient to achieve Attention will now be focused on the the desired welfare-maximizing solution. first-order (necessary) conditions as giv- We shall argue at a later point in this en in (4). Note that if either (0C2)/ paper that a tax-subsidy approach is not (Oqi) 5 0 or (0C1)/(aq2) 5 O.then condi- sufficient to guarantee the attainment of tions (2) and (4) will not coincide. Due a welfare maximum and, furthermore, to the technological externalities, profit that in some cases it is not even clear maximization by the individual firms that it will lead to an improvement in will not give the greatest social benefit social welfare. We shall attempt now to that is possible.6 show that this scheme is not necessary Marshall and Pigou, considering the for the optimal welfare solution. In contrast to the above authors, we 6 The usual discussion of technological externalities deals in terms of production functions. We work do not take the firms as given. Rather, with cost functions merely for convenience. Identical we shall argue that there is a "natural results are achieved when production functions are unit" for decision-making and that this considered. For example, consider a single firm in a competitive industry, with the production function unit is responsive to market forces. A "natural unit" is defined as one which q =f(L, q2), (a) results after sufficient mergers have where qi represents the output of firm 1, L represents taken place to produce a certain "mini- an input of labor, and q2 is an output of firm 2 which mal" set of interrelationships with other affects the production of qi; q2 is a non-priced and uncontrolled "input" of firm 1. By assumption the units in society. In the context of this firm desires to maximize the following profit func- discussion, these "interrelationships" tion, where P represents profits, p the price of out- may be thought of as external economies put, and w the wage rate. and diseconomies. P = pq 1-wL. (b) As might be expected, the formation

Note that q2 does not enter "directly" into (b), but of natural units poses certain problems. it does affect (b) since we can write These range from the question of how a P = pf (L, q2)-wL (c) competitive market structure is main- of to by making a simple substitution. tained in the face mergers the ques- In its attempt to maximize profits the firm would tion of the terms on which such mergers hire labor up to the point where may be arranged. For the moment, how- -p -wOf 0. (d) ever, let these problems be waived. Then 3L dL.( if either or both However, by the externality assumptions, firm 1 does not control the output q2 of firm 2. Therefore if -$C2 0 -$C 0 (6) 0q, ~(q2O dP plf wok (e) aq2 Oq2 7 James E. Meade, "External Economies and Diseconomies in a Competitive Situation," Economic firm 1 would not be able to achieve its over-all Journal, LXII (March, 1952), 54-67. See also chap. profit maximum. This means, under the assumed x of his Trade and Welfare (London: Oxford Uni- conditions, that welfare is not maximal. versity Press, 1955). 244 OTTO A. DAVIS AND ANDREW WHINSTON obtains, it seems clear that either (a) schemes for welfare maxima when the there would be some price at which one natural unit has not been realized. firm would be willing to acquire the other or (b), more generally, gains to III. SEPARABILITY AND DOMINANCE both firms can be secured by effecting a Since we shall argue that the classical merger. These two cases are lumped into tax-subsidy prescription will not achieve one here simply to argue that a tendency the optimal welfare solution in all cases, toward such mergers is consistent with, it seems desirable to try to determine the if not implied by, the idea of profit max- conditions under which the scheme can imization. Consequently, there would be or cannot be expected to work. In this a tendency for such mergers to occur, for regard it is convenient, in order to make production externalities to be internal- clear the distinction between what at a ized, and for joint profits (hence social later point we call the "separable" and welfare, if competition is maintained) the "non-separable" types of externali- to be maximized. Insofar as this occurs, ties, to introduce the following definition. a natural unit for decision-making would A function, f(x1, x2), is said to be "sepa- be realized and, assuming competition, rable"9 if and only if welfare maximized without the use of f(Xl, X2) =fl(Xl)+f2(X2). (7) externally imposed subventions or penalties.8 In other words, separability means that We do not claim, of course, that the it must be possible to express the func- natural unit will always be achieved. In- tion, (xi, x2), as a sum of two functions stead, we are content to point out that each of which involves only one variable motivation exists for the formation of in its argument.'0 natural units, and we argue that there As a case in point, we may consider should be a tendency toward such a specific example of two interrelated, mergers. However, realism compels us to but separable, cost functions: recognize that such problems as might C1 ( q1, q2) = A l '+Blq1 q2 be associated with decentralized adminis- ~ (8) tration within the merged entity might C2( q1, q2) = A2q2-B2q's- prevent the achievement of the natural The profit maximization condition (2) unit in some instances. In addition, from then gives the standpoint of social policy, some mergers might result in a change of p= a~l= nA1 in market structure and, therefore, be deemed socially undesirable. Hence, it (9) seems appropriate to analyze the ex- p a2=rA= 2qV'-l. ternality question further in order to de- I termine whether there exist workable See A. Charnes and C. E. Lemke, "Minimiza- tion of Non-linear Separable Convex Functionals," 8 The idea of natural units is not original with us. Naval Research Logistics Quarterly, Vol. II (June, George J. Stigler, commenting on the production 1954). theory of Alfred Marshall, observed that when pro- 10Note that transformations of the kind dis- duction functions are technically related there may cussed in A. Charnes and W. W. Cooper, "Non-linear exist motivation for combination and merger (Pro- Power of Adjacent Extreme Point Methods in duction and Distribution Theories [New York: Mac- Linear Programming," Econometrica, Vol. XXV millan Co., 19461, p. 75). (January, 1957), are also admitted. EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 245 Note in particular that the marginal cost ly, each firm may calculate its marginal of each firm is given entirely in terms of cost unambiguously at every output, and, its own output variable. therefore, it can also determine its out- The typical cases with which the put unambiguously in accordance with classical analysis has been concerned the stipulated rule. have, in fact, assumed the condition of To bring out the importance of this separability." When this assumption is consequence of separability, it is desir- dropped, there is inevitably introduced able to reformulate the problem in terms an element of uncertainty which be- of a two-person non-zero sum continuous comes rather difficult to deal with in game.'2 To accomplish this reformulation terms of traditional tools and concepts. we note that by definition We shall also attempt to show at a later Pi =Pi( qi q2) - Ci( q2) point that the absence of this condition =pql qj, complicates policy choices and, in par- P2 =P2( qj, q2) =pq2 -C2( qln q2)X ticular, renders the tax-subsidy scheme where p is taken as given by the market. practically useless. The game aspect of this problem is the For the moment, however, let us con- fact that the profit level of each firm de- tinue our analysis under the traditional pends upon the output (strategy) se- assumption of separability. Consider the lected by the other firm. Consonant with rule "price equals marginal cost" for the assumptions of classical analysis, it each firm as represented in (9). Evident- is assumed that the game is non-co- 11 It may seem presumptuous of us to assert that operative. Neither firm communicates the typical cases in the classical analysis implicitly with or consults the other while making have assumed separability. After all, if the discussion its choice of output. Then the rule of is verbal, then it is difficult to determine the implicit assumptions underlying the analysis; and if the dis- profit maximization gives "price equals cussion is mathematical but utilizes only general marginal cost" for each firm. But since functional notation, then implicit assumptions are the marginal cost of each firm as stated equally obscure. Both these cases have been the rule in the literature. Witness, for example, P. A. Samuel- in (9) is defined entirely in terms of its son, Foundations of Economic Analysis (Cambridge, own output, this rule means that whatever Mass.: Harvard University Press, 1947); W. J. the output chosen by firm 2, there is a Baumol, Welfare Economics and the Theory of the State (Cambridge, Mass.: Harvard University Press, unique output which maximizes firm l's 1952); and J. de V. Graaff, op. cit. However, as will profit. Similarly, for firm 2 there exists a become clear from our later discussion, the conclu- unique output at which, whatever the sions of these analyses necessarily require the separability assumption if one presumes that they output of firm 1, its profits are maximal. are correct. However, it is only fair to point out that In the context of game theory, this is the K. J. Arrow, while discussing consumption externali- Von of ties, expressed some doubts about the classical tax- Neumann-Morgenstern concept subsidy scheme. He observed, "The general feeling dominance. is that in these cases, optimal allocation can be We shall shortly state this result in achieved by a price system, accompanied by a terms of mathematical theorems and ex- system of taxes and bounties. However, the problem has been only discussed in simple cases and no sys- plore more fully its implications within tem has been shown to have, in the general case, the the relatively simpler context of discrete important property possessed by the price system" ("An Extension of the Basic Theorems of Classical 12 For a brief discussion of continuous games, see Welfare Economics," Second Berkeley Symposium on I. L. Glicksberg, "A Further Generalization of the Mathematical Statistics and Probability, ed. J. Ney- Kakutani Fixed Point Theorem with Application to man [Berkeley: University of California Press, 1951], Nash Equilibrium Points," Proceedings of the A neri- pp. 528-29). can Mathematical Society, III (1952), 170-74. 246 OTTO A. DAVIS AND ANDREW WHINSTON games. Before leaving the continuous Hence, the cost functions differ from case, however, it seems appropriate to each other by the value of a constant describe what separability means in (the value of the externality) since the terms of the familiar graphical ap- slopes of the tangent lines (marginal proach. In the particular case which we cost) of all possible total cost functions are examining, the total cost of each at any specified level of output must be firm is a function of two variables, its equal to each other. Thus, given some own output and the output of the other specified price and total revenue curve- firm, and can be represented on a two- say Op'-an alteration in the value of the dimensional graph by a family of curves externality, q2, will not change the opti- relating the firm's costs to its own output mal output ql of the firm.'3 The effect of

COST AND REVENUE

q(2)

cIl ql OUTPUT,q

FIG. 1

for various given levels of the other separable externalities is strictly intra- firm's output. "Separability" means that marginal. They affect the over-all profit a change in the other firm's output position of the firm but do not alter simply shifts the cost curve of the firm marginal cost and, hence, do not alter vertically upward or downward. the optimal output choice of the firm. This case is depicted in Figure 1. We are now able to state precisely the Here two cost functions are drawn- major results obtained thus far: Aq(') and Bq2'). If the externality is an Theorem I: The presence of separable external economy, then q(l) > q 2); and if externalities in a firm's cost function im- the externality is an external disecon- plies, under the usual convexity assump- omy, then the inequality is reversed. As tion, that the firm must follow the "mar- we have shown, separability means in ginal cost equals price" rule in order to the context of this discussion that ex- 13 This assumes, of course, that average variable ternalities do not affect marginal cost. costs are covered. EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 247 maximize profits. Conversely, an opera- variation. Accordingly, we have drawn tional "marginal cost equals price" rule two total revenue curves, Op' and Op*, for profit maximization implies that any corresponding to the two prices which technological externality must be sepa- we wish explicitly to consider. Let us rable.14 consider now the price represented by Our previous discussion has dealt ex- the slope of Op'. Call this price p'. If the tensively with the first part of this firm follows the "marginal cost equals theorem; we content ourselves with two price" rule, then it must equate the further observations. First, note that as tangent to its cost curve, say AqM'),with long as the firm remains in operation, the price p'. Therefore, the firm would value of the externality is irrelevant to choose output q* since the slope of T2T' the firm's attempt to maximize profits equals the slope of Op' at this point. But because (a) by assumption the possibili- for the marginal cost curve to be unique- ties of merger, collusion, or co-operation ly defined, any other cost curve (caused are not admitted here and (b) the firm by by a change in the value of the external- definition cannot exercise control over ity), say Bq 2), must have a tangent T'T' the externality in the absence of these with the same slope at q*. This means possibilities. Second, we note that by that the total cost curves differ from one definition separability means, in this another by some constant amount at ql*. context, that marginal cost is defined in Now let us consider any other price, terms of the firm's own output variable. p*, which is represented by the slope of Consequently, if Op*. The same argument applies here as was used above. The firm must equate the slope of the tangent to its cost curve, say T*T*, to price p*. Therefore, output profits are not maximal and are made so, q is chosen. But in order for the marginal by well-known theorems, only when this cost curve to be unambiguously defined,'5 equality is established. the tangents to all other total cost curves The second part of the theorem has -T*T*, for example-must have the not been explicitly considered. Again we same slope at q' as T *T *. Therefore, the choose to proceed along intuitively ap- total cost curves differ from each other pealing routes rather than with nothing by some constant amount at q. but mathematical rigor. Accordingly, we Since, as long as the firm remains in again turn our attention to Figure 1. We operation, the above argument applies wish to show that, if the firm can follow for all prices, it follows that the total operationally the "marginal cost equals cost curves differ from each other only price" rule, then the externality must be by some constant amount. This can only separable. be the case if the externalities are for the equals marginal First, "price separable. cost" rule to be operational, the firm Theorem II: Separable externalities must be able to equate marginal cost to imply the game theoretic concept of price over all relevant ranges of price dominance. 14 We use the term "operational" in a special sense here. Our usage requires that the firm be 16 The terms "uniquely defined" and "unambigu- able to know its own marginal cost curve in the ab- ously defined" are used in a special manner here. sence of knowledge of the other firm's output We mean that the marginal cost curve is defined if decisions. one firm does not know the output of the other firm 248 OTTO A. DAVIS AND ANDREW WHlNSTON We have shown that separable ex- maximization problem. This fact is the ternalities imply the "price equals mar- essence of our argument for the discrete ginal cost" rule for profit maximization. case. Each firm is assumed to attempt to But this means, as long as the firm re- pick that (discrete) output for which its mains in operation, that the same output profits are maximal. What is required is is optimal for the firm no matter what that we formalize this condition-the value the externality takes on. This analogue of the marginal-cost-pricing means that one firm's output decision is rule-for the discrete case. independent of the other firm's decision Let us consider first the simpler prob- and this is, by definition, the game- lem where externalities are not present. theoretic concept of dominance in the Suppose that the cost function C(q) of a context of our approach. firm is defined for only certain values of Let us now turn our attention to dis- q. In other words, this firm produces only crete cases in order that we may explore at discrete levels of output. We now the welfare implications of separable ex- define a set Q ternalities with the analytical tools of . (11) game theory without having to resort to Q= [qIC( q)defined] the mathematical complications of con- In other words, the set Q is composed of tinuous games. It can be argued, of all the values of q which the firm could course, that most firms will not know choose. Still using P to represent profits their marginal cost functions exactly. and p price, we note that the firm would But it is usually assumed that to a satis- desire to maximize factory degree of approximation they can P( q) = pq-C( q) (12) at least determine noticeable stepped increments in cost for discrete variations over the set Q. Suppose that q* is that in production. And if this argument is feasible output for which (12) is maximal. accepted, then the discrete formulation is Then it is obvious that entirely appropriate. We prefer the discrete formulation, however, for addi- P( q* +Aq) -P( q*) < 0 (I13) tional reasons. First, it entails no real must hold for all admissible choices for loss of generality and avoids resort to Aq since otherwise q* could not be the more complicated mathematics. Second, most profitable output as was assumed. the theory of discrete games is easily Now we may use the profit function (12) available and widely known.16 in order to rewrite (13) in the following Before proceeding to the discrete form: game formulation, however, it is neces- -C( sary to digress briefly on the meaning of P( q* +Aq) q* +Aq) (14) the "price equals marginal cost" rule for -pq*+C(q*) <0 discrete cases. From (14) we can obtain We first note that, under the usual convexity assumptions, the "marginal plvq

for Aq > O PAq,< j( q* +Aqj)- e( q* ) (21) (16) >C( q* +Aq) -C( q*) from which it follows that for Aq < 0. < c ( l1Aa )c(a In other words, the optimal output q* Aql for > must satisfy the condition that if output forzq1>O(22)Aq, were to be admissible increased by any cq* +A qj)- e( q*) amount Aq, then price would be less than the slope of the line segment joining the for Aq1 < . two points C(q* + Aq) and C(q*). Con- These are, of course, the discrete ana- if versely, output were to be decreased logue of the "marginal cost equals price" by any admissible amount Aq, then price rule for the case with separable externali- would be greater than the slope of the ties, and, as was expected, there is a line segment joining these two points. similarity between (22) and (16). Note Let us now examine this discrete that the externality, e(qO), does not ap- problem when externalities are present. pear in (22). Evidently, in the case of Suppose that the cost function C(q1, q2) separable externalities, discreteness does of the firm under consideration is not affect the results which were ob- separable, so that tained for the continuous case. The firm C(q1, q2) = j(q1) +c(q2). (17) may calculate the cost associated with Since this firm is assumed to produce each discrete change in output and, it can determine its output un- only certain (discrete) outputs, C(qi, q2) therefore, is defined only for these values of q1. Let ambiguously in accordance with the stip- a set Q1, composed of all values of qi ulated rule (22). In particular, it is inter- which can be chosen, be defined as in esting to note that no matter what value (11). Note that the firm would desire to the externality takes on, the firm will maximize a profit function still select, as long as it remains in operation, the output q*. P( q1, q2) = pq1-C( q2) (18) q1, It should be obvious that both over the set Qi. Assuming that q* is a Theorem I and Theorem II apply for feasible output for which (18) is maximal, the discrete case with appropriate modi- given any particular value of q2, then it fications in wording. In other words, follows that Theorem I can be restated as follows: the presence of separable externalities in P(q*+Aql, q2)-P(q*, q2) ? 0 (19) a firm's discrete cost function implies must hold for all admissible choices of that the firm must follow the discrete Aq1. As was done for the previous ex- analogue (22) to the "marginal cost ample, we may substitute the profit equals price" rule in order to maximize function into the above in order to profits. Conversely, the discrete analogue obtain to the "marginal cost equals price" rule for profit maximization implies that any p( q* +Aql) - J( q* +Aql) technological externality must be sepa- -(q2 )-pq*+c(q*) (20) rable. Theorem II is applicable as it +c(q2) < . stands. No rewording is needed. 250 OTTO A. DAVIS AND ANDREW WHINSTON Since the proofs for the discrete case exists a column for which, given any follow the same form as those for the particular output of firm 1, its profits are continuous case, we choose to omit them greater than at any other output level. here. The theorems are intuitively obvi- These results are apparent from the dis- ous in any event. cussion concerning Theorem I where it We are now able to formulate this was shown that the value of the exter- problem in terms of a discrete, two-per- nality (or the output decision of the other son, non-zero sum game in order to ex- firm) was irrelevant for the optimal out- plore the welfare implications of sepa- put decision by either of the firms under rable externalities. Thus let us represent consideration. Theorem II tells us that the various combinations of outputs this is the Von Neumann-Morgenstern (strategies) of the two firms by the fol- concept of dominance. lowing game matrix: It is true, of course, that individual Firm 2 maxima need not-indeed, in general will not-be equal to the maxima for both firms considered together as a co- ordinated unit. This is apparent from and Firm 1 . t (aij, bij) ]. (23) (18) was the cause of our concern in section 2 when we attempted to show n that, under competition, joint maximization is a necessary condition for a welfare The profit accruing to firm 1 when it optimum. A simple example, attributed chooses the output associated with row i, to A. W. Tucker and developed in an and firm 2 chooses the output associated entirely different context, may be helpful with column j, is represented by aij. here."8 Similarly, bij represents the profit accru- Assume that the payoff (profit) matrix ing to firm 2 when the indicated output for the two "interrelated" firms is as choices are made.'7 follows: Consonant with the assumptions of Firm 2 classical analysis and with our assumptions of this section, it is assumed that Firm 1 R, (0.9 0.9) (0,1) 1 (2) the game is non-co-operative. Then, R2L( 1,O) (0.1, 0.1)] since we assume separability and a competitive market, Theorem I tells us that Clearly, row R2 is dominant for firm 1 the supposition of profit maximization since 1 > 0.9 and 0.1 > 0. Column Q2 gives "price equals marginal cost" for is dominant for firm 2 since 1 > 0.9 and each firm. But, since separability re- 0.1 > 0. Hence, the non-co-operative quires that the marginal cost of each solution is R2, Q2, which yields a profit of firm be given entirely in terms of its own 0.1 to each firm."9 output, this rule means that for firm 1 18 This example, known as the "Prisoner's there exists a row for which, given any Dilemma," is adopted from Luce and Raiffa, op. cit., particular output of firm 2, its profits pp. 94-102. are maximal. Similarly, for firm 2 there 19Some empirical data confirm the hypothesis that, if communication does not take place, players 17 It is assumed that bji, aji > 0 for all relevant continually choose the individual rational strategy. i, j choices so that there is no problem of covering In terms of the above example, this choice would average variable cost. mean R2 and Q2.For the results of the laboratory EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 251

Since a competitive market structure (24)-into a position of dominance. Al- is assumed here, the appropriate welfare though it might appear a simple task for solution is the joint maximum R1, Q1, the policy-maker to accomplish this re- with an individual profit to each firm of sult in the simple example (24), since an 0.9. The problem of social policy is how exact knowledge of the costs and profits to bring about this solution. of each firm would not be required, but only sufficiently large tax-subsidy ar- IV. SOME POSSIBLE APPROACHES rangements to reverse the indicated Let us now discuss possible ways in dominance, the general cases are more which the desired welfare solution could complicated and demanding. Therefore, be accomplished. First, the government it seems entirely appropriate to consider could adopt a "planning" approach and the conditions which the tax or subsidy impose direct constraints so that the would have to meet in order to accom- appropriate outputs-R1, Q, for the plish the desired solution. previous example-would be chosen. Let us consider, first, the continuous While we shall postpone a detailed dis- case. The policy-maker must be able to cussion of this "constrained-game" determine (at least approximately) ap- formulation until a later point in the propriate outputs for each firm in order paper, it is appropriate that we point that proper taxes and subsidies can be out here some of the difficulties associ- levied. This could be accomplished in our ated with this approach. In a more com- two-firm examples by solving equations plicated situation than that represented (4) simultaneously in order to obtain the by our simple example (24), such as the qi and q2 that achieve joint maximiza- more general cases conceptualized in the tion. Designate these welfare-optimal matrix (23) or the continuous game outputs q* and q*. Then, using t to formulation, the governmental policy- represent both taxes and subsidies (a maker must possess some knowledge of positive t indicates a subsidy and a nega- the cost functions of the individual tive t a tax), the proper subvention or firms, or at least some knowledge of the penalty would be given by entries in the payoff matrix, in order to aC1 ( q1, q2) I * accomplish this solution. Since many P+ ti- Oq, firms and multitudes of externalities (25) may be present in the real world, the P+ t2 = aC2 ( q1, q2 )q problem of gaining adequate information appears to be very great.20 Of course, p is easily available. But the Second, the welfare solution could be partial derivatives which have to be achieved by imposing a tax-subsidy ar- evaluated here (and available for a solu- rangement that brought the appropriate tion to may not be so readily ob- output decisions-for example, R1, Qi in [4]) tainable. To say the least, the policy- experiments, see A. Scodel, J. S. Minas, P. Ratoosh, maker would have to make an intensive and M. Lipetz, "Some Descriptive Aspects of Two- study in order to obtain the desired in- Person Non-Zero Sum Games," Journal of Conflict Resolution, III (1959), 114-19. formation. 20 Throughout this paper we assume that the We now turn our attention to the dis- government desires to maximize welfare, an assump- crete case. Here, too, the policy-maker tion which may not always be completely appropriate. must be able to determine appropriate 252 OTTO A. DAVIS AND ANDREW WHINSTON welfare outputs-perhaps by simultane- ever, that in both the discrete and con- ously solving discrete equations for a tinuous cases the tax-subsidy solution joint maxima-in order to determine does not possess one of the most im- proper taxes and subsidies. Designate a portant characteristics of a perfectly welfare optimal output for firm 1 by functioning market mechanism. Each q'%2'Since the firm is free to choose any time there is a technological change output q, which it pleases, it is obvious which affects the firms under considera- that, if the welfare optimal output qg is tion, there would have to be a recompu- to be chosen, the per unit tax or subsidy tation and adjustment of the taxes and must be such that (19) is satisfied. If t subsidies. A perfectly functioning mar- is used as it was in the previous discus- ket, on the other hand, automatically sion, then p + tl can be used instead of adjusts for these changes (at least from p in the derivation (equations [20] and the point of view of comparative stat- [21]) of condition (22) with the result ics). that As a final policy approach, we note that, provided that the market structure A P + tJ< 6 (qi' + qL) -c (*ql ) can remain competitive, forces may exist within the price system that will for Aq1 >0 (26) tend to produce the optimal solution even with no action by the government. j - ( q* +Aql) ( q* ) For merger might be mutually beneficial P +tl- - _zAq1 to both firms and could be expected to for Aq1 < 0 occur if either the rules of society or pos- must obtain. Once again, p is easily sible internal problems of decentralized available for the policy-maker, but the administration within the merged entity slopes of the line segments joining did not prevent such action.23 c(q* + Aq1) and c(q*) may not be so It is interesting to note that Meade, readily obtainable although, of course, 22 Many of our externality problems appear to fall an evaluation may be possible. The inter- into the discrete case. For example, problems associ- esting point concerning the discrete case, ated with plant location, municipal zoning, or even minimum building codes can be viewed as discrete. however, is that taxes and subsidies are (see Tjalling C. Koopmans and Martin Beckmann, not necessarily determined uniquely. "Assignment Problems and the Location of Eco- Instead, there may be limits, which de- nomic Activities," Econometrica, XXXIV [January, 1957], 53-76; and Otto A. Davis and Andrew B. pend upon the relative slopes of the line Whinston, "The Economics of Complex Systems: segments, between which the taxes and The Case of Municipal Zoning" [O.N.R. Research subsidies can vary.22 Memorandum, Graduate School of Industrial Ad- ministration, Carnegie Institute of Technology]). It is almost trivial to point out, how- Unfortunately, it appears as if many of these externalities are not separable. 21 We use "a welfare optimal output" here because with discreteness it is likely that more than 23 A further problem for research would be to one optimum output exists. Also note that we use q1 determine the "fair" terms under which the merger (and q*) here and in the previous discussion of the would take place, with reference to the division of continuous case as welfare-optimal outputs which the gains among the individual stockholders in the are determined by the policy-maker. Since these be- two firms. While this is not the appropriate place to come optimal outputs for the individual firms only go into this problem, it appears that a possible after proper taxes and subsidies have been levied, method of analysis would be along the lines out- they are not to be confused with the firm maximal lined by L. S. Shapley, "A Value for n-Person outputs of the previous section which were desig- Games," Annals of Mathematics, Study No. 28 nated by the same symbols. (Princeton, N.J.: Princeton University Press). EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 253 who produced the modern restatement C1(qj, q2) =)A1ql+Blqlq(2 of the classical tax-subsidy prescription ( 2 7 ) for the externality problem, does not C2( qI, q2) = A2q2-B2qtq' even consider the possibility of the by each individual merger solution in his "unpaid factors Profit maximization relationships: case."24 In fact, in the particular ex- firm implies the following ample that Meade uses, this solution ap- p nAjqj +B 2q2 pears possible. The example involves aq,= (28) apple growers and honey producers. The nectar from apple blossoms is a scarce aC2= rA2 q'-tB2q-7' qu commodity which, it is postulated, cannot be priced. (Later in his analysis Now note that, from the individual Meade also assumes that the bees help firm's standpoint, marginal cost is de- to pollinate the apples, thus creating a fined not only in terms of the variable mutual externality.) Thus Meade advo- which it can control-its own output- cates the classical solution of taxes and but also in terms of the variable which subsidies. If, however, there exists a it cannot control-the other firm's out- spatial distribution of apple growers and put. Both q, and q2enter into each equa- honey producers such that after merger tion. How, then, can the firm choose an one firm's bees would not be expected to output which will maximize its profit wander over into some other firm's apple when its own marginal cost depends orchard, the externality would be in- upon the decision of the other firm? ternalized and, if competition could be Let us now compare and contrast the maintained, the optimal solution would continuous cases of separable and non- result. separable externalities in order to bring out the effects of non-separability. It will V. NON-SEPARABLE COST FUNCTIONS be recalled from Figure 1 that a change Let us now consider the non-separable in the value of a separable externality type of externality where it is not clear had the effect of vertically shifting the that the usual solution of taxes and sub- total cost curve upward or downward. sidies will work, even at the conceptual The curves differ from one another only level. The difference between the sepa- by the value of a constant. Thus at any rable and the non-separable cases lies in given output the slopes of the tangent the fact that externality enters the cost curves-that is, marginal cost-were not function in a "multiplicative" manner affected by alterations in the value of the rather than in a manner which is strictly externality. The non-separable case does "additive." In other words, the sepa- not have this property. While altera- rability condition (7) is not satisfied. tions in the value of the externality cause Once again, it is necessary for us to the total cost curve to shift upward and discuss both continuous and discrete downward, there is no reason to expect cases. We consider the continuous case that this shift will be a simple vertical first. And it also seems completely ap- displacement. In general, total cost propriate, because of the greater clarity curves generated by changes in the value achieved, to proceed by assuming specific of a non-separable externality will not cost functions: simply differ from each other by the 24Meade, op. cit., pp. 56-61. value of a constant. Rotation or some 254 OTTO A. DAVIS AND ANDREW WHINSTON other type of alteration is likely to take that C(qj, q2) is a non-separable cost place when the externality changes in function which is defined for a set Qi of value. This fact is, of course, obvious discrete outputs, we may derive from the fact that the separability condition (7) is not satisfied. It can also be C(q* +Aqj, q2)-C(q ,_q2) seen from observation of the marginal P_ - - cost curves (28) of our special example. forA q1 >0 It is now obvious that the "marginal (29) cost equals price' rule is affected by non- C(q* +A qj, separable externalities. For whereas q2)-C(ql, q2) separable externalities had a strictly Aq1 intramarginal effect, non-separable ex- for Aq1 < 0 ternalities affect the margin. In (28) the as the externality enters into the definition of desired rule.5 But note that, since marginal cost. Since by definition the C(qj, q2) is non-separable, the terms in- firm cannot control the value of the ex- volving the externality cannot be can- celed out and that ternality, it clearly follows that the firm condition (29), unlike its separable will find it difficult operationally to fol- counterpart (22), involves low the "marginal cost equals price" q2. Thus q2 affects the discrete analogue to rule of profit maximization. The fact is marginal cost. This means that the that, for the non-separable case, marginal output q* which satisfies (29) must de- in q2 cost is ambiguously defined in terms of pend, general, upon the value which the firm's own output. assumes. Therefore, in the discrete as in From the game-theoretic standpoint, the continuous case, non-separable ex- this type of externality suggests the ab- 25We present this derivation in a footnote since sence of dominance. This point, too, is it is similar to that used for the separable case. The obvious from the separability condition firm desires to maximize the profit function (7) and from the marginal cost curves P(qj, q2 )=pqi-C(ql, q2) (a) (28) of our example. For, supposing that over the set Q, of possible outputs. By assumption the firm desires to maximize profits, it the firm cannot control the value of the externality must alter its output with every change q2. But for some given q2 it is obvious that any profit in the value of the externality in order to maximizing output ql must satisfy attempt to equate marginal costs with P( < 0 price. This means that the optimal out- q? +Aq, q2)-P( q', q2) (b) put (strategy) of one firm depends upon for all admissible choices Aqx. Substituting the the output (strategy) selected by the actual profit function (a) into (b) we obtain other firm. Such interdependence is the p(ql +Aql)-C(ql +Aqb, q2) essence of non-dominance. C) Now let us examine the discrete case, -pq*+C(ql, q2) <0 in order to utilize the analytical tools of Since C(qj, q2) is non-separable, collecting terms the theory of games in exploring the wel- gives fare implications of non-separable externalities. It is necessary, of course, pAqj?C(qg+,Aq, q2) (d) that we state the discrete analogue for the "marginal cost equals price" rule for -C( q* , q2) the non-separable case. Then, assuming from which (29) follows directly. EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 255 ternalities introduce an interdependence situation? One possible approach would in decision-making. be for each firm to attach subjective From the standpoint of discrete games, probability to its set of possible outputs the presence of non-separable externali- and select that output which would ties suggests that there is no row-column maximize its expected profits.28 A max- dominance. In other words, in a matrix min approach might be another possi- representation such as (23), for firm 1 bility. Or one can make various other there does not exist a row in which, for assumptions concerning how the firms every output of firm 2, its profits are might act and react. But there seems to maximal. Similarly, for firm 2 there does be no a priori method for determining the not exist a column in which, for every outputs (strategies) selected. Non-sepa- output of firm 1, its profits are greater rable externalities raise the possibility of than at any other output level.26 Non- the non-existence of equilibrium. dominance is evident, of course, from The important point here, as in the the fact that the externality enters the separable case, is that there is no reason discrete analogue to marginal cost. to expect the output which maximizes It seems clear that in both the con- social benefit (meaning the solution tinuous and discrete cases non-separable which maximizes joint profits in the as- externalities introduce an interdepend- sumed competitive market) to be chosen. ence between decision-making units. We Again it seems desirable from the stand- have here, even in what is usually con- point of society that either the game be sidered the certain world of competitive constrained, the scores altered, or other price theory, an example in which de- changes be affected so that the appropri- cisions must be made under uncertainty. ate welfare solution will emerge. But It is this aspect of the externality prob- whereas the separable case raised only lem which is roughly analogous to the problem of the misallocation of re- duopoly theory.27 How can a firm deter- sources, the non-separable case raises mine its profit-maximizing output in this both the problem of the misallocation of resources and the problem of mal-co- 26 A very simple example of non-dominance is the following: ordination of decision-making because of Firm 2 the interdependence between marginal Qi Q2 cost curves.

Firm 1 R L (0, 10) (.1, 19) VI. POLICY APPROACHES AND EQUILIBRIUM It is not clear whether firm 1 will choose strategy (output) R1 where .9 > 0 or strategy R2 where We now discuss possible ways in 1 > .1. Similarly, it is not clear whether firm 2 will which the welfare objective might be ac- choose strategy (output) Q2 where .9 > 0 or Q where 1 > .1. complished in the case of non-separable 27 It is to be emphasized, however, that this re- 28 Such an approach would be similar to that of semblance to duopoly theory is somewhat super- L. J. Savage, The Foundations of Statistics (New ficial. Although we have developed our analysis in York: John Wiley & Sons, 1954). On the other terms of two firms, this approach has been used only hand, Von Neumann and Morgenstern assert that for expository convenience. It is obvious that, with the difficulties inherent in situations in which neither non-separable externalities, interdependence be- participant controls the relevant variables cannot tween decision-making units is the source of the be obviated by recourse to statistical assumptions difficulty, and this interdependence exists if the and analysis (The Theory of Games and Economic Be- number of firms is two, three, or n, where n can be havior [Princeton, N.J.: Princeton University an indeterminately large number. Press, 19471, p. 11). 256 OTTO A. DAVIS AND ANDREW WHINSTON externalities. The first possibility is the that the firms will not make a decision proposed merger solution. With no ac- on each play, but only that the decisions tion by the government and provided cannot be predicted). Thus it seems im- that the market can remain competitive, probable that the governmental policy- forces may exist within the price system maker would know the strategies which which will tend to produce the optimal the individual firms would play since, as welfare solution, since merger might be was pointed out above, there is no a mutually beneficial to firms operating priori method for determining the out- under the postulated types of externali- puts which might be selected. In fact, ties. It merits repeating here that, if for the policy-maker to be able to de- problems such as might be associated termine the strategy which individual with decentralized administration do not firms might be playing, it would seem prevent merger, firms are motivated to necessary, in the absence of a priori merge until the postulated externalities methods, to obtain informationconcern- which can be "internalized" are elimi- ing the psychologies of the managers, nated; that is, until the "natural unit" their "taste" for risk, and so on. Of for decision-making is reached.29 course, this knowledge is not readily Second, the government could try the available; and if the policy-maker did classic prescription of levying special not know these strategies, he could not taxes and subsidies. This solution, how- possibly predict the reaction of the firms ever, appears even less feasible in this if he tried to rotate the total revenue case, and this point can be seen clearly by curve (or, what is analytically the same, a comparison with the separable ex- shift the price line; shift the marginal ternality example. In the latter case a cost functions; or, in game-theoretic dominant solution existed. The govern- terms, alter the payoffs) through the mental policy-maker, if he knew the tax-subsidy method. Thus, even if the relevant cost functions, could levy excise policy-makerdetermined what might be taxes and give subsidies on output as a considered"appropriate" subventions or constant function for each firm accord- penalties by methods analogousto those ing to rules (25) and (26) so that profit suggested by (25) and (26) for the maximizing firms would be induced to separable case, there is no assurance- choose the optimal welfare outputs. and even little likelihood-that the firms However, in the non-separable external- would voluntarily choose the welfare- ity case, even assuming that the govern- optimal outputs. Non-separable ex- mental policy-maker knows the relevant ternalities affect firms' marginal costs cost functions and desires to maximize and thus create interaction between the welfare, there seems to be no dominant decision-making efforts of individual solution to aim at. It is well known that firms.30 there does not exist a known, unique 30It might be suggested that the tax-subsidy equilibrium solution in pure strategies scheme could be made appropriate even in this case for this type of game (which is not to say by having the policy-maker simply solve for the optimal outputs and offer a subsidy conditional 29 The notion of a natural unit can be taken as upon firms producing those specified outputs. But roughly corresponding to the concept of a stable this would entail abandoning the advantages of a coalition in n-person game theory. In this respect it decentralized market system, since the policy- might be noted that throughout this analysis we maker must actually specify acceptable outputs. have chosen to ignore the possible instabilities which Also note that if acceptable outputs are specified by might be associated with entry into the industry. policy, then there would seem to be little reason to EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 257

It follows from the above analysis that which necessarily contain externalities the classical tax-subsidy solution, origi- of the non-separable type because of the nally stated by Marshall and Pigou and "multiplicative" terms Al (x2) and A2(xl). recently restated by Meade, breaks down It is true, of course, that Meade is deal- for the case of this non-separable type ing with industries rather than firms and of externality.3' that he (at least implicitly) assumes that Meade's analysis, although it is care- the firms in each of these industries takes fully developed and illuminating, is es- the output of the other industry as a pecially interesting in this respect. Like parametric constant. On the basis of most of the other writers on this subject, this assumption it might be argued that he uses for much of his analysis a gen- a tax-subsidy scheme could work in eral functional notation which makes it principle, although it would require which would impossible to determine accurately elaborate computations have to be repeated each time a price whether the externalities are separable alteration, or a technological change oc- or non-separable.32 But, perhaps because curred. However, the major objective of he deals more thoroughly with the prob- our game-theoretic analysis of non-sepa- lem other than writers, Meade intro- rable externalities has been to show that duces in his "Atmosphere Case"33 the this type of interdependence creates un- production functions certainties which, in turn, make such an x-= HJi(l,, ci)A,(X2) assumption arbitrary and unwarranted. And our analysis, which has been de- X2= H12(12, C2)A2(x,), veloped for two firms but which certainly can be extended to cover any number offer conditional subsidies. Why not simply dictate of firms, shows that in the non-dom- to the firms that certain outputs must be produced? is unlike- This, however, weakens the case for private owner- inance case a stable equilibrium ship of the facilities under consideration. ly to be achieved. There seems to be no 31 Our results here hold for the case of reciprocal, a priori method for determining what non-separable externalities. If the externalities are strategies (output choices) firms will not reciprocal in any sense-that is, if firm 1 im- select in the presence of non-separable poses a non-separable externality upon firm 2 but not vice versa, and if there are no other externality- externalities. creating firms which are relevant-then our analysis Since our discussion of the non- does not hold. For a given output of firm 1, firm 2's dominance case has been based upon a marginal cost curve will be precisely determined so that, at least conceptually, the policy-maker can game-theoretical analysis, an approach compute the appropriate rate of the tax or subsidy; which we feel is to be preferred to the but, of course, this rate will have to be recomputed approach used by Meade and others, it each time firm 1 alters its output. If the situation is more complex (as for example, firm 1 imposing a might be argued that our conclusions non-separable externality upon firm 2, firm 2 im- that there is no obvious equilibrium solu- posing one upon firm 3, and firm 3 upon firm 1), then tion in pure strategies are based solely our analysis does hold. It is the necessity for "simultaneous" decision-making in the presence of upon our choice of tools of analysis and this type of interdependence that creates the un- that the "usual" tools do logically show certainty and difficulty here. that an equilibrium must be achieved. 32Meade makes the usual assumption of linear The usual tools involve the solution of a homogeneous production functions. It can be shown, however, that this assumption does not rule out the set of simultaneous equations. We shall possibility of non-separable externalities. show that an analysis with these usual 33 Meade, op. cit., pp. 61-66. tools need not imply the existence of an 258 OTTO A. DAVIS AND ANDREW WHINSTON equilibriumsolution. Ourmain point will data about cost functions? But if they be that the assumptions needed for an exchange data about cost functions (or equilibrium solution are not consistent even if they just communicate), why with the other assumptionsof the model. would they not make the most of the Let us consider the simple example ability to communicate by exchanging upon which we based much of our information that would lead to a joint analysis of the non-separable case. The maximization,since both would stand to usual method would be to solve the fol- gain thereby? Thus, on the basis of the lowing equations: simple communication assumption it would seem that they would exchange lo _ p- nA Ij1 -B1qq= 0 such information as would make them (30) act as if they were merged instead of = O trying to seek the so-called"equilibrium" -P2 -p_ rA2 q7-1 + tB2qtj1ql O q2 solution. Under the other set of assumptions, Let q* and q2 be the "equilibrium"solu- the firms strive to reach "equilibrium tions to these two equations: if both outputs in the long run." Since one firm firms happened to select the outputs q* cannot be assumed to know the other and q2*,neither firm would have any de- firm's cost function, there is little reason sire to change its output plans upon to suspect that in the initial period the hearing of the output plans of the other "equilibrium"outputs would be chosen. firm. We assume that these solutions are Each period each firm observes the unique and non-negative. Suppose that, other's behavior, which it desires to take for some reason, some other outputs q' into account in its own decisions. Thus and q' were chosen. Then each firmwould each firm would be led to try to predict, desire to alter its output in order to on the basis of past data, the other firm's adjust for the externality in its efforts to output for the next period. The firms ob- maximizeprofits. But when the new out- serve, predict, make their decisions ac- puts were chosen, say qI' and q"',qi' s cordingly, and produce. This process qI , qg' # q2*,then it would be desirable goes on period after period. As we saw to alter outputs again and so on ad earlier,unless the equilibriumvalues are infinitum. chosen, both outputs will change. A We now distinguish two sets of as- (somewhat naive) application of Muth's sumptions that might lead one to infer "Rational Expectations Hypothesis""4 that the process of "an infinite progres- suggests that the one hypothesis that the sion" of mutual adjustments and read- firmswould not be expected to use would justments would lead to an "equi- be that the other firm would not change librium" solution. its output in the next period. Change First, the firms might be assumed to would be expected. Thus, under this communicatewith each other, announc- hypothesis, even if the two firms do ing tentative outputs but not producing happen to reach the unique "equi- until the "equilibrium"outputs q* and librium" outputs after "an infinite pro- q2 are announced.But if the firms are al- gression"of periods of adjustment, each lowed to communicate,it seems unlikely "John F. Muth, "Rational Expectationsand that they would just exchange output the Theory of Price Movements," Econometrica, data. Why would they not also exchange XXIX, No. 3 (July, 1961),315-35. EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 259 firm would expect the other to alter functions, but also the psychological output in the following period; this ex- characteristics of the decision-makers pectation would be taken into account within the firm or at least the general in making decisions for that next period, qualitative properties of their preference- and equilibrium would not be main- satisfaction functions to be able to pre- tained. For it is only at one unique point dict the strategies which might be chosen that the equations in the model predict by the players. no change for the next period, and this Finally, the governmental policy- point is unknown to the firms. maker might adopt the method of the It follows from the above line of constrained game approach. While this reasoning that the only way in which approach is subject to the difficulties of the firms would reach and maintain the gaining adequate information that were equilibrium solution would be by one pointed out previously, it does seem to firm assuming that the other firm would offer hope for some type of solution not change its output from period to either where merger will not work be- period even though observations re- cause of the impossibility of "internaliz- vealed otherwise. ing" the externalities, or where it will However, the game-theoretic formu- result in the creation of monopoly. lation shows clearly that there is unlikely The government could try, of course, to be a unique equilibrium solution (the to "strictly constrain" the game by case of non-dominance) for this type of dictating appropriate outputs to the externality.35 In order to know how the firms. However, granted our ethical bias firms would react to the possible taxes against such direct planning, we assume and subsidies, the policy-maker would that decentralized (non-governmental) have to know, as has been pointed out, decision-making is desirable wherever it not only the exact nature of the cost is possible. Thus we propose to discuss now some cases where it may be possible The simultaneous solutions q*, q* to the set of equations (30) correspond to a Nash equilibrium to employ a combination of various con- point in a two-person, non-zero-sum game. Since we straints and the pricing mechanism in do not assume that one player knows the payoffs order to obtain an approximation to the to the other player, expectations are all-important for the attainment of an equilibrium. In this respect, appropriate welfare solution. a comment by Luce and Raiffa on the Nash equi- Let us consider a simple case where librium seems especially relevant here. "These there is "almost" row-column domi- strategies will be in equilibrium provided that no player finds it is to his advantage to change to a nance: Firm 2

Ql Q2 Q3 Q4 R [-(10, 10) ( 4,17) (2, 12) ( 6, 4)1 (31) Firm 1 R2 (12, 3) (15, 8) (3, 6) ( 8, 9) R3 ( 3, 4) ( 4, 15) (2, 10) ( 9, 7) R4 L(13, 8) ( 7, 3) (5,14) (11, 11)

different strategy so long as he believes that the other Row R2 dominates row R1 and column Q3 players will not change" (op. cit., pp. 170-71). (Our dominates column Q1. The remaining italics.) For a discussion of further difficulties associated with this type of equilibrium notion see rows and columns introduce a non- ibid., pp. 171-73. dominance situation. If the governmen- 260 OTTO A. DAVIS AND ANDREW WHINSTON tal policy-maker imposes regulations so pected payoff associated with the opera- as to constrain behavior to rows R1 and tion of each type of plant at its indi- R2and columns Qi and Q3, then the row- vidually optimal output level is similar to column dominance case is achieved. that represented in (31), where the sub- Once choices are so constrained, the use scripts now refer to the two entre- of the price mechanism will result in preneurs; but here entrepreneurs, in- strategies R2, Q3 being chosen. It would stead of firms, make choices and the R's now be possible to use taxes and sub- and the Q's represent types of plants sidies in order to cause the "optimal" rather than output levels. Obviously, strategies R1, Qi to be achieved.6 In from the viewpoint of social policy it is other examples, of course, the simple im- desirable that this game be constrained, position of the direct constraints might and a method actually used in modern result in an optimal, dominant solution. municipal zoning is to place use restric- The phenomenon of municipal zoning tions upon an area so that certain seems to afford a practical example of a property uses (plants in this example) case where "partial constraints" are are excluded. In this illustration, uses used. The existence of height, area, and R3, R4, Q2, and Q4 would certainly be use restrictions may be viewed as evi- forbidden in the area by any rational dence of the fact that these property zoning ordinance. Granted these restric- features impose externalities upon cer- tions, the price mechanism would be al- tain types of other property.37 As a very lowed to operate, and each entrepreneur simple illustration, let us consider two would be able to pick the most profitable entrepreneurs who own adjacent lots and type of plant not excluded by the restric- who are trying to decide what types of tions. Of course, in this particular ex- plant they should erect upon their lots. ample, the unhappy choice R2, Q3would Assume that for some reasons (noise, result; so regulations might exclude smoke, vibrations) the payoff to each these possibilities also. In other ex- entrepreneur depends upon the decision amples the simple elimination of rows of the other entrepreneur. The game and columns which cause non-dominance situation is evident. Assume, as seems might produce a more desirable result; reasonable in this instance, that the ex- but this particular example does have 36 The Ri, Q, solution is not the over-all opti- the merit of suggesting the possibility of mum, which is, of course, R2, Q2. This latter solution, using taxes and subsidies in order to rely however, would be impossible to achieve unless less there was "complete" regulation. more on the pricing mechanism and constraints in municipal zon- 37 See Otto A. Davis, "The Economics of Munici- on direct pal Zoning" (unpublished Ph.D. dissertation, Uni- ing.38 versity of Virginia, 1959), for a more complete analysis of the zoning phenomenon which is not in 38Koopmans and Beckmann have shown, in a game theoretic terms. See also Davis and Whinston, theoretical analysis of plant location, that in the op. cit., for a discussion which utilizes game theory absence of interaction a pricing mechanism will and programming. It also might be pointed out that be given an optimal solution; but if interdepend- the phenomenon of urban renewal involves ex- encies via transportation costs on "intermediate ternalities and that a "preventative" solution can commodities" exist, no optimal equilibrium solution be obtained throughout a constrained game ap- can be expected (see Tjalling C. Koopmans and proach (see Davis and Whinston, "The Economics of Martin Beckmann, "Assignment Problems and the Urban Renewal," Law and ContemporaryProblems, Location of Economic Activities," Econometrica, Vol. XXVI [Winter, 1961]). XXIV [January, 19571, 53-76). EXTERNALITIES, WELFARE, AND THE THEORY OF GAMES 261

VII. CONCLUDING REMARKS course, mergers may not always be Throughout much of the analysis of technically best. An implication of our this paper we have assumed that the argument is that it is less likely that market structure remains competitive the government will need to be concerned even after a sufficient number of mergers about externalities on the production have occurred for the natural unit for side than is often thought, as long as the decision-making to be achieved. This as- market is and remains competitive. sumption has been necessary, of course, The difficult problem for the policy- to permit welfare statements to be made. maker when there are externalities on We do not propose to debate here the the production side arises when the question of whether or not markets in market is not competitive, or when pos- the "real world" are competitive. Yet, sible mergers which would "truly in- since it might be inferred that a logical ternalize" the externalities would result implication of our argument is that the in a change in the market structure. Here natural unit for decision-making will be some measures must be devised to indi- so large that the market structure will cate the possible welfare gains from change, it must be pointed out that this merger and the welfare loss that would implication is not necessarily true. The result from the divergence from the importance and extent (or "scope") of competitive situation. externalities is an empirical fact of the Let us now consider briefly some pos- real world, and we have as yet no sible methods of estimating the effects of problems systematic evidence about this. A priori, externalities. Admittedly, the it does seem plausible that in some in- here are very difficult, and this is pre- stances the natural unit for decision- cisely the point which we have tried to of making will be so large that a competi- emphasize in our previous discussion tive market structure will not exist, but possible solutions. It is not easy for the this is not a novel conclusion. It has long governmental policy-maker to obtain been recognized that natural monopolies needed information on the nature of the exist, and it is equally well known that cost functions and thus the entries in the some competitive markets exist. payoff matrix. But, presumably, after of The main point of our argument has study of each particular instance been that the "classical" tax-subsidy externalities, some estimates could be and solution to the problem of externalities made. So let us assume discreteness on the production side would be difficult consider the problems the policy-maker ex- to achieve in the dominance case and would face in the case of separable impossible in the non-dominance case ternalities when a possible merger which effects even if the government could be as- would truly internalize external from sumed to be trying to maximize welfare. might change the market structure The We have argued that a much easier solu- competitive to non-competitive. tion may exist in the case of competitive policy-maker could use the estimated markets (and it is in this context that payoff matrix to determine the differ- the problem has been analyzed by Meade ence in total profits between the domi- and others) simply by allowing mergers nant solution and the maximum-profit to take place until the "natural unit" for solution. This difference in profits could decision-making has been achieved. Of be used as a crude measure of the change 262 OTTO A. DAVIS AND ANDREW WHINSTON in welfare which is associated with a ods of approximation are necessary. change from a situation of competitive Again assuming discreteness, statistical markets and externalities to a situation analysis of variance suggests one such of competitive markets and no externali- possibility. Since, if there are no ex- ties. Assuming merger is feasible, one ternalities present, the payoff matrix is must subtract from this difference a sum composed of constant elements, a vari- which would reflect the welfare loss as- ance analysis would give a zero value sociated with the alteration of the here. Thus a variance analysis of the market to a situation of no externalities payoff matrix which gave a non-zero and a non-competitive market structure. value could be taken as an approximation One approach toward the estimation of to the welfare value of the change from this latter magnitude might be to con- a situation of externalities and competi- sider this part of the welfare change as tion to a situation of no externalities and some function of the change in output a competitive market. Thus the policy- and price that would result from the maker might be able to make the ap- change in market structure."9 propriate comparisons as in the previous The policy-maker must also compare case. In situations where externalities with this estimated net gain the net wel- exist and the market structure is non- fare gains that might result from direct competitive to begin with, the measure- regulation, inequality constraints, and ment problem is even more difficult. other alternatives. These gains could be Yet, our a priori judgment is that this estimated by deducting from the esti- may be the more important area for mate of the welfare gain calculated from policy choice by a welfare-maximizing the payoff matrix an estimate of the government. costs of the constraints themselves. This paper has been limited largely to With an externality of the non- externalities on the production side. separable type the measurement prob- Other important externality problems lem is even more difficult. Cruder meth- associated with, for example, interre- 39Franco Modigliani has suggested methods for lated utility functions have not been predicting the change in output and price which ac- treated here, although the game-theo- company change in market structure ("New De- retic approach does seem promising for velopments on the Oligopoly Front," Journal of Political Economy, LXVI [June, 1958], 215-32). future research in these areas.