CHAPTER 6 THE EFFECT OF NUMBERS UPON COMPETITIVE EFFORT Julian L. Simon and Salim Rashid "The economic role of competition is to discipline the various participants in economic life to provide their goods and services skillfully and cheaply...When one asks...whether the competition of three merchants will serve better than two...the answers prove to be elusive." George Stigler (1968, p.181) INTRODUCTION Under which conditions (if any) does more competition--say three firms rather than two, or two rather than one--produce beneficial results? Or to put it differently, how do two or more firms behave differently than does one alone? The answers are indeed elusive. And as we shall see, under at least some circumstances the result of additional competitors may _n_o_t be better service or greater competitive effort under the discipline of competition than without it. With respect to price, there are various conjectures, going back to Cournot (l838), about mechanisms that lead to a duopoly price lower than a monopoly price, and the conclusion is then simply extrapolated to triopoly and beyond, without formal argument. But even in the monopoly-duopoly price comparison, a good many assumptions must be made before one can even be sure theoretically that the duopolists will not settle at the monopoly price,1 and these assumptions are often not appropriate. *** With respect to other competitive behavior, there exists even less economic explanation of how and why two or three competitors will serve the public differently than will one firm alone--if they do. Economists _b_e_l_i_e_v_e that service will be better if there are three restaurants or airlines than if there is only one. Certainly there is lots of casual evidence that competition improves service. The competition of Hertz and Avis is a legendary example; Macy's versus Gimbels is another. And the outpouring of new telephone products with the onset of competition to AT&T following the court decision allowing people to buy their own phones is a recent illustration. The usual explanation offered is that competitors are forced to try harder--that is, to exert more effort--than is a monopolist who supposedly can enjoy the "quiet life" that Hicks (1935/1952, p.369) said is the best of all monopoly profits. But this explanation is far less precise than the sort of explanation economists are accustomed to accept. By analogy, one could say about price, too, that it is expected to be lower in duopoly than in monopoly because competitors will "try harder," and let it go at that; such an answer would be below professional contempt, however. The aim of this chapter is to construct an apparatus that will provide a reasonably systematic analysis of some non-price behavior in monopoly compared to duopoly. The analysis applies to individuals and business organizations which are assumed to differ in their human and economic characteristics of the decision-makers (including their wealth), rather than where the organizations are conceived of as identical machines.2 *** But this is not different from satisfactorily-rich models of price competition; such models also necessarily involve such human qualities as probability assessment and learning (unless one believes that rational- expectations models which depend upon a stable equilibrium with an infinite horizon would say all that needs to be said about duopolistic competition). Effort need not be measured directly in this conceptual framework, any more than such intervening variables as utility or risk preference need to be measured directly in the economic analyses in which they are commonly used. But if one did wish to measure effort directly, it is certainly measurable in a variety of ways, e.g., objectively by the number of smiles exhibited, the sound level of ragslaps by a bootblack, or subjectively on a self-rating scale, even if it cannot be measured as simply as can money or time inputs. The logic underlying measurement of a similar concept that is commonly thought to be more difficult to measure interpersonally--utility-- is discussed elsewhere--Simon (1974). We will first consider the impact of an additional (second) competitor in a given market, as compared to the situation where just one "firm" is operating. This is the comparison which economists seem to have in mind when contrasting monopoly with duopoly. The number of competitors allowed is assumed to be regulated (by law or otherwise) independently of the competitors' actions, and hence the effects of firms' behavior upon prospective entry need not be considered. We will next consider another prototypical comparison: two artificially-separated identical markets each with a single competitor, versus the same two no-longer-separated markets with the barriers removed so that both firms compete for all of the combined market. We shall first show the analysis with informal geometry; this will be crude and numerical, and will require a good many steps. The reason for this laborious and inelegant presentation is that the concepts being worked with here are deceptively slippery, and experience shows it necessary to nail down their interpretations in great detail if confusion is to be avoided. Hence we hope that the reader will bear with us and not be put off by this plodding prolix approach. If patience is lacking or insight is plentiful, one may proceed immediately to the short proof given after the long geometric analysis. Eventually the central idea in this chapter should link up with the rapidly burgeoning new literature on industrial structure.3 But though *** that body of work includes advertising and modes of competition other than price in its research program, it has so far concentrated upon price. And the outcome of the analysis for non-price competition is quite different than for price competition, as this chapter shows, and as may also be seen elsewhere.4 The work on this chapter also has an intellectual link *** to Leibenstein's X-efficiency and to the "slack" concept, slack being the opposite side of the coin from effort. ELEMENTS OF THE ANALYSIS Imagine a bootblack A having the sole right to operate in a large office building. (We will shortly imagine a second bootblack B also given the right to operate in the same place.) Assuming price fixed and no paid advertising, the owner-operator bootblack's own services are the only significant input. Therefore "profit" (which equals sales revenue in this case) depends only upon effort. Effort is the amount of energy A puts into snapping the rag, the amount of warmth in A's smile, and other activities which affect consumer reaction. The number of clients that A serves during the eight hours the building is open depends upon the amount of effort only. When not actually shining shoes, A waits for clients and loafs, so time is not a variable input in this case. Figure 6-l shows the Incremental Drive-Effort (IncDrEf) function which portrays the additional effort that A is _w_i_l_l_i_n_g_ _t_o_ _e_x_p_e_n_d at various levels of wealth in hand in order to obtain an additional dollar of revenue, at two levels of initial wealth. The declining convex function indicates that the more assets A has in hand at any given moment, the larger the increment of revenue that would be required to induce him/her to exert an additional unit of effort. ----------- Figure 6-l ---------- The IncDrEf function is hypothesized to have the form Wn+1 - Wn Incremental payoff) IncDrEf = a(---------) = a(---------------------- -----------) Wn Wealth in hand at decision moment where Wn is the wealth prior to undertaking the incremental opportunity,and Wn+l is the expected wealth after the incremental effort is exerted.In other guises, this function has a long theoretical history, and itseems to fit the results of various psychological experiments. Butthough _s_o_m_e shape must be chosen for illustrative purposes, _t_h_e_ _e_x_a_c_t_s_h_a_p_e_ _o_f_ _t_h_e_ _f_u_n_c_t_i_o_n_ _h_a_s_ _a_b_s_o_l_u_t_e_l_y_ _n_o_ _b_e_a_r_i_n_g_ _u_p_o_n_ _t_h_e_ _a_r_g_u_m_e_n_t_g_i_v_e_n_ _h_e_r_e; the only requirement is that the incremental effort func-tion be monotonically decreasing with wealth in hand. Wealth would generally be defined as future lifetime earnings contingent on some prior set of decisions, plus present assets. But in the present case, where we will study a situation sufficiently simple so that the decisions about the present day's effort and opportunities can be considered independent of all similar future decisions, and where the day's income can be considered small relative to present assets, we can for convenience consider only the existing assets as wealth. (We might think of this day as the last working day in A's life.) Becker's analysis of the allocation of time yields the proposition that, under plausible assumptions, an increase in wealth leads to a decrease in the amount of time that the person would spend on work. This implies a function that has the same declining characteristics as does the Incremental Drive-Effort function in Figure 6-l. Therefore, as long as the analysis does not depend upon the specific exponential form, we may broaden the effort concept to include time and managerial attention as well as other effort. One may then either regard the function as referring to effort alone, or to time alone, or to both. This may be reassuring to the reader who worries about effort being too evanescent a concept for economic analysis. More generally, the analysis fits any quantity which has an increasing cost of supply with increasing quantities of the input. But this is likely to be true only of inputs which are limited to the resources of a given individual. Other inputs can ordinarily be purchased at constant or declining prices as volume increases, and plant size can be chosen so that larger quantities can be supplied at a cost that is at least not increasing if not actually declining. Calculating the Drive-Effort function should be done marginally, in accordance with the facts about how the decision-maker sequentially arrives at the decision. The bootblack first inquires whether a first unit of effort should be expended. If the answer is "yes", then inquiry is made about an additional unit of effort, and so on until the answer is "no." The wealth in each calculation includes the results of all decisions up to that point. This is in contra-distinction to asking about various _t_o_t_a_l amounts of effort; this latter procedure would only be appropriate for an opportunity which must be accepted or rejected as a whole, on a take-it-or- leave-it basis, rather than for an opportunity where one is able to choose whichever amount of effort one wishes. Further discussion of the creation of the IncDrEf function is in the Appendix. Next we must bring the _e_f_f_e_c_t_s of effort upon sales into the analysis. To explicate as clearly as possible, let us portray this relationship in several ways. To begin, Figure 6-2 shows the volume of sales that will be produced by various amounts of effort. The solid line shows the cumulative sales response (TR) at various effort levels, and the dotted line shows the incremental sales response per unit of effort (MR) at successive effort levels. This figure is similar to the graph of an advertising-to-sales relationship, which can be seen as a special case of the effort-sales relationship. ----------- Figure 6-2 ---------- Now we determine the amount of effort that will be chosen by A under existing circumstances. We begin by mapping the marginal data in Figure 6-2 into the form of Figure 6-3, where the MR relationship is shown by what we may now call the Incremental Necessary Effort (IncNecEf) line, that is, the additional effort required at each level of sales to obtain an additional dollar of sales. IncNecEf is the transformation of the marginal relationship derived from the cumulative effort-sales relationship (TR) shown in Figure 6- 2. The TR curve is drawn concave, just as the revenue function is concave with respect to advertising.5 The TR function cuts the *** vertical axis above the origin because the bootblack will get some business even if he/she makes no effort at all to do so, but instead simply "provides the basic goods," i.e., shines the shoes mechanically with no flourish or conversation, and without making his/her kiosk bright and attractive. From this is calculated the marginal sales response function (MR). Also shown in Figure 6-3 is an IncDrEf function redrawn from Figure 6-l. The point of intersection Z of the IncNecEf and IncDrEf functions in Figure 6-3 determines the amount of effort that A will expend. Up until point Z, the required effort for an additional dollar of sales (Incremental Necessary Effort, or IncNecEf) has been less than the effort that A would be willing to expend (IncDrEf) to obtain an additional dollar; at effort levels above Z, the opposite is true. The difference between the IncNecEf and the IncDrEf is similar to producer surplus in conventional analysis. ONE VERSUS TWO COMPETITORS IN THE SAME MARKET Now we add a second competitor B to the same building. The cumulative sales response (TR) function in Figure 6-2 may be understood as the sales that will result from various amounts of _t_o_t_a_l effort in the market, that is, the sum of the efforts by A and B (or by A alone when A is a monopolist). To bring out the nature of the process most clearly, let us shift temporarily to the case in which TR goes through the origin--that is, where no sales are made without some effort. On this assumption, Figure 6-4 shows the monopolist's function, as well as A's function computed for two levels of effort by B. (Zero effort by B would in this case lead to a function for A identical to the monopolist's function.) The amount of sales obtained by each competitor is assumed proportional to the effort exerted by that firm relative to the sum of the effort by the two, and the total sales by the two are equal to the sales that would be obtained by a monopolist exerting that much effort. ---------- Figure 6-4 --------- _P_r_o_p_o_s_i_t_i_o_n_ _l_:_ _ _I_n_ _a_ _g_i_v_e_n_ _m_a_r_k_e_t_ _w_i_t_h_ _t_w_o_ _c_o_m_p_e_t_i_t_o_r_s_ _r_a_t_h_e_r_ _t_h_a_n_a_ _m_o_n_o_p_o_l_i_s_t,_ _t_h_e_r_e_ _w_i_l_l_ _b_e_ _l_e_s_s_ _e_f_f_o_r_t_ _e_x_e_r_t_e_d_ _p_e_r_ _f_i_r_m_,_ _b_u_t_ _g_r_e_a_t_e_r_t_o_t_a_l_ _e_f_f_o_r_t. IncNecEf functions calculated in rough increments from Figure 6-4 are shown in Figure 6-5. There we see that the IncNecEf functions for A are higher than for the monopolist, because the slopes of competitor A's TR functions in Figure 6-4 are lower at each effort level than are the monopolist's, and A will therefore exert _l_e_s_s_ _e_f_f_o_r_t in a competitively- divided market than he/she will as a monopolist in the same market. This can be seen even for the case in which A's IncNecEf effort function would be perfectly horizontal (as is the function for the incremental cost of advertising, aside from discounts), no matter what its height. ---------- Figure 6-5 ---------- The _t_o_t_a_l effort exerted by the two competitors A and B together, however, is _g_r_e_a_t_e_r than would be exerted by a monopolist alone, and in that sense the public is better served; this total effort is the central issue which this chapter addresses. Ultimately we want a neat formal proof of this proposition, but the following argument should suffice to make the point for now: By eye alone we can see that the intersections of A's IncNecEf function with the IncDrEf function are at more than half the effort levels of the monopolist's intersection, and hence two competitors' efforts add to more than the monopolist's. Furthermore, it is rather obvious that this would even be so if the IncDrEf function were horizontal. _C_o_r_o_l_l_a_r_y_ _a_:_ _ _T_h_e_ _p_r_o_p_o_s_i_t_i_o_n_ _h_o_l_d_s_ _e_v_e_n_ _i_f_ _t_h_e_ _c_o_m_p_e_t_i_t_o_r_s_c_o_o_p_e_r_a_t_e_ _t_o_ _t_h_e_ _u_t_m_o_s_t. To make the argument more conclusive, consider the "cooperative" function in Figure 6-5, representing A's effort and sales results on the assumption that A and B always exert the same amount of effort (which they would do only if they have the same IncDrEf functions). Furthermore, consider how things would be if the IncDrEf function were horizontal, as it would be for advertising. If so, and if the competitors exerted (and planned to exert) the same amount of effort (on advertising) as each other, they would expend just as much effort (or spend just as much for advertising) in total as would a monopolist. It would then be as if they were each half- shareholders in the monopolist; they would be moving up the same TR function as the monopolist, with the same objectively-caused stopping point. In contrast, with a function for A where B's effort is assumed fixed at any particular level, A's IncNecEf function lies below the cooperative function, as seen in Figure 6-5. The reason is that with any increase in A's effort in the cooperative function, something "bad" (an increase in B's effort) is assumed to happen along with the "good" result of an increase in sales to A, when A exerts an additional unit of effort. This "bad" does not occur in functions where B's effort is fixed and hence A will exert more effort with the latter. And this proves that A will exert more than half the effort of a monopolist, and hence competition produces more total effort, even with a horizontal DrEf function. (One might imagine some perverse functions that would induce less effort than the "cooperative" function, but these implausibilities need not be considered here.) Furthermore, since the IncDrEf function is sloped downwards rather than being horizontal, the distance between the cutting points on the monopolist's and the competitors' MR function is less than with a horizontal line, and the cooperative function then produces more than half the effort of the monopolist's function. A fortiori, then, the two competitors' efforts add to more than a monopolist's effort. Also, the amount of total effort per dollar of sales is obviously greater with two competitors than with the monopolist. It is illuminating that the concavity of the response function and the convexity of the IncDrEf function are _b_o_t_h needed to produce the result that competition leads to more effort in the "cooperative" case. If the IncDrEf function were horizontal, total effort in that case would be no more than in monopoly. But if their IncDrEf functions are convex, this will not be true for the cooperative function; the competitors will be "poorer" than the monopolist, and hence will exert more effort than "half" a monopolist. Now let us analyze this case in a more rigorous fashion, which also brings out the driving mechanism more clearly. Consider that the monopolist's chosen equilibrium is where (6-2) d(IncDrEf) = (IncNecEf) d(TR) d(TR) which is at the effort level E**, as seen in Figure 6-5. Now assume that an additional competitor enters the same market, and that the ^ ^ chosen effort levels for firms A and B are EA and EB. Suppose that ^ ^ the level of total effort remains the same, that is EA + EB = E**. ^ ^ ^ If so, at least one of EA and EB must be less than E**, say EA. If so, ^ (6-3) d(IncDrEf) E > d(IncDrEf) d(TR) A d(TR) because A's effort is less than the monopolist's; and A is therefore further back toward the origin, and therefore higher, on the IncDrEf ^function in Figure 6-5. But now we notice that, on the assumption that EB is fixed, the return to additional effort by A would be the same as it ^ ^ ^ would for the monopolist at E**. (That is, if EB is fixed, and EA + EB = ^E**, then A's response function IncNecEf at E would be the same as IncNecEf for the monopolist at E**.) If so, ^ (6-4) d(IncDrEf) E > d(IncNecEf) E = d(IncNecEf) E** d(TR) A d(TR) A d(TR) ^ That is, A would then not be in an equilibrium at E , and would exert ^ Aadditional effort. (In other words, A at EA would get the same returnfrom a unit of additional effort as would the monopolist at E**, butwould "value" it more because of being at a higher level ofwillingness to exert effort to gain an additional unit of revenue, dueto having less wealth.) This is also true for B. Hence * * * ^ (EA + EB) > E**, where EA > EA by the argument just given. We have shown that the total effort expended by A and B willexceed that of a monopolist in the same market (assuming all have thesame IncDrEf functions). But the increase in effort expended arisesfrom a single simple cause: The existence of competitor B makes Apoorer--that is, he/she has less wealth--than a monopolist whenthe total effort expended (and payoffs obtained) by A and B are equalto those of a monopolist, and therefore either or both of them hasgreater willingness than the monopolist to exert additional effort atthat point. It should be noted that the result obtained does not depend uponconcavity of the TR function; it would hold even if TR were linear. The case in which the competitors' IncDrEf functions differ has notyet been analyzed. But we doubt that the results will differ fromthose discussed above. One may wonder how allowing for more fully interactive competition would affect the results. A long-run multi-period dynamic analysis, bringing in many of the relevant competitive considerations, would be much more complex. And there will not be any one single determinate result; this we can conclude with surety from the Simon-Ben-Ur (l98l) analysis of duopolistic competition in advertising, as well as from the similar Simon-Puig-Aschoff (l973) analysis of price competition. Even in the case of advertising competition, where (unlike the situation in which the competitive variable is the amount of effort) there can be no difference with respect to the cost of inputs (that is, where the counterpart of the IncDrEf function is the same for both firms), the outcome differs depending upon initial conditions, the discount factor, probabilistic assessments of the competitor's behavior, etc. With the additional complication of differences in IncDrEf functions, the range of indeterminacy must be even greater. Interactive competition is not really important in this context, however. The analysis at hand involves only the ex post effort positions of the competitors, and it does not matter by what gaming process they arrive at those positions. The discussion so far has dealt with the optimizing position for A, given some level of effort by B, which could be assumed to be the optimizing effort level for B given the effort level by A. There could be more than one set of mutually-optimizing positions, though the analysis has not yet reached that far. But the conclusions reached above seem to hold for each pair of competitive/outcome positions, no matter how arrived at, and that is all that seems to matter for now. _C_o_r_o_l_l_a_r_y_ _b_:_ _ _T_h_e_ _p_r_o_p_o_s_i_t_i_o_n_ _h_o_l_d_s_ _w_h_e_t_h_e_r_ _t_h_e_ _m_a_r_k_e_t_ _T_R_ _f_u_n_c_t_i_o_n_d_o_e_s_ _o_r_ _d_o_e_s_ _n_o_t_ _p_a_s_s_ _t_h_r_o_u_g_h_ _t_h_e_ _o_r_i_g_i_n. If we now shift back to a market in which the monopolist's TR cuts the vertical axis _a_b_o_v_e the origin (Figure 6-6), we see that all the conclusions drawn for the through-the-origin function hold, a fortiori, because the TR functions are steeper for A in the above-the-origin case than in the through-the-origin case, whereas the monopolist's function has the same slope at all effort points for both cases. This conclusion can be seen in a rough way by comparing the two figures visually. It also follows from the fact that A's function passes through the origin at even the slightest effort by B--say .00000l of a unit of effort--but at positive effort by A, the function then rises almost to what it would have been if B exerted zero effort, in which case A's function is the same as the monopolist's function. Hence A's function must be steeper in the early range than the monopolist's function, while it has almost the same slope in the later range; in comparison, at .0000l units of effort by B in the through-the-origin case, A's function would be the same as the monopolist's throughout. Even in the absence of a general proof, this should satisfy the reader that the conclusions drawn from the through-the-origin case apply to the above-the-origin case. ---------- Figure 6-6 ---------- As suggested earlier, advertising can be seen as a special case of the analysis given here, that in which the input cost function is linear and horizontal (a given amount of advertising space or time costs almost the same no matter how much you advertise), and is the same for both competitors (who therefore will advertise the same amounts under such simple conditions), whereas the IncDrEf function is not likely to be constant, and may differ for the two competitors. Therefore, if the monopolist's TR function with respect to advertising passes through the origin, competitors will advertise the same amount in total as will a monopolist, on the same reasoning given above in discussion of a "cooperative" TR-effort function with a horizontal effort curve.6 ****Otherwise, the conclusions drawn above are consistent with the results if the concept of effort is replaced by the concept of advertising.7 ***** This general line of analysis enables us to determine the effect of one competitor's wealth, and his/her consequent willingness to exert effort, upon the other competitor's actual effort. The greater the wealth of B, the less effort B is willing to exert, and the higher and steeper will be A's response function. Hence A will exert more effort if B is wealthier and exerts less effort. This is the opposite of the effect that one expects in sports competition where greater effort by one competitor is thought to induce greater effort by the others, under most circumstances (but not if the greater effort by one dispirits the others). There may also be a sport-like competitive effect in business, perhaps opposing the effect analyzed here, and perhaps even dominating this one empirically. But that phenomenon would require a very different sort of theoretical analysis. One may wonder about the social effects of there being more competitors who expend more effort in total but less per competitor. The public gains by the greater effort, but might lose through the lessened output per competitor. This is exactly the common argument against allowing an unlimited number of taxicabs on the streets of New York, or shops in the bazaar of an Indian city, or bag-hustlers outside a poor country's airport. Presumably, however, each of the competitors in such a situation could not be more productively employed elsewhere in the economy, or else he/she would move into other situations. So a general-equilibrium analysis gives no reason to worry about this effect. Furthermore, if the firms under discussion are larger than a single operator, if the Drive-Effort effect seeps through to the employees as it affects the owners, there will be only public gain. There will be no social loss from diminished time input, because the number of employees would be appropriately smaller in competitive firms than in a monopolist firm. ONE VERSUS TWO COMPETITORS IN A DUOPOLY MARKET TWICE THE SIZE OF THE MONOPOLY MARKET Imagine now two contiguous identical market areas--two towns, or two entry corridors of an office building--which are separated by fiat. Each of two firms has a license to operate in each separated market, and is forbidden to compete in the other's market. We will compare such a situation with that in which the barrier to competition between the two markets is removed, and both firms (but only these two firms) are then permitted to compete in both the contiguous markets. The combined market is simply twice as large as each original uncombined market. For specificity we will think about two bootblacks A and B operating in the separated corridors of an office building, and we will again assume price fixed and no advertising. We shall also assume no economies of scale in effort. More specifically, if the TR function in a separate market yields S sales at E effort, in the merged market a total effort of 2E yields 2S total sales. ---------- Figure 6-7 ---------- Unlike the divided-market analysis discussed earlier, we find that the conclusion differs depending upon the conditions of competition. Let us first examine the case where the TR function cuts the vertical axis above the origin. Figure 6-7 shows the TR function for the combined market, that is, the sales response to the sum of the two competitors' efforts. _P_r_o_p_o_s_i_t_i_o_n_ _2_:_ _ _I_f_ _c_o_m_p_e_t_i_t_o_r_s_ _i_n_ _m_e_r_g_e_d_ _m_a_r_k_e_t_s_ _c_o_o_p_e_r_a_t_e_ _f_u_l_l_y_,_t_h_e_y_ _w_i_l_l_ _e_x_e_r_t_ _n_o_ _m_o_r_e_ _e_f_f_o_r_t_ _t_h_a_n_ _w_h_e_n_ _e_a_c_h_ _o_p_e_r_a_t_e_s_ _a_s_ _a_ _m_o_n_o_p_o_l_i_s_t _ _i_n_ _a_ _s_e_p_a_r_a_t_e_ _s_i_n_g_l_e_ _m_a_r_k_e_t. If A and B decide on effort levels cooperatively, they will produce the same amounts of effort in total as they would when operating as monopolists in separated markets. The explanation is that their wealth levels would also be the same at those points in both cases. That is, in Figure 6-7 the curve marked "cooperative TR" shows the response either to a monopolist in a separate market, or to one competitor in the combined market assuming the other competitor exerts equal effort. This result is sharply at variance with the divided-market case, where cooperation among two competitors leads to larger total effort than a monopolist exerts, because the two competitors are "poorer" than a single monopolist in the same market. It was pointed out in the divided-market cooperative case that the result of greater effort depends upon both concavity in the TR function and curvature in the IncDrEf function. In the merged-markets case, the curvature in the IncDrEf function has no differential effect because the function is effectively the same for firms as cooperating competitors and as monopolists. (The above discussion assumes the same IncDrEf functions for the two competitors. Differences in the function are explored in a richer context below.) The outcome that the two firms may reach a cooperative outcome which is no better for the public than a monopolist in the merged markets resembles the situation in price competition. But in competition with effort (as with advertising) there is less incentive for each competitor to cheat on a cooperative agreement than there is with price, because an across-the-board price reduction (the sort usually discussed) affects all customers and not just the marginal customers, whereas a change in effort only works at the margin.8 Additionally, price cuts can be secret, ***whereas competition with effort on advertising is revealed publicly. Therefore, the potential benefits of competition with effort on advertising are more likely to be frustrated by agreement among the competitors, explicit or implicit, than in price competition. For example, competing stores are more likely to cut prices surreptitiously or with various gimmicks than they are to sneak a little bit later closing time. _P_r_o_p_o_s_i_t_i_o_n_ _3_:_ _ _I_f_ _t_h_e_ _f_i_r_m_s_ _d_o_ _n_o_t_ _c_o_o_p_e_r_a_t_e_,_ _t_h_e_r_e_ _w_i_l_l_ _b_e_ _m_o_r_e_t_o_t_a_l_ _e_f_f_o_r_t_ _w_h_e_n_ _m_a_r_k_e_t_s_ _a_r_e_ _m_e_r_g_e_d_ _than when separate. Unlike the single firm's TR function in the cooperative situation when the TR function does not go through the origin, the function for A with a _f_i_x_e_d effort by B goes through the origin, and then rises more steeply than the combined function crossing it at the point where A's effort is equal to B's. Because the slope is greater, and marginal response is greater, the marginal function must cross the IncDrEf function at a higher level of effort for A than when in a separate market. For example, assume that in a separated market, where sales are greater than zero at zero effort, A's *optimum effort is DrEf = l0 units. With the two markets combined, if A and B each exert ten units of effort, A will get the same response as in the separated market. But with B's effort fixed at l0 units, the marginal response to A's effort is greater at DrEf = l0 units in the combined markets than in the separate market. To prove this rigorously, let us first recall our assumption about the distribution of sales among the competitors: Of the sales resulting from the total effort exerted by the two competitors, each competitor receives sales in the same proportion as its effort is to the total. By analogy, two persons putting fishing lines into a lake at random places will catch fish in proportion to other numbers of lines, while the total catch is a concave-downward function of the total number of lines. To continue the analysis in this analogy: Imagine a rectangular pond divided into two halves by a fish-proof barrier. Each half is fished by a monopolist. We will consider simply one day's fishing, without worrying about depletion effects in the future. Each person fishing has the same Drive-Effort function, we assume. Each puts in lines until the marginal return from an additional line equals the marginal Drive-Effort from putting in an additional line. Now imagine that the barrier is removed. If the two fisherpersons collaborate as a firm, they would put in the same total number of lines that they did as separate monopolists, as discussed above. Now imagine that they do not act cooperatively. If A assumes that B will put in as many lines as B did as a separate monopolist, which is half the number of lines they would put in as a joint firm, A will now reason as follows: If I put in one additional line, the return to that line will be greater than if both of us put in additional lines, because the probability of getting a catch on one of two additional lines is less than the probability of getting a catch on just one additional line due to there being fewer total lines in the pond in the latter case. Therefore, if earlier I was indifferent about putting in an additional line conditional on B also putting one in, I will no longer be indifferent about putting in an additional line when I assume B does not. Therefore, A's imagined response function is different than either the function for the joint firm or for the separate-market monopolists. The geometry implies more effort for A, and hence more total effort, when the two are not cooperating than when A and B are operating separately or when they are competing. To prove this proposition rigorously, we compare the slope of the TR function rho for A with B's effort fixed, against the slope of the function R for the two competitors taken together, at the effort level X* for A at which A would stop when operating in a separated market; this latter is the same as the slope r for A when A and A are acting in concert, and is the slope for the total TR function at twice A's effort when acting alone. That is, (6-5) l/2 R(2X*) = l/2[2r'(X*)] = r'(X*) where an apostrophe indicatesa derivative The TR function rho at that point for A when B's effort is fixed is (6-6) rho(2X* + delta) = R(2X* + delta) _X*_+_delta 2X* + delta That is, A's proportion of the sales resulting from A's increment of effort delta above the joint effort total when operating alone in separated markets, or in concert in the merged markets, is A's proportion of the total sales resulting from delta effort. Now differentiating, we find that (6-7) rho'(delta) = (2X*_+_delta)[R().+(X*_+_delta)R'()]_-_R()(X*_+_delta) 2 (2X* + delta) = R(X*)_+_(2X_*+_delta)(X*_+_delta)R' 2 (2X*_+_delta) = r(X*) + _X*_+_delta R'(2X* + delta) 2X* 2X* + delta = r(X*) + r'(X*) 2X* 2 = 1 [r(X*) + r'(X*)] > 1 2r'(X*) = r'(X*) 2 X* 2 * Now what about when B's effort is above or below DrEf, A's chosenlevel of effort in the separated market? The conclusion is not sostraightforward or conclusive. We must first ask _w_h_y B might be aboveor below DrEf. The simplest assumption is that B has the same IncDrEffunction as does A but decides--and signals so to A--that he will exert *effort greater than DrEf simply because of miscalculation or otherrandom influence. If A now optimizes, its decline in effort below*DrEf will be less than the increment that B is above it because of the *slopes of the IncNecEf function above and below DrEf; the sum of efforts byA and B will then be greater than in separated markets. For example, *if both A and B would optimize at DrEf = l0, but B exerts 20 units ofeffort by mistake, A would optimize by dropping effort about 6 unitsif the IncNecEf function were horizontal, and therefore would reduce effort by even less than 6 units even less because IncNecEf is upward-sloped *instead of horizontal. On the other hand, if B is below DrEf by random *occurrence, then A will move less far above DrEf than B is below it, by the same reasoning, and the total effort exerted will be _l_e_s_s than in separatemarkets. However, after B discovers the miscalculation, he/she will move *somewhat toward DrEf (but not all the way to it) as long as A is above * *DrEf. But then A would move toward DrEf, there would be iterations, and *they would then both end up near DrEf. If A and B have different IncDrEf functions, the matter is even morecomplex. Assume that B's IncDrEf function leads to less effort than *does A's. B will then exert effort above DrEf, but not as much as B *is below it. But the difference between DrEf and B's effort is not alldue to competition, but rather some of the difference is due to B's IncDrEf function. Here is one scenario of what might happen. * * Assume A plans effort DrEf and B plans effort (l/a)DrEf where a >l, because B is wealthier than A. They meet and tell their trueplans. They then calculate new optimizing points, both responding to * *the difference [DrEf - (l/a)DrEf]. B moves back on his DrEf--that is,upwards -- and also up the marginal response function, both of whichbuffer each successive reduced unit of effort more strongly. A movesdown his DrEf function toward more effort, and also down the marginalresponse function, both of which buffer each additional unit of effortmore strongly. Which competitor would move further in response to thegap, i.e., whether B's reduction in effort would be greater or lessthan A's increase in effort, may depend upon the particular shape ofthe TR function. And there would then be subsequent adjustments. Butit is probable that each of the adjustments would have the same direction of effect as the first one. (The tell-the-truth sort of interactive process discussed here, without gaming, might serve very well as a simple model of the non-price interactive process, in contrast to the analytic needs in a price- adjustment process. The comments made earlier about the possibility of creating a dynamic model of interactive competition apply here as well as to the one-market case.) _P_r_o_p_o_s_i_t_i_o_n_ _4_:_ _ _I_f_ _t_h_e_ _T_R_ _f_u_n_c_t_i_o_n_ _g_o_e_s_ _t_h_r_o_u_g_h_ _t_h_e_ _o_r_i_g_i_n_,_ _t_h_e_r_e_w_i_l_l_ _b_e_ _m_o_r_e_ _e_f_f_o_r_t_ _e_x_p_e_n_d_e_d_ _b_y_ _t_w_o_ _f_i_r_m_s_ _c_o_m_p_e_t_i_n_g_ _i_n_ _a_ _m_e_r_g_e_d_m_a_r_k_e_t_ _t_h_a_n_ _w_h_e_n_ _t_h_e_y_ _a_r_e_ _m_o_n_o_p_o_l_i_s_t_s_ _i_n_ _s_e_p_a_r_a_t_e_d_ _m_a_r_k_e_t_s_,_ _b_u_t_ _t_h_e_e_f_f_e_c_t_ _w_i_l_l_ _n_o_t_ _b_e_ _a_s_ _m_a_r_k_e_d_ _a_s_ _w_h_e_n_ _t_h_e_ _f_u_n_c_t_i_o_n_ _d_o_e_s_ _n_o_t_ _p_a_s_s_t_h_r_o_u_g_h_ _t_h_e_ _o_r_i_g_i_n. This proposition may be seen intuitively in the geometry of thetwo cases. DISCUSSION 1. The results we get for the merged-market case are in no way dependent upon the shape of the Drive-Effort function, or even on it being a declining function. Therefore, the result immediately generalizes to advertising, for which the cost function corresponding to the effort function is flat. In the divided-market case, the driving force is the effort function, because there is a difference in the wealths of monopolists and competitor. But there is no such wealth difference between competitor and monopoly in the merged-markets case. Rather, the moving force is the change in opportunity between the separate-market and the merged-market cases. Therefore, together the two cases illustrate the operation of both bladesof the Drive-Effort function's scissors--the effort scissor in thedivided-market case, and the wealth scissor in the merged-marketscase. 2. Absent from this discussion of effort exerted in competition is the sense of struggle to outdo the competition before he/she outdoes you. This is the feeling of the heat of battle, to be compared to the monopolist's quite life. Upon reflection, it seems to us that the operational counterpart of this phenomenon is founded in expectations about the competitor's behavior, with actions taken contingent in these other expectations. This interaction can only be comprehended scientifically with a rich simulation model, such as Simon, Puig, and Aschoff (l973) or Simon and Ben-Ur (l982). But though the subjective reflection of these activities may be the most noticeable aspect of the competition, but that is of relevance to pscyhologists rather than to economists. 3. The effects of market structure upon the adoption of innovation illustrate the Drive-Effort hypothesis. Primeaux's work (1977) on competitive electric monopolies shows that competition-induced effort reduces cost by about 10% of average cost, though adoption of innovations is not separated from other managerial efforts in that analysis. Cases of adoption of innovations that have no cost except the effort of making the change (and perhaps a relatively tiny expenditure of time) should be particularly relevant. For example, a historical examination of the adoption of the now-universal "January White Sale" of linens by department stores (Simon and Golembo, 1966) showed that stores in small towns were much slower than were stores in big cities to adopt the practice, though there was no installation cost and the practice quickly proved profitable (as shown by continued use by almost all stores after initial use). The only likely explanation is the stronger lash of competition in large cities, which fits with the analysis of monopoly- duopoly effort offered above. SUMMARY AND CONCLUSIONS The central question addressed in this chapter is the amount oftotal effort that will be expended if there are two competitors in amarket rather than one. The core of the analysis is the hypothesizedrelationship between (a) wealth, and (b) the effort that a competitoris willing to exert: With respect to a given economic opportunity,the more wealth a competitor has in hand at a given moment, the lessDrive-Effort the opportunity will evoke. The particular functionused for illustration is exponential, in accordance with the standardassumption about the incremental function of (Benthamite) diminishingmarginal utility. But any monotonic downward- sloped function--equivalent to an increasing "cost" of additional inputs of effort withincreasing quantities of that input--would produce similar results. We first compare a given market when monopolized or as competed for by two firms. A downward-sloping incremental Drive-Effort function with respect to additional net revenue, and a (wholly-plausible) concave-downward total-revenue sales-response function with respect to successive amounts of effort expended (together with the plausible assumption that competitors obtain sales in proportion to the amounts of effort they expend) combine to imply that two competing firms will expend more effort than will one firm, even if the overall sales-response function passes through the origin. (This is unlike the case with advertising, which can be thought of as similar to the Drive-Effort analysis except that the "cost" of incremental input is constant rather than increasing as is the case with effort; if the total-revenue function with respect to advertising passes through the origin, two competitors may be expected to advertise in total only as much as would a monopolist in this simple case.) The reasons for the greater expenditure of effort in the divided-market case are two: l) The division of the market pushes competitors back to ranges of operation where they are poorer and therefore have a higher propensity to exert effort in order to obtain a given amount of net revenue. This conclusion follows from the combination of the Drive-Effort function and the total revenue function; neither alone is sufficient. If the TR function cuts the vertical axis above the origin, then the conclusion follows a fortiori, and then the concave TR function is enough to lead to the conclusion, as is the case also with advertising. 2) True competition rather than perfect cooperation gives each firm an incentive to expend effort in excess of the amount that would be expended in a cooperative solution. This follows from the definition of cooperation as firms acting in concert in such fashion as will maximize the welfare of the two taken together. The absence of such cooperation implies that one firm will assume that the competitor will _n_o_t act in concert, and therefore the firm treats the competitor's position as at least temporarily fixed. If so, a move by one firm even from the joint-maximization point will be (at least temporarily) profitable for the other firm because the latter will receive returns as if it were a monopolist having the whole market; with this outlook, the marginal function at the joint-maximization point is steeper than if the outlook is for the firm with the competitor acting in concert. This is the mechanism of "cheating" described and analyzed convincingly by Stigler (l964). In the case where the sales in a merged market are split by two competitors who each formerly monopolized the separated markets, if the two competitors cooperate perfectly there will be no increase in effort expended. But in the absence of perfect cooperation, each competitor has reason to expend effort beyond the cooperative amount when cooperating, by exactly the same reasoning as in the divided-market case. Concerning the outcomes if the firms have different Drive-Effort functions: We have no theory about the equilibrium the firms would reach even if they were to communicate with total honesty about future interactions and actions. Hence we cannot say whether the total effort under those circumstances would be greater, less, or the same as if the firms operated as monopolists in separated markets. But we can say that no matter what the equilibrium reached under such conditions, there would then be incentive for each of the competitors to expend greater effort if there is an absence of perfect cooperation. The reason for the great difference in results for the divided-market case versus the merged-market case is simply that in the former the monopolist is made poorer by the change, and this impoverishment leads rather straightforwardly to more effort. In the latter case, no such impoverishment takes place, and hence the outcome depends on various other conditions The chapter does not deal with the situation in which there may be a continuing attachment of customers to a seller from period to period, and the effect this may have upon a new entrant. This factor would be a considerable complication; it is difficult to analyze even where effort is not considered to be a variable. Eventually, however, it could be included in the analysis. A major aim of this chapter is to demonstrate the usefulness of theDrive-Effort concept. The same concept seems to be useful for analyzing issues as various as the rise and fall of empires, and the relationship of the number of children in a family to the amount of work done in the labor force by the father and the mother at various children's ages, as for the analysis of monopoly versus duopoly in this chapter. D/129A85- 37 Effort6 6/6/85 Figure 6-l's principle of construction: (TRi+1 + W0) - (TRi + W0) IncDrEf = ------------------------- (TRi + W0) Figure 6-2's principles of construction: TR | Ef = l0 = $l00 (TR | Ef = 20) - (TR | Ef = l0) = .8 x $l00 (TR | Ef = 30) - (TR | Ef = 20) = .8 x .8 x $l00 . . . (TR | Ef = 10) - (TR | Ef = 0) MR at Ef = 5 = ------------------------------ x 10 10 (TR | Ef = 20) - (TR | Ef = 10) MR at Ef = l5 = ------------------------------- x 10 10 . . . Figure 6-3's principles of construction: IncDrEfA from Figure 6-l. IncNecEf from MR in Figure 6-2. $100 Figure 6-4's principles of construction: monop TRDrEf=l0 = $l00 monop TRDrEf=20 = .8 x $l00 monop TRDrEf=30 = .8 x .8 x $l00 . . .TR for A in the presence of B exerting ten units of effort (A | B = l0):For each level of A's effort, DrEfA, find the results for the monopolist at (DrEfA + l0), and attribute to A sales (TR) in the same proportion as A's effort to A's plus B's effort [DrEfA/(DrEfA + DrEfB)] = [DrEfA | (DrEfA + l0)]. Similar procedure followed for DrBf = 20. Figure 6-5's principle of construction: IncNecEfmonop, IncNecEfA | B = l0 and IncNecEfA | B = 20 from Figure 6-4. IncNecEf cooperative represents IncNecEfA taken from the TRmonop function monop in Figure 6-4 in this fashion: IncNecEfj/2A = l/2(IncNecEfj) at l/2(TRmonop | DrEfj). Figure 6-6's principles of construction: TRmonop (5 + TRmonop in Figure 6-5) (TRA | DrEfB = l0) = same principle as in Figure 6-4. Figure 6-7's principle of construction:For the merged markets, double each effort level for the monopolist inFigure 6-6 yields, and double the sales results. For A in the presenceof B exerting DrEfB = l0 units of effort; for each level of A's effortfind the results for the monopolist at (DrEfA + l0), and attribute toA sales in the same proportion as A's effort. Appendix to Chapter 6 Notes on the Construction of the Incremental Drive-Effort Functionl. The results of the calculations obviously depend on the procedure.Consider a decision-maker with W0 = $l00, and a = 2000, decidingwhether to expend l500 units of effort which will produce $l00. Thecalculation is IncDrEf = 2000($200_-_$100) = 2000, $l00 and the opportunity will be accepted because 2000 > l500. If an additional l500 units of effort will produce $50, the calculation for that decision will be IncDrEf = 2000($250_-_$200) = 500, $200and the second unit of effort will not be expended because 500 < l500.(Notice, please, that the second calculation had Wn = $200, includingthe $l00 at the start of the decision-making plus the $l00 obtainedfrom the first unit of effort. But now consider the calculation if the decision had been 3000units of effort or none to obtain $l50. IncDrEf = 2000($250_-_$100) = 3000, $100 and since IncDrEf is just equal to the necessary effort, the decision toexpend both units of effort would be on the knife-edge. 2. For a situation in which any amount of effort can be chosen, it isappropriate to use a continuous function, constructed operationally byconsidering successive very small increments of effort; this is thenature of the IncDrEf function in Figure 6-l. (One might just as well dothe calculations in fairly large increments, for convenience andbecause this is the way decision-makers act in most situations, in thesame way that they decide, for example, whether to spend $500,000, or$750,000, or $l,000,000 for advertising, and not whether to spend$500,000, or $500,00l, or $500,002...).3. It does not make sense to draw a cumulative function for DrEf theway one constructs a function for the relationship between actualeffort and sales. This is because there is no basis for adding theincremental DrEf that one is willing to expend to go from, say, $l00to $l80 sales revenue, to the DrEf that one owuld have been willing toexpend to get from $0 to $l00 sales. The sum of those quantities is_n_o_t the same as the amount of DrEf that one would be willing to expendto get from $0 to $l80. There is no sensible way that one canconstruct a cumulative function from which one could determine theDrEf to get from any one revenue point to another. FOOTNOTES 1This was shown by: Stigler, l964, pp. -6l; and Simon, Puig, and Aschoff, l973. 2Alfred Marshall's comment about the classical writers isrelevant in this connection: "For the sake of simplicity of argument,Ricardo and his followers often spoke as though they regarded man as aconstant quantity...It caused them to speak of labour as a commoditywithout staying to throw themselves into the point of view of theworkman; and without dwelling upon the allowances to be made for hishuman passions, his instincts and habits, his sympathies and anti-pathies, his class jealousies and class adhesiveness, his want ofknowledge and of the opportunities for free and vigorous action. Theytherefore attributed to the forces of supply and demand a much moremechanical and regular action than is to be found in real life."(l920, pp. 762-763) 3For example, see: Baumol, l982; and Smith, Williams, Bratton, and Vannona. 4See Stigler, l964; Simon, l970, Chapter 4; and especially Simon, l982. 5For summaries of the shape of the analogous advertising- responsefunction, see Simon; l965, and Simon and Arndt, l980. Contrary tocommon belief, the function is not S-shaped but rather is simply concave downward, as shown conclusively by a wide array of data of many kinds. 6This latter case was incorrectly analyzed in Simon, Julian L., l967; and in Simon, Julian L., l970, Chapter 4. 7As shown in: Simon, l967; and in Simon, l970, Chapter 4. 8For discussion of this, see Stigler, l964. 1This was shown by: Stigler, George J., "A Theory of Oligopoly," _J_o_u_r_n_a_l_ _o_f_ _P_o_l_i_t_i_c_a_l_ _E_c_o_n_o_m_y, 72, February l964, pp. -6l; and Simon, Julian L., Carlos Puig, and John Aschoff, "Duopoly Simulations and Theory: An End to Cournot," _T_h_e_ _R_e_v_i_e_w_ _o_f_ _E_c_o_n_o_m_i_c_ _S_t_u_d_i_e_s, Vol. XL, l973, pp. 353-366. 2Alfred Marshall's comment about the classical writers isrelevant in this connection: "For the sake of simplicity of argument,Ricardo and his followers often spoke as though they regarded man as aconstant quantity...It caused them to speak of labour as a commoditywithout staying to throw themselves into the point of view of theworkman; and without dwelling upon the allowances to be made for hishuman passions, his instincts and habits, his sympathies and anti-pathies, his class jealousies and class adhesiveness, his want ofknowledge and of the opportunities for free and vigorous action. Theytherefore attributed to the forces of supply and demand a much moremechanical and regular action than is to be found in real life."(l920, pp. 762-763) 3For example, see: Baumol, William J., "Contestable Markets: An Uprising in the Theory of Industry Structure," _A_m_e_r_i_c_a_n_ _E_c_o_n_o_m_i_c_ _R_e_v_i_e_w, 72, March, l982, l-l5; and Smith, Vernon K., Arlington W. Williams, W. Kenneth Bratton and Michael G. Vannona, "Competitive Market Distinctions: Double Auctions vs. Sealed Bid-Offer Auctions," _A_m_e_r_i_c_a_n_ _E_c_o_n_o_m_i_c_ _R_e_v_i_e_w, 72, March, l982, 55-77. 4See Stigler, George J., "A Theory of Oligopoly," _J_o_u_r_n_a_l_ _o_f_ _P_o_l_i_t_i_c_a_l_ _E_c_o_n_o_m_y, 72, February l964, pp. 44-6l; Simon, Julian L., _I_s_s_u_e_s_ _i_n_ _t_h_e_ _E_c_o_n_o_m_i_c_s_ _o_f_ _A_d_v_e_r_t_i_s_i_n_g (Urbana: University of Illinois Press, l970), Chapter 4; and especially Simon, Julian L., "Firm Size and Market Behavior," mimeo, l982. 5For summaries of the shape of the analogous advertising- responsefunction, see Simon, l965, and Simon and Arndt, l980. Contrary tocommon belief, the function is not S-shaped but rather is simply con-cave downward, as shown conclusively by a wide array of data of manykinds. 6This latter case was incorrectly analyzed in Simon, Julian L., "The Effect of Competitive Structure Upon Advertising Expenditures," _Q_u_a_r_t_e_r_l_y_ _J_o_u_r_n_a_l_ _o_f_ _E_c_o_n_o_m_i_c_s, Vol. LXXXI, November, l967, pp. 6l0-62; and in Simon, Julian L., _I_s_s_u_e_s_ _i_n_ _t_h_e_ _E_c_o_n_o_m_i_c_s_ _o_f_ _A_d_v_e_r_t_i_s_i_n_g (Urbana: University of Illinois Press, l970), Chapter 4. 7As shown in: Simon, Julian L., "The Effect of Competitive Structure Upon Advertising Expenditures," _Q_u_a_r_t_e_r_l_y_ _J_o_u_r_n_a_l_ _o_f_ _E_c_o_n_o_m_i_c_s, Vol. LXXXI, November, l967, pp. 6l0-62; and in Simon, Julian L., _I_s_s_u_e_s_ _i_n_ _t_h_e_ _E_c_o_n_o_m_i_c_s_ _o_f_ _A_d_v_e_r_t_i_s_i_n_g (Urbana: University of Illinois Press, l970), Chapter 4. 8For discussion of this, see George J. 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