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Polymorphism in competitive strategies: Trading stamps
J ECO BUSN 1985; 37:1-17
1
Polymorphism in Competitive Strategies: Trading Stamps William HaUagan and Wayne Joerding
This paper uses the example of trading stamps to examine a model of competitive nonprice marketing strategies. A concept borrowed from ethology, polymorphic equilibrium, is developed and used to explain salient features of trading stamp use by retail firms. The results contribute to an understanding of why virtually identical firms may optimally choose quite different competitive strategies. The paper also demonstrates the possibility of cyclical movements in the proportion of firms that adopt nonprice strategies such as the use of trading stamps.
Introduction Several issues of general interest can be addressed by an analysis of trading stamp use as a type of nonprice competition. While we focus on trading stamps, the analysis applies to a wide variety of promotional strategies like free gifts and quasi-lotteries. In the most general terms, we are interested in the conditions under which a competitive environment can sustain the allocation of resources by some (but not all) firms to activities which only serve to determine relative market shares. In practice, we observe a dual market structure with respect to promotional strategies like trading stamps. For example, during most of our history with trading stamps, retail markets have been composed of stamp and nonstamp firms. In the analysis of market structure, it is most common to argue that the coexistence of dual strategies represents a disequilibrium, supported by mobility barriers, and that over time all firms will be forced to adopt the single most efficient strategy. 1 In this paper we examine the conditions under which the competitive solution, unhampered by mobility barriers, will generate a market structure in which otherwise identical firms adopt widely divergent promotional strategies. 2 It is recognized that a market structure in which no firms adopt promotional strategies to increase their market share can be unstable. 3 However, if the promotional strategy does not i See, for example, Caves and Porter (1977) and Porter (1979). 2 Hallagan and Joerding (1983) consider the case in which advertising affects both the share and the market size. Here we concentrate on the case in which nonprice competition is strictly a battle over the market share. However, whereas advertising expenditures are usually treated as a fixed cost, the promotional expenditure treated in this article varies with sales. 3 Scherer (1980), p. 389. Address reprint requests to Dr. Wayne Joerding or Dr. William Hallagan, Department of Economics, Washington State University, Pullman, Washington 99164-4860.
Journal of Economics and Business © 1985 Temple University
0148-6195/85/$03.30
2
William Hallagan and Wayne Joerding increase the size of the market, a market structure in which all firms adopt identical promotional strategies will result in all firms having the same sales (but higher costs) then they would have had if no firms had adopted a promotional strategy. Under some conditions, this pure strategy structure is also likely to be unstable. Thus, a priori, the case can be made that we are likely to observe dual market structures, although these structures may, in turn, also be unstable. In the field of biology, the term "polymorphism" is used to describe populations in which otherwise identical individuals adopt noticeably different strategies. 4 For example, it is argued that a population in which all males fight to increase their share of the pool of females will be invaded by males who allocate their energies to nonfighting activities more directly related to reproduction. Similarly, it is argued that a population composed entirely of nonfighting males will be invaded by males who adopt the fighting strategy. Since neither pure strategy population structure is likely to be stable, we will observe species in which some otherwise identical males fight while others do not. In this paper we investigate whether the concept of polymorphism has a plausible counterpart in the promotional strategies adopted by firms. Like the example cited above, we concentrate on strategies which do not affect the size of the market but instead represent a battle over market share. Additionally, a rather puzzling phenomenon that arises from an examination of trading stamp use is the existence of cycles. At the national level, the prevalence of trading stamp use reached a local maximum during the period 1910-1916, followed by a period of relatively low use, only to be followed by another boom in the late 1950s and early 1960s, which again was followed by a period of declining use. Results of our model offer a tentative explanation of this cyclical behavior. The rest of this paper is organized as follows. The next section provides a brief description of the history and practice of trading stamps. This is followed by the development of a formal model of trading stamp use in which otherwise identical firms are allowed to adopt the stamp or no-stamp strategy. From this model we derive conditions under which the market will have a polymorphic equilibrium in which some, but not all, firms adopt the stamp strategy. We extend the results by noting that price cuts enter our model in exactly the same manner as stamp intensity. Interpreted this way, our model can exhibit both the single- and two-price equilibria (SPE and TPE) studied by Salop and Stiglitz (1977, 1982). The next section presents a simple model of cyclical stamp use that is intended to be suggestive rather than definitive. The final section summarizes our results and discusses the broader implications of this research. It is worth pointing out that this paper does not attempt to explain why consumers engage in stamp collecting; rather, we skirt this interesting issue by simply assuming that some consumers desire to collect stamps.
The History and Practice of Trading Stamp Use 5 The present practice of trading stamp use evolved from the initial efforts of individual store owners during the late 1800s, who placed coupons inside the packages of particular commodities like soap. Coupons were collected and redeemed at the issuing store for small gifts. In 1869 Sperry and Hutchison (S&H) saw the advantage 4 See Ayala and Kiger (1980) for a textbook treatment of this concept, 5 The material in this section is drawn primarily from FTC (1966), Breem (1957), and Fox (1968).
Polymorphism in Competitive Strategies
3
of selling stamps which were issued by a variety of stores and were redeemable at a central location. Following their introduction, the prevalence of stamp use grew until stores issuing stamps represented 7% of all retail sales in 1914. 6 Thereafter it declined in importance. In 1950, only 1% of all retail sales were generated by stamp issuing stores, but the incidence of stamp use accelerated again until it reached a peak in 1965, when 16% of all retail sales came from stamp-issuing stores and 55 % of all grocery stores issued trading stamps. 7 During this period, it was estimated that 84 % of all families were actively collecting stamps. Thereafter, the practice declined in importance, but in 1973, 31% of all grocery stores still issued trading stamps. 8 The basic principles of trading stamps have, however, remained relatively constant over time. Retail outlets purchase stamps from stamp companies and issue them to their consumers, who then collect the stamps and redeem them for commodities at redemption centers operated by the stamp company. The value of the stamps, in terms of the goods for which they can be redeemed, is typically about 2 % of the consumer's purchases; although retail stores will, at times, choose to offer multiple-stamp days in an effort either to stimulate off-peak use of the store or as a response to the actions of rivals. To promote the value of stamps sold by stamp-issuing stores, the stamp companies agree not to sell stamps to any retailers competing in the same market. Thus, retail outlets have an exclusive franchise with respect to a particular stamp plan in their market. During the 1950s and 1960s, more than ten major stamp companies competed for retailer business. Grocery stores represented the majority of clients for the stamp companies, with gas stations comprising the majority of the remaining market. Although markets where stamps are issued are generally characterized by a large proportion of consumers who collect stamps, these consumers differ with respect to the intensity of their preferences for stamp plans. According to one survey, stamp collectors can be classified into ~hree broad groups: 1) the eager (10%), 2) the interested (35 %), and 3) the indifferent (55%). 9 One interpretation of these classes is that the " e a g e r " will actively search out stamp-issuing retail outlets, the "indifferent" save stamps only because the store where they traditionally shop happens to issue stamps, and the "interested" are split in some undetermined proportion between the "stamp-seeker" and "store-loyal" groups, depending on the stamp intensity of the market. Considerable research has been done on the effect of trading stamp use on prices, market shares, and profits. Surprisingly, there does not seem to be a consensus on the effect that trading stamps have had on prices. Although they represent an additional cost, trading stamps also offset other promotional expenditures and allow the store to spread fixed expenses across greater sales. Empirical comparisons of prices for stamp and nonstamp stores have not permitted any definite conclusion. 10 The effect on market shares and profitability is clearer. There seems to be little doubt that the decision to issue trading stamps will increase the market share, but the effect on profitability will depend on when the firm adopts the stamp-issuing strategy. Firms 6 FTC (1966), p. 2. 7 Boone, Johnson, et al (1979), p. 72. 8 Ibid. 9 FTC (1966), p. 33. ~o FTC (1966), pp. 29-31. Also see Churchill, et al. (1971) for a review of the empirical evidence on the stamp-price relationship. In their study of gasoline prices, they found no evidence that stamp use was associated with higher prices.
William Hallagan and Wayne Joerding adopting the stamp strategy early in the cycle, when few other firms issue stamps, experience the most significant gain in market share and profitability, while firms adopting the strategy later in the cycle have less favorable experiences, t t
A Model of Trading Stamp Use In this section, we develop of model of trading stamp use to examine the conditions determining the equilibrium proportion of stores adopting the trading stamp strategy. There are three steps in the development of our model. First, we allow stamp-issuing stores to select the profit-maximizing stamp intensity. Second, we derive an equal profit condition by assuming that stamp and nonstamp stores adopt and abandon strategies until the profits from each strategy are equal. Finally, we combine the equal profit condition with the profit-maximization condition to define the equilibrium proportion of stores adopting the stamp strategy. The notion of equilibrium that we use is consistent with firms' maximizing behavior, in that they cannot increase profits either by changing their stamp intensity or by switching to a no-stamp strategy. The interesting characteristic of a polymorphism is that otherwise identical entities select divergent strategies. Incorporating this feature in our model requires us to assume that firms are identical in every way, except that some firms adopt the stamp strategy while others do not. This assumption implies that the profit-maximizing level of stamp use will be the same for all firms adopting the stamp strategy, t2 We assume that firms are price takers and that price is independent of stamp use. 13 This assumption has three rationales. First, it simplifies the analysis considerably. Second, it retains the focus of our model on strategies representing a struggle over market share, and allows us to maintain our assumption that firms are identical except with respect to the promotional strategy they adopt. Finally, the assumption does not violate the empirical evidence of the relation between stamp use and prices. We have also assumed that sales are characterized by constant average variable costs in the relevant range, so that the firm's price cost margin (P) is constant. On the demand side, we first assume that firms consider the market demand (Q) at the given price as fixed. ~4 This is probably a realistic assumption in that a promotional strategy like stamp use, unlike advertising, does not increase consumer information. Although the practice of issuing stamps, given a fixed price, does lower the net price, stamps are typically tied to commodities with inelastic demands like groceries and gasoline, so that stamps with a redeemable value equal to 2 % of sales are most likely to be viewed by retailers as having no effect on total market demand. This assumption also maintains the focus of this paper on promotional strategies which affect only relative market shares. Given this demand structure, the profits for firm i are given by equation (1), in which qi is the quantity sold and P is the price cost margin, which is constant given ~ Fox (1968), pp. 30-50. J2 Since all firms are identical, we will not be able to address the issue of what types of stores are most likely to adopt stamps. Instead, we focus on the proportion of stores which adopt stamps. ~3If trading stamp use is interpreted as a price cut, this assumption means that the high price is fixed, which is analogous to the Salop and Stiglitz (1982) assumption of a fixed monopoly price determined by consumer preferences. The low price is endogenously determined. 14This is also characteristic of the Salop and Stiglitz (1982) model.
Polymorphism in Competitive Strategies
5
our previous assumptions. On the cost side, Si is the stamp intensity, defined as the average stamp cost per unit of sales, which is subtracted from P to calculate the profits for stamp-issuing stores. Fixed costs arising from store operation are F, while fixed costs associated with issuing stamps (e.g., advertising informing consumers of stamps and stamp-issuing equipment) are C.
IIi=(P -
Si)qi - F -
C
(1)
To provide added structure to the variable qi, we adopt the following approach. Two types of consumers are envisioned. Type I consumers are "stamp seekers," who will switch from nonstamp stores to stamp stores if given the chance. Let ct = ct(•) be the proportion of sales to type I consumers, where S is the average stamp intensity of stamp-issuing stores. 15 The function or(. ) is assumed to be increasing and continuous with continuous first derivatives, or(0) ~ c~0 > 0, and a ( P ) _< 1. The most restrictive aspect of this specification is the condition that at least one consumer will seek stampissuing stores even when the stamps are valueless: ct(0) > 0. This specification greatly simplifies the analysis and does not affect the main results in any important way. t6 Type II consumers are "store loyal," and collect stamps only if their preferred store adopts the stamp strategy. The proportion of type II consumers is noted as 1 - ct, and all firms are assumed to have equal sales to type II consumers. For convenience, all consumers are assumed to generate one unit of sales. These assumptions imply that firm i's sales can be expressed as equation (2), where Q is market demand and N is the total number of stores in the market, both of which we have assumed to be fixed. 17In equation (2), Mi = Mi(Si, S) is the firm's share of type I consumers.
qi=Q(cv.Mi+~--~)
(2)
Profit Maximization Given the structure developed above, we can specify the firm's optimization problem as equation (3). Maxs, 1-Ii=(P-
(
lo /
Si)Q (xMi'Jt"7
- F- C
(3)
The profit-maximizing stamp intensity is restricted to the interval (0, p).~8
msThis is an ad hoc assumption about consumer behavior, but can be motivated in several ways. As stated in Salop and Stiglitz (1977), it may be that only a proportion of the consumers spend the resources necessary to inform themselves about the location of stamp-issuing stores. Alternatively, one might suppose that there is a distribution of the propensity to collect among consumers. Finally, as stated in Salop and Stiglitz (1982), it may be that consumers choose stores randomly but purchase more in a stamp store than a nonstamp store. 16 An alternative would be to allow a(S) = 0 for all ~ _< S for some ~ _> 0. Although this specification considerably complicates the analysis, variations on the main results can be proved. See note 28. m7Fixed N is not crucial, but it considerably simplifies the analysis. At any rate, it seems sensible to assume that stamping policies will not affect entry and exit decisions, especially since Q is fixed and stamp policies only determine market shares. ~s Given these assumptions, it should be clear that S can be thought of as the shadow price of stamps, and P - S can be interpreted as a discounted price. The variable C becomes a fixed cost to the low-price strategy, and again can be justified by the need to advertise the lower price.
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William Hallagan and Wayne Joerding F u r t h e r m o r e , in o r d e r to e x a m i n e p o l y m o r p h i s m , we have p u r p o s e l y assumed that all firms are identical so that p r o f i t - m a x i m i z i n g stamp intensity satisfying equation (3) will be the same for all stores adopting the stamp strategy. As a consequence, Si is the same for all stamp-issuing stores, and ~¢ = Si so that the subscript and b a r notation is superfluous and is hereafter dropped. It is a s s u m e d that in solving this p r o b l e m , firms c o n s i d e r only the effect that stamp intensity has on their costs and market share. Although ~ is a function o f the average stamp intensity o f the market, each firm is assumed to act as i f its own choice o f stamp intensity has no effect on the m a r k e t stamp intensity (i.e., a s m a l l - f i r m assumption), so that firms treat a as a m a r k e t parameter. Solution o f equation (3) implies that:
[
-
s ~(s) -~ M(A-N J - ( P
- sO~(s0E=0
(4)
where E ~ ( d M / d S ) ( S / M ) is the f i r m ' s expectation o f the elasticity o f its share o f type I consumers with respect to its own stamp use. This elasticity contains the direct effect o f its own stamp use as well as the indirect effect on the stamp intensity o f other firms, and is assumed to be a constant. 19 Condition (4) can be used to derive a function o f S which is useful in our analysis. Let Zm(S), for Se(0, P ) , be the p r o p o r t i o n o f firms issuing stamps that is consistent with the equilibrium condition (4) and with identical stamp intensities o f all stampissuing firms. Identical stamp intensities implies that M = I / N s , and (MAr) = l = N s / N ~ Z where N s is the number o f stamp-issuing firms. A f t e r substituting these restrictions into equation (4), Zr~(S) must satisfy the resulting condition identically on a set o f S such that Ze(0, I). W e can derive 2° undefined if S e A l = [Se(0, P ) : a ( S ) > S / ( P -
Zm(S)~
(P-
S)a(S)E-offS)S [l
-
o~(s)]s
S)E]
if SeA - - [Se(0, P):S not in A~ or A0]
undefined if SeA0-----[S~(0, P ) : S > P E / ( E
+ 1)]
(5)
a real valued function defined on (0, P ) . 21 Concentrating for a m o m e n t on the interior solution for Zm, we can m a k e some rough calculations which will p r o v i d e some insight into the relation b e t w e e n Zm and E . F o r g r o c e r y store operations, the gross m a r g i n ( P ) is about 20% o f sales, and the most frequently o b s e r v e d stamp intensity was two cents p e r d o l l a r o f sales. 22 F r o m our discussion o f the history o f stamp use, it appears that the proportion o f consumers that are stamp seekers (o0 is s o m e w h e r e between 0.10 and 0.45, say, c~ = 0.30 as an intermediate value for the calculations. Substituting for the interior solution o f ~9Since M(., •) is a function with a range restricted to the unit interval, it is not possible in general for E to be a constant. Thus, more accurately, we assume that/14(., .) is a hyperbola for relevant values of S and ~. 20 This is easily done by substituting Z = MN-1 into equation (4) and solving for Z. Then use the resulting condition to define sets of S for which Z > 1 and Z < 0. Thus, Zm(') is continuous, but its derivative is not. 2J A , A0 and At are sets on the real line. If sup At < I, then sup A t < inf Ao, since Pc~E/t~E + 1 <
PE/E + 1. 2~ In 1982 the gross profit divided by sales was 18.5% for grocery stores and 15.3% for gasoline stations, as reported in Industry Norms and Key Business Ratios, Dun and Bradstreet (1982).
Polymorphism in Competitive Strategies
7
equation (5) we get Z = 3.14E - 0.36. Thus, setting the roughness of the calculations as well as the equal profit condition aside, polymorphism (0 > Z > 1) is consistent with expected stamp elasticities between 0.11 and 0.43. Although the range is narrow, the values are plausible. The earlier observation that 55 % of all grocery stores issued stamps at the peak of the trading stamp boom (Z = 0.55) suggests that profit-maximizing grocery stores expected the stamp elasticity to be about 0.29 at that point in time. These calculations apply only to the profitmaximizing decision and are not about an equilibrium. A plausible configuration for Zm(S) is shown in Figure 1. Of course, without further restrictions, there are many different possible shapes for Zm(S). The example shown in Figure 1 would apply if the slope of eft. ) at S = 0, e ' (0) is not too large and declines monotonically with increasing S. In the example, AI = (0, Sin) and A0 = [PE/(E + 1), P]. In this example, an exogeneous increase in either the price cost margin or the expected stamp elasticity shifts the Zm function out and, for any interior values of Z, increases the profit-maximizing stamp intensity. These results hold only for the firm's profit-maximizing decision. To analyze the market equilibrium, we must also consider the equal profit condition, to which we now turn.
Z
z=(s}
Se Figure 1.
Sm
S*
P'E. E+I
P
S
$
William Hallagan and Wayne Joerding
Equal Profits Imposing the equilibrium condition that M = I / N s for all firms adopting the stamp strategy, the profit functions for representative stamp-issuing firms (IIs) and nonstamp firms (HN) are given by equations (6) and (7). I-Is = (P -
S)Q
-~
~
rlu=PQ[ ('-N
c0]-F
- F- C
(6) (7)
At an equilibrium, firms will adopt and abandon strategies until the profits for stamp and nonstamp firms are equal, l-Is = l-IN, which implies S
~
P-
S=(1
-
ot)Z
(P-
NC S)Q(I -
or)
(8)
The equal profit condition can be rearranged to calculate how much the sales of stamp-issuing firms must exceed the sales of the nonstamp firms in order for each strategy to generate equal profits. For the case in which C = 0, the equal profit condition is satisfied when the ratio of the sales of the stamp-issuing firms to the nonstamp-issuing firms equals P / ( P - S). Roughly, using the market norms for grocery stores, the equal profit condition is satisfied if stamp-issuing stores have about 10% higher sales than non-stamp-issuing stores. Again, this result is within the plausible realm. However, equation (8) need not hold for single-strategy equilibria since profits for one of the strategies is irrelevant. Let Z~(S), for S~(0, P ) , be the proportion of firms issuing stamps, which is consistent with the equilibrium condition that all firms earn the same profits and all stamp-issuing firms have the same stamp intensity. 23 Then Z = Ze(S) satisfies equation (8) as an identity for 0 < Z < 1, which implies that Z~(S) can be written as 1 if S e B t ~ [ S e O , P):ot(S)>(SQ + N C ) / P Q ] Ze(S)=
(P - S)Qa(S) Qs[1 - . ( s ) ] + NC
if SeB=[Se(O, P):S not in Bz or B0]
0 if SeBo~[Se(O, P):A(S)<0]
(9)
a real valued function on (0, P), where A(S) is the excess profits of stamp firms, 7rs 24 --
~'N.
A possible configuration for Z e ( ' ) is shown in Figure 1; many other shapes are possible depending on the restrictions placed on a ( ' ) . As before, the illustrated Ze(" ) would result from an ct(.) with or'(0)small and declining with increases in S. Note that Z , ( S ) is not a function of E. The set Bi = (0, Se) and Ze(.) reaches a minimum of zero at S = P; thus, the set B0 = {P}. 23This condition applies to single- as well as multiple-strategyequilibria. 24B, Bo, and B, are sets on the real line. Bj is definedby noting that (P - S)Qa(S)/{QS[I - ot(S)l + NC} ~ I as a(S) f (SQ + NC)/PQ; consequently,Z,(S) is continuous, but its derivative is not. B0 is the set of S for which the stamp strategy is less profitable than the nonstamp strategy.
Polymorphism in Competitive Strategies
9
Polymorphic Equilibrium An equilibrium is defined by the condition that all stamp-issuing firms are at the profit-maximizing stamp intensity, and all firms are equally profitable. That is, S* will be an equilibrium level of stamp intensity by stamp-issuing stores if S* satisfies Zm(S*) ~ Ze(S*). The equilibrium proportion of firms choosing the stamp strategy can then be determined by either equation (5) or equation (9). Such a solution is illustrated in Figure I. Since a closed-form solution for S* is not possible, we first examine a special case in which c~ is a positive constant and C = 0; then propositions characterizing the general case of ~ = c~(S) are proved. C o n s t a n t ~ a n d C = 0:
An equilibrium Z which is interior to the unit interval can be obtained directly by solving the maximum profit condition (4) and the equal profits condition (8) simultaneously for Z*. This results in 1
1 if E < 1
Z~ ~
and S*_
--
ot
Ot
(1
-
ot)(g
0 if S*>So
-
1)
1
if E > 1
-
and S*_< So ot
(lo)
where So ~ a P ( N - 1)/[1 + c¢(N - 1)]. 25 The point So is easily seen to be less than 1, and is the inf Bo. That is, So is the minimum stamp level for which the profits of the nonstamp strategy exceed or equal the profits of the stamp strategy, A(S) < 0 for all S > So. Intuitive insight into the specification and implications of equation (10) can be obtained from Figure 1. Also, since the figure is not dependent on the specific simplifying assumptions of this section, the discussion is useful in understanding the results of the general model. Remember, a decrease in E causes Zm(') to shift down, while Z e ( ' ) is unaffected. Consider that E > 1/(1 - ct), so that the equilibrium Z is as shown in Figure 1. As E declines, Z* increases and S* decreases until Sm coincides with Se [at E = 1/(1 - ot)] and Z* just equals 1. For all values of E less than or equal to 1/(1 - a), but such that S* < So, the profit of stamp firms is greater than the profit of nonstamp firms, and so Z* = 1. Whenever S* > So, the profits of nonstamp firms are greater than those of stamp firms, and a single-strategy equilibrium occurs at Z* = 0. The intuition behind these results is that an increase in E causes profitmaximizing stamp intensity to increase. This means that the costs of stamp-issuing stores are higher and implies that fewer such stores can survive (if they are to profitable as non-stamp-issuing stores). Consequently, Z* must fall. In order for equation (10) to generate a polymorphic equilibrium (0 < Z* < 1), highly implausible conditions, E > 1, are required. We turn to analysis of the general model. 26
25 N o t i c e that Z* in equation (10) is not a function of P or Q, w h i c h is a result of setting C = 0. 26 T r e a t i n g S as a price cut, even this highly restrictive model can generate s i n g l e - p r i c e e q u i l i b r i a (SPE) at both the low and high prices ( Z = 1 or 0). This contrasts to the w o r k of Salop and Stiglitz (1982), w h o had an SPE only at the high price (Z* = 1).
10
William Hallagan and Wayne Joerding
The General c a s e In this section, w e are concerned with the distinction between two types o f equilibrium, a single-strategy equilibrium (SSE) and a polymorphic or two-strategy equilibrium (TSE). We present conditions relating to the existence o f each o f these equilibria. An SSE occurs when Z* = 0 or 1. Note that an SSE trivially satisfies the equal profits condition (9) but may not satisfy (5), the profit-maximizing condition. A TSE occurs when Z*e(O, 1). An SSE will occur at Z* = 0 if the profits of a single stamp-issuing store, Z = 1/ N, are less than the profits of nonstamp stores. Conversely, an SSE will not occur at Z* = 0 if a single stamp-issuing store can earn more profits by switching to the stamp strategy. This occurs if and only if A(S*m)-~(P -
S*)Qot(S*)- S'Q[1 -
e t ( S * ) l / N - C>_O,
(11)
where S* is the profit-maximizing stamp intensity when Z = 1/N, and thus is a function o f parameters. 27 Condition (1 1) is difficult to interpret, but some insight into the plausability o f an SSE can be gained from the following proposition. Proposition 1 : If up _> C / Q ( P - S*), a necessary condition for the existence of an S S E a t Z * = 0 i s E > 1. Proof: If an SSE exists at Z* = 0, then A(S*) -< O. Since a ( S * ) _< a ( P ) ~ ap, observe that A(S*) _< 0 only if
(P -
S*)Qotp-S*Q(1 -
Ctp)N-~-C<_O.
(12)
Rearrange equation (12) to obtain
olpN i
CN
ap
-
Q(P-
S*)(1
S* -
otu)
P
-
(13) S*
Setting M = 1 (since there is only one stamp firm) in the maximum profit condition (4), one can derive
S* P-
Not(S*)E S*=Not(S*)+[1 - e~(S*)]
(14)
The right side o f equation (14) is an increasing function o f e~, which means that
ap I -
CN Otp Q ( P -
s*)(1
agE
-
_< Up) N a p + ( 1
-
Up)
(15)
is a necessary condition for equation (13) to hold. This condition can be rearranged to obtain
E>ap(N - 1)+1 1 - otp
C[(n -
Q(P-
1)otp+l]
S*)(I -
--- T(aA etp)eLp
(16)
27 Equation (9) is obtained from 17s - lIjv where lqs and l-INare evaluated at Z = I/N and the profitmaximizing stamp intensity S*.
Polymorphism in Competitive Strategies
11
Rearrange T(ap) to see that T(t~p) _> 1 as ~(Otp)~NQ(P -
S*)ot2p-C(N-
1)ap-C_>0
The quadratic ~'(. ) is a parabola which opens upward and has a positive root at & < 1 if Q ( P - S*) > C (since ~'(0) = - C and ~'(1) = N [ Q ( P - S* - C]), which must hold if the stamp strategy is feasible. The value ~'(. ) is not positive semidefinite, but from the quadratic equation C(N-
&=
1) + ~/{C2(N - 1) + 4 N Q ( P - S * ) C } 2 N Q ( P - S *)
Since x/{x + y} < x/x + ~/y and C / N Q ( P - S*) < 1, it can be shown that 6~ < C~ Q(P S*). But ap >_ C / Q ( P - S*), by assumption, which implies that ~'(C~p) _> 0 and E _> 1 is a necessary condition for equation (12). [] Proposition 1 is quite counterintuitive, since it seems to say that as the expected marginal impact on the market share of the stamp strategy increases, then the likelihood of an SSE in which nobody issues stamps increases. This result can be shown for the case represented in Figure 1. Recall that for any interior value of Z, an increase in the stamp elasticity causes an increase in the profit-maximizing stamp intensity (shifts the Zm function to the right). The stamp elasticity does not enter the equal profit condition. Thus, for sufficiently high levels of E, the profit-maximizing stamp intensity approaches the profit margin, and it becomes impossible for a stampissuing store to earn profits equal to those of a store that doesn't issue stamps. It should be remembered that this proposition is about an equilibrium. It is entirely possible that for a large enough E the SSE at Z* = 0 is very unstable; nevertheless, it is an SSE. Proposition 1 is useful because Otp >_ C / Q ( P - S * ) is plausible, while E > 1 is implausible. For example, if C is considered an advertising cost, then the condition Olp >~ C / Q ( P - S*) requires that the maximum proportion of type I customers, c~(P), be greater than or equal to the advertising sales ratio. In general, one would expect C to be very small; thus, this condition is easy to satisfy. If C = 0, then any positive Otp will do. The other condition is implausible, E > 1 implies that agents expect their share of type I customers to increase proportionally more than an increase in stamp intensity. Certainly, this condition cannot be satisfied by all agents in equilibrium. Furthermore, if C is small, then the right side of equation (16) will be considerably larger than 1, and the necessary condition of proposition 1 could be considerably strengthened. I r E > 1 or equation (16) is implausible, then proposition 1 implies that one should not expect to see an SSE at Z* = 0, a result consistent with the empirical observations reviewed in the second section. Consequently, for the rest of this section, we will restrict our attention to the case of no SSE at Z* = 0. It is also possible to have an SSE at Z* = 1. Informally, an SSE at Z* = 1 could occur if, for all combinations of Z and S which satisfy profit maximization, all stampissuing stores make greater profits than nonstamp stores. Unlike the potential SSE at Z* = 0, there are no plausible reasons to reject the possibility of a n S S E at Z* = 1. This is because many plausible specifications of c~(.) can satisfy the conditions of the following lemma. -
Lemma: Given no SSE at Z* = 0, an SSE exists at Z* = 1 if and only ifPQot(Sm) QSm >__ N C , where Sm =- inf A.
12
William Hallagan and Wayne Joerding Proof: The condition E < 1 implies that we only need to consider possible SSEs at Z*~(0, 1). Note that Sm is the smallest S which just satisfies ot(S,n) = Sm/(P - S,,)E. (See the definition of Al.) Also, Zm(Sm) = 1. Consequently, if Ze(Sm) = 1, then Z,n(S,n) ~ Ze(S=) and an SSE will occur at Z* = 1 and S* = Sin. But Ze(Sr,) = 1 if and only if Sr~Bl or, equivalently, PQot(S,n) - QSm >- N C . [] There clearly exist a large number of plausible specifications for a ( ' ) which could satisfy the lemma. However, the condition of the lemma does constrain the permissible set somewhat, as shown in the next proposition, which is analogous to proposition 1. Proposition 2: Given no SSE at Z* = 0, a necessary condition for the existence of an SSE at Z* = 1 is either ao >- N C / P Q or ot'(O) ~ doddS(O) > l / P . Proof: Define O(S) = p Q a ( S ) - QS. Then the condition for the lemma requires either 0(0) = PQoto > N C (which is the first condition above) or, if O(0) < N C , 0 (.) must have a positive slope at S = 0. This latter case will hold if ct'(0) > I / P . D The intuition behind the lemma and proposition 2 is that an SSE can exist at Z* = 1 only if stamp-issuing stores are more profitable than nonstamp stores. But this will occur only if the proportion of the consumers who collect stamps is large at low stamp intensities. The large ot can occur either with a large ct0 or a high rate of increase in ~x for small increases in the stamp intensity. As one would suppose, the necessary size of a0 or or' (0) depends on the fixed cost associated with adopting the stamp strategy. I f C = 0, then any positive a0 may be sufficient to allow an SSE at Z* = 1 to exist. We next consider the main result of this article: conditions for the existence of a TSE or polymorphic equilibrium. Proposition 3: Given no SSE at Z* = 0, a sufficient condition for the existence of a TSE is PQct(S,,) - QSm < N C . 28 Proof: An equilibrium S* is defined by the condition F(S*) =-- Ze(S*) - Zm(S*) ~ O. Let # = P E / ( E + I) < P. From equation (5), it is easy to see that l i m s ~ _ Zm(#) = 0.29 Also, it is easy to see that Ze(l~) > 0, since/~ < P. Consequently, we have F(/~) > 0. Note Zm(Sm) = 1 by definition of the set A. Next, Ze(Sm) < 1 since
o~(s.,) <
QSm + N C
PQ
(17)
or
PQa(Sm) - QSm < N C
(18)
2s A similar proposition holds for the case a ( S ) = 0 for all S < ~ > 0. The restrictive conditions are that ff~(0, ,if'), where if' is the positive root of $(x) = - E Q x 2 + I ( P Q - N C ) E - ( N C + P Q ) l x + P E N C = 0, and 0 < E < ( P Q + N C ) / ( P Q - N C ) > 1. It can be shown that there exists a TSE at
S*~(g, p). 20 lims.~_ Z,n(S) is the left-hand limit of Z,,,(.) at #.
Polymorphism in Competitive Strategies
13
Thus, under the conditions o f the theorem, F(Sm) < 0. Since F ( ' ) is continuous on the connected set (Sin, #), the intermediate-value theorem can be used to conclude that there exists an S* such that F(S*) =- O. [] The condition o f the proposition 3 is simply the negative o f the condition for the lemma preceding proposition 2. The reason P Q a ( S ~ ) - QSn < N C is not a sufficient condition is that not enough restrictions have been placed on or(.). For example, if o~(.) has a monotonically decreasing slope, then one can easily see from Figure 1 that P Q a ( S ~ ) - QS~ < N C becomes a sufficient condition as well. It amounts to a condition in which Sm > So. 30 A number of interpretations are possible. An increase in the number o f firms or the cost of the stamp strategy makes a TSE more likely to occur by increasing the right side of equation (18). In the case o f N, the rationale for this result is that an increase in N raises the maximum profit stamp intensity making it more difficult for the market to support a large number of stamp-issuing firms. Thus, a high enough N would make the stamp strategy less profitable than the rronstamp strategy. In the case o f C, an increase in the cost of the stamp strategy makes the non-stamp strategy more attractive. Since the possibility of an SSE at Z* = 0 is eliminated by the condition E < 1, this means that the likelihood of a TSE at Z* < 1 increases as C increases. Finally, an increase in P raises the likelihood of observing a TSE by increasing the potential profits of the nonstamp strategy while increasing the profit-maximizing stamp intensity. Thus, an increase in P benefits the nonstamp firms more than stamp firms.
A Digression on P r i c e D i s c o u n t s Our model needs no adjustment to serve as a model o f sales. The stamp intensity S can be interpreted as a price discount, and the variable C would then represent the fixed costs associated with implementing a sale (e.g., remarking prices, advertising). While we offer no answer, it is interesting to speculate on why the trading stamp strategy would ever dominate the price discount strategy. Since the stamp strategy restricts consumer choice to those goods offered in the stamp catalog, an equivalent price discount would appear to be preferred by consumers. Yet, after the Tennessee legislature banned the use o f trading stamps in 1957, it was flooded with over 40,000 letters of protest, and a group o f housewives picketed the capitol. 31 While some argue that the stamp strategy is a price discrimination mechanism, this argument does not apply to the national stamp plan described in the second section. 32 Under the usual trading stamp practices, retail stores purchased stamps from national firms and gave these stamps to all customers. Thus, the retail store issuing stamps earned the same net profit ( P - S) from each consumer, regardless of whether or not the consumer
30 A necessary and sufficient condition could be stated as inf A > sup B~, but yields little insight. 31 Fox (1968), p. 16. 32 The usual treatment of consumer behavior predicts that consumers would prefer an equally valued price discount, since it does not constrain their budget allocation, whereas trading stamps are redeemable only for the limited group of goods offered by trading stamp redemption centers. Most existing economic analyses of this puzzle model trading stamp use as a practice adopted to facilitate price discrimination between consumers, with differing values of time (Clarkson and Miller 1982, pp. 269-271; Alchian and Klein 1976; Sherman 1968; and Davis 1959). However, there is still some shadow price which measures the value of stamps as a price cut.
14
William Hallagan and Wayne Joerding eventually redeemed the stamps from the national firm's catalog. Since the retail outlet receives the same net price from each consumer, the stamp strategy does not represent price discrimination from the store's viewpoint. It has been suggested that trading stamps may be more effective in attracting consumers than equivalent price discounts because trading stamps are more visible and credible than a claim that all prices have been cut by 2%. Although this explanation of trading stamp use may operate to some extent, empirical studies suggest that, when explicitly offered trading stamps and an equivalent amount of cash, many consumers chose the stamps. For example, in 1963 the Burgoyne Study of Supermarket Shoppers found that a majority of consumers "prefer to shop in a store giving trading stamps rather than an otherwise identical store that is 2 % cheaper on all g o o d s . " 33 Another study found that "when the offer was one dollar cash or four hundred stamps (presumably equivalent value), 37% chose cash and 62% chose stamps. 34 Thus, by itself, the visibility argument cannot explain the use of stamps rather than price discounts. Market researchers have put forth other explanations for the popularity of trading stamps. Two hypotheses dominate. The first is that trading stamps, due to their shortrun illiquidity (until a sufficient number of books are filled), force consumers to save. Since, for some, forced saving is a good, they are willing to restrict themselves to the types of semi-durable goods offered in the stamp catalog. The second hypothesis is that trading stamps were useful allies for housewives in the battle over the household budget. According to this argument, housewives sought out stamp-issuing stores so that they could redeem the stamps " o n goods for which their husbands may be unwilling to incur a financial expenditure." 35 Consistent with this argument is the fact that the popular premiums were automatic coffee makers, toasters, steam irons, household lamps, and bedspreads. 36 This argument presumes that housewives had less control over the money saved from equivalent price discounts. Since we have not set out to explain why trading stamps are sometimes preferred to price discounts, let us presume that the alternative strategies are roughly equivalent. We now turn to a brief comparison of our model and an existing model of price discounts by Salop and Stiglitz (1982). Our model (HJ) differs from the Salop and Stiglitz model (SS), since in our model it is possible to have a single-price equilibrium (SPE) at the low price (Z* = 1). Indeed, in our model, the SPE at the high price (Z* = 0) is possible but not plausible, so the most likely SPE to occur is at the low price. This contrasts to the SS model, in which a low-price SPE cannot occur. The difference in models occurs despite the presence of C > 0, which should make the low-price SPE less tenable compared to C = 0, as implicitly assumed in SS. The source of this difference in the models is the treatment of excess sales by low-price firms. To compare the two models, let the number of firms and total industry sales be the same in HJ and SS, 37 and set C = 0 and M = 1/Ns. Using HJ notation, the excess sales of the low-price firm is Q/2N and Q~/ZN in the SS and HJ models, respectively. First, note that the extra sales is a
33 Study cited in Fox (1968), p. 25. 34Ibid., p. 26. 3s Ibid., p. 22. 36Ibid., p. 98. 37 Using HJ (SS) notation on the left (right) o f the equal sign, this means that N = n and Q = 2L.
Polymorphism in Competitive Strategies
15
constant in the SS model. Now consider the effect on extra sales in the HJ model of an increase in S, a price cut. Both the zero profit and the maximum profit conditions imply that Z will decline, while or(.) is assumed to be an increasing function of S. Thus, a price cut in the HJ model will increase extra sales of the low-price firms, in contrast to the SS model. In this sense, price cutting is more profitable to low-price stores in the HJ model than in the SS model, which leads to the greater plausibility in the HJ model than in the SS model of an SPE with all low-price firms.
Cyclical Stamp Use As discussed in the second section, the proportion of firms issuing stamps has experienced cycles. It is well recognized in the retailing literature that most promotional strategies are characterized by cycles, a phenomenon called "the wheel of retailing." 3s In applying the wheel of retailing to trading stamp use, one writer has enumerated four stages of the wheel. 39 In the first stage (labeled the "stimulation stage"), early users of stamps find that store sales are relatively elastic with respect to stamp use despite an underlying inelastic demand for its products. In the second stage ("stabilization"), rival stores imitate the stamp strategy, and the stamping strategy's "power to amass a disproportionate increase is sales is neutralized." In the third stage ("saturation"), stamp-issuing firms begin to experience lower profits. In the saturation stage, " i f all major competitors have them and if they no longer attract additional customers, trading stamps have many disadvantages." In the fourth stage ("separation"), firms abandon the stamp strategy. After the separation stage, the cycle is complete, and begins again as the wheel of retailing continues to turn. While the primary concern of this article is the existence of a polymorphic equilibrium in nonprice promotional strategies, the wheel of retailing story, with its implication that firms overshoot the equilibrium point, suggests an ad hoc explanation for the observed cycles in stamp intensity. Assume that the firm considers the stamp elasticity to be a function of the expected proportion of firms issuing stamps. Let Et = f ( Z ~ ) where Et and Z~ is the firm's belief about the elasticity ( d M / d S ) ( S / M ) and expectation of Z in period t, respectively. Further assume that, optimizing over information costs, it turns out that at the margin it just pays to collect information on the previous period's value of Z; that is, firms use the previous period's Z only to estimate the current Z. Then, assuming that firms have rational expectations of Z, Z~ = E ( Z t : Z t _ l ) ~- g ( Z t _ l ) . From the previous section we know that, barring a stochastic shock, the equilibrium Z for period t is given by Z * = h(Et) for some function h(. ). Pulling these ideas together and adding a stochastic element to h(. ), we have Z* = h{f[g(Zt*_ 0]} + ~ t ~ ( Z *- ~)+/~t
(19)
a nonlinear first-order stochastic difference equation. Linearizing ~(. ) with a Taylor expansion yields a linear first-order stochastic difference equation which is quite capable of modeling a wide range of cyclical processes (Sargent 1979). In particular, such a first-order stochastic difference equation can exhibit overshooting as Z moves about its steady state. 3~McNair (1958) and Hollander (1960). 39Fox (1968), pp. 30-50.
16
William Hallagan and Wayne Joerding The resulting cyclical behavior of Z can be seen by referring to Figure 1 and remembering that the effect of changing E is to shift the Zm(') locus up and down. This will cause the equilibrium Z* to rise and fall as E changes over time. If the actual level of Z moves with Z*, then the result will be cyclical behavior in the proportion firms following the stamp strategy.
Conclusion Market structures, like biological systems, are sometimes characterized by a dualism in which individuals follow widely divergent strategies. Biologists have called this " p o l y m o r p h i s m . " In economics it is commonly argued that the persistent existence of strategic dualism (polymorphism) is a symptom of mobility barriers inhibiting the adoption of the best strategy by all firms. In this paper we have argued that, in some cases, polymorphism can arise in a competitive market in which no mobility barriers exist. Using trading stamp strategies as an example, we have provided an explanation for the persistent coexistence of markets composed of stamp-issuing firms and nonstamp firms. Our model implies that polymorphism can represent an equilibrium which is consistent with profit maximization and equal profits for each strategy. The second result of interest in this paper is that a simple ad hoc extension of our model can account for the observed cycles in trading stamp use. Further research in this field should place a high priority on a better theoretical understanding of the cyclical nature of promotional strategies.
The authors are grateful to the editor and anonymous referees for helpful suggestions.
Bibliography Alchian, A. and Klein, B. 1976. "Trading Stamps" (mimeograph). Los Angeles: UCLA. Ayala, F., and Kiger, J. 1980. Modern Genetics. Menlo Park, Calif. : Benjamin/Cummings. Boon, L., Johnson, J., et al. Winter 1978. Trading stamps: Their role in today's marketplace. Journal of the Academy of Marketing Science, 6:70-76. Breem, E. Nov.-Dec. 1957. Who profits from trading stamps? Harvard Business Review, 35:123-136. Caves, R., and Porter, M. May 1977. From entry barriers to mobility barriers. Quarterly Journal of Economics, 91:241-262. Churchill, G., and Ford, N., et al. Feb. 1971. The trading stamp-price relationship. Journal of Marketing Research, 8:103-106. Clarkson, K., and Miller, R. 1982. Industrial Organization. New York: McGraw-Hill. Davis, O. 1959. The economics of trading stamps. Journal of Business, 32:141-150. Dun and Bradstreet. 1982. lndustry Norms and Key Business Ratios. New York: Dun and Bradstreet Inc. Fox, H. 1968. The Economics of Trading Stamps. Washington, D.C.: Public Affairs Press. Federal Trade Commission. 1966. On the Use and Significance of Trading Stamps. Washington, D.C.: U.S. Government Printing Office. Hallagan, W., and Joerding, W. Spring 1983. Polymorphic equilibrium in advertising. Bell Journal of Economics, 14:191-201. Hollander, S. 1960. The wheel of retailing. Journal of Marketing, 25:37-42.
Polymorphism in Competitive Strategies
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McNair, M. 1958. Significant trends and developments in the postwar period, in A. B. Smith (ed.), Competitive Distribution. Pittsburgh, University of Pittsburgh Press. Porter, M. May 1979. The structure within industries and companies' performance. Review o f Economics and Statistics, 61:214-227. Salop, S., and Stiglitz, J. Oct. 1977. Bargains and ripoffs: A model of monopolistically competitive price dispersion," Review o f Economic Studies, 44:493-510. Salop, S., and Stiglitz, J. Dec. 1982. The theory of sales: A simple model of equilibrium price dispersion with identical agents. American Economic Review, 72:1121-1130. Sargent, T. 1979. Macroeconomic Theory. New York: Academic Press. Scherer, F. 1980. Industrial Market Structure and Economic Performance. Chicago: Rand-McNally. Sherman, R. Nov. 1968. Trading stamps and consumer welfare. Journal o f Industrial Economics, 17:29-40.
1
Polymorphism in Competitive Strategies: Trading Stamps William HaUagan and Wayne Joerding
This paper uses the example of trading stamps to examine a model of competitive nonprice marketing strategies. A concept borrowed from ethology, polymorphic equilibrium, is developed and used to explain salient features of trading stamp use by retail firms. The results contribute to an understanding of why virtually identical firms may optimally choose quite different competitive strategies. The paper also demonstrates the possibility of cyclical movements in the proportion of firms that adopt nonprice strategies such as the use of trading stamps.
Introduction Several issues of general interest can be addressed by an analysis of trading stamp use as a type of nonprice competition. While we focus on trading stamps, the analysis applies to a wide variety of promotional strategies like free gifts and quasi-lotteries. In the most general terms, we are interested in the conditions under which a competitive environment can sustain the allocation of resources by some (but not all) firms to activities which only serve to determine relative market shares. In practice, we observe a dual market structure with respect to promotional strategies like trading stamps. For example, during most of our history with trading stamps, retail markets have been composed of stamp and nonstamp firms. In the analysis of market structure, it is most common to argue that the coexistence of dual strategies represents a disequilibrium, supported by mobility barriers, and that over time all firms will be forced to adopt the single most efficient strategy. 1 In this paper we examine the conditions under which the competitive solution, unhampered by mobility barriers, will generate a market structure in which otherwise identical firms adopt widely divergent promotional strategies. 2 It is recognized that a market structure in which no firms adopt promotional strategies to increase their market share can be unstable. 3 However, if the promotional strategy does not i See, for example, Caves and Porter (1977) and Porter (1979). 2 Hallagan and Joerding (1983) consider the case in which advertising affects both the share and the market size. Here we concentrate on the case in which nonprice competition is strictly a battle over the market share. However, whereas advertising expenditures are usually treated as a fixed cost, the promotional expenditure treated in this article varies with sales. 3 Scherer (1980), p. 389. Address reprint requests to Dr. Wayne Joerding or Dr. William Hallagan, Department of Economics, Washington State University, Pullman, Washington 99164-4860.
Journal of Economics and Business © 1985 Temple University
0148-6195/85/$03.30
2
William Hallagan and Wayne Joerding increase the size of the market, a market structure in which all firms adopt identical promotional strategies will result in all firms having the same sales (but higher costs) then they would have had if no firms had adopted a promotional strategy. Under some conditions, this pure strategy structure is also likely to be unstable. Thus, a priori, the case can be made that we are likely to observe dual market structures, although these structures may, in turn, also be unstable. In the field of biology, the term "polymorphism" is used to describe populations in which otherwise identical individuals adopt noticeably different strategies. 4 For example, it is argued that a population in which all males fight to increase their share of the pool of females will be invaded by males who allocate their energies to nonfighting activities more directly related to reproduction. Similarly, it is argued that a population composed entirely of nonfighting males will be invaded by males who adopt the fighting strategy. Since neither pure strategy population structure is likely to be stable, we will observe species in which some otherwise identical males fight while others do not. In this paper we investigate whether the concept of polymorphism has a plausible counterpart in the promotional strategies adopted by firms. Like the example cited above, we concentrate on strategies which do not affect the size of the market but instead represent a battle over market share. Additionally, a rather puzzling phenomenon that arises from an examination of trading stamp use is the existence of cycles. At the national level, the prevalence of trading stamp use reached a local maximum during the period 1910-1916, followed by a period of relatively low use, only to be followed by another boom in the late 1950s and early 1960s, which again was followed by a period of declining use. Results of our model offer a tentative explanation of this cyclical behavior. The rest of this paper is organized as follows. The next section provides a brief description of the history and practice of trading stamps. This is followed by the development of a formal model of trading stamp use in which otherwise identical firms are allowed to adopt the stamp or no-stamp strategy. From this model we derive conditions under which the market will have a polymorphic equilibrium in which some, but not all, firms adopt the stamp strategy. We extend the results by noting that price cuts enter our model in exactly the same manner as stamp intensity. Interpreted this way, our model can exhibit both the single- and two-price equilibria (SPE and TPE) studied by Salop and Stiglitz (1977, 1982). The next section presents a simple model of cyclical stamp use that is intended to be suggestive rather than definitive. The final section summarizes our results and discusses the broader implications of this research. It is worth pointing out that this paper does not attempt to explain why consumers engage in stamp collecting; rather, we skirt this interesting issue by simply assuming that some consumers desire to collect stamps.
The History and Practice of Trading Stamp Use 5 The present practice of trading stamp use evolved from the initial efforts of individual store owners during the late 1800s, who placed coupons inside the packages of particular commodities like soap. Coupons were collected and redeemed at the issuing store for small gifts. In 1869 Sperry and Hutchison (S&H) saw the advantage 4 See Ayala and Kiger (1980) for a textbook treatment of this concept, 5 The material in this section is drawn primarily from FTC (1966), Breem (1957), and Fox (1968).
Polymorphism in Competitive Strategies
3
of selling stamps which were issued by a variety of stores and were redeemable at a central location. Following their introduction, the prevalence of stamp use grew until stores issuing stamps represented 7% of all retail sales in 1914. 6 Thereafter it declined in importance. In 1950, only 1% of all retail sales were generated by stamp issuing stores, but the incidence of stamp use accelerated again until it reached a peak in 1965, when 16% of all retail sales came from stamp-issuing stores and 55 % of all grocery stores issued trading stamps. 7 During this period, it was estimated that 84 % of all families were actively collecting stamps. Thereafter, the practice declined in importance, but in 1973, 31% of all grocery stores still issued trading stamps. 8 The basic principles of trading stamps have, however, remained relatively constant over time. Retail outlets purchase stamps from stamp companies and issue them to their consumers, who then collect the stamps and redeem them for commodities at redemption centers operated by the stamp company. The value of the stamps, in terms of the goods for which they can be redeemed, is typically about 2 % of the consumer's purchases; although retail stores will, at times, choose to offer multiple-stamp days in an effort either to stimulate off-peak use of the store or as a response to the actions of rivals. To promote the value of stamps sold by stamp-issuing stores, the stamp companies agree not to sell stamps to any retailers competing in the same market. Thus, retail outlets have an exclusive franchise with respect to a particular stamp plan in their market. During the 1950s and 1960s, more than ten major stamp companies competed for retailer business. Grocery stores represented the majority of clients for the stamp companies, with gas stations comprising the majority of the remaining market. Although markets where stamps are issued are generally characterized by a large proportion of consumers who collect stamps, these consumers differ with respect to the intensity of their preferences for stamp plans. According to one survey, stamp collectors can be classified into ~hree broad groups: 1) the eager (10%), 2) the interested (35 %), and 3) the indifferent (55%). 9 One interpretation of these classes is that the " e a g e r " will actively search out stamp-issuing retail outlets, the "indifferent" save stamps only because the store where they traditionally shop happens to issue stamps, and the "interested" are split in some undetermined proportion between the "stamp-seeker" and "store-loyal" groups, depending on the stamp intensity of the market. Considerable research has been done on the effect of trading stamp use on prices, market shares, and profits. Surprisingly, there does not seem to be a consensus on the effect that trading stamps have had on prices. Although they represent an additional cost, trading stamps also offset other promotional expenditures and allow the store to spread fixed expenses across greater sales. Empirical comparisons of prices for stamp and nonstamp stores have not permitted any definite conclusion. 10 The effect on market shares and profitability is clearer. There seems to be little doubt that the decision to issue trading stamps will increase the market share, but the effect on profitability will depend on when the firm adopts the stamp-issuing strategy. Firms 6 FTC (1966), p. 2. 7 Boone, Johnson, et al (1979), p. 72. 8 Ibid. 9 FTC (1966), p. 33. ~o FTC (1966), pp. 29-31. Also see Churchill, et al. (1971) for a review of the empirical evidence on the stamp-price relationship. In their study of gasoline prices, they found no evidence that stamp use was associated with higher prices.
William Hallagan and Wayne Joerding adopting the stamp strategy early in the cycle, when few other firms issue stamps, experience the most significant gain in market share and profitability, while firms adopting the strategy later in the cycle have less favorable experiences, t t
A Model of Trading Stamp Use In this section, we develop of model of trading stamp use to examine the conditions determining the equilibrium proportion of stores adopting the trading stamp strategy. There are three steps in the development of our model. First, we allow stamp-issuing stores to select the profit-maximizing stamp intensity. Second, we derive an equal profit condition by assuming that stamp and nonstamp stores adopt and abandon strategies until the profits from each strategy are equal. Finally, we combine the equal profit condition with the profit-maximization condition to define the equilibrium proportion of stores adopting the stamp strategy. The notion of equilibrium that we use is consistent with firms' maximizing behavior, in that they cannot increase profits either by changing their stamp intensity or by switching to a no-stamp strategy. The interesting characteristic of a polymorphism is that otherwise identical entities select divergent strategies. Incorporating this feature in our model requires us to assume that firms are identical in every way, except that some firms adopt the stamp strategy while others do not. This assumption implies that the profit-maximizing level of stamp use will be the same for all firms adopting the stamp strategy, t2 We assume that firms are price takers and that price is independent of stamp use. 13 This assumption has three rationales. First, it simplifies the analysis considerably. Second, it retains the focus of our model on strategies representing a struggle over market share, and allows us to maintain our assumption that firms are identical except with respect to the promotional strategy they adopt. Finally, the assumption does not violate the empirical evidence of the relation between stamp use and prices. We have also assumed that sales are characterized by constant average variable costs in the relevant range, so that the firm's price cost margin (P) is constant. On the demand side, we first assume that firms consider the market demand (Q) at the given price as fixed. ~4 This is probably a realistic assumption in that a promotional strategy like stamp use, unlike advertising, does not increase consumer information. Although the practice of issuing stamps, given a fixed price, does lower the net price, stamps are typically tied to commodities with inelastic demands like groceries and gasoline, so that stamps with a redeemable value equal to 2 % of sales are most likely to be viewed by retailers as having no effect on total market demand. This assumption also maintains the focus of this paper on promotional strategies which affect only relative market shares. Given this demand structure, the profits for firm i are given by equation (1), in which qi is the quantity sold and P is the price cost margin, which is constant given ~ Fox (1968), pp. 30-50. J2 Since all firms are identical, we will not be able to address the issue of what types of stores are most likely to adopt stamps. Instead, we focus on the proportion of stores which adopt stamps. ~3If trading stamp use is interpreted as a price cut, this assumption means that the high price is fixed, which is analogous to the Salop and Stiglitz (1982) assumption of a fixed monopoly price determined by consumer preferences. The low price is endogenously determined. 14This is also characteristic of the Salop and Stiglitz (1982) model.
Polymorphism in Competitive Strategies
5
our previous assumptions. On the cost side, Si is the stamp intensity, defined as the average stamp cost per unit of sales, which is subtracted from P to calculate the profits for stamp-issuing stores. Fixed costs arising from store operation are F, while fixed costs associated with issuing stamps (e.g., advertising informing consumers of stamps and stamp-issuing equipment) are C.
IIi=(P -
Si)qi - F -
C
(1)
To provide added structure to the variable qi, we adopt the following approach. Two types of consumers are envisioned. Type I consumers are "stamp seekers," who will switch from nonstamp stores to stamp stores if given the chance. Let ct = ct(•) be the proportion of sales to type I consumers, where S is the average stamp intensity of stamp-issuing stores. 15 The function or(. ) is assumed to be increasing and continuous with continuous first derivatives, or(0) ~ c~0 > 0, and a ( P ) _< 1. The most restrictive aspect of this specification is the condition that at least one consumer will seek stampissuing stores even when the stamps are valueless: ct(0) > 0. This specification greatly simplifies the analysis and does not affect the main results in any important way. t6 Type II consumers are "store loyal," and collect stamps only if their preferred store adopts the stamp strategy. The proportion of type II consumers is noted as 1 - ct, and all firms are assumed to have equal sales to type II consumers. For convenience, all consumers are assumed to generate one unit of sales. These assumptions imply that firm i's sales can be expressed as equation (2), where Q is market demand and N is the total number of stores in the market, both of which we have assumed to be fixed. 17In equation (2), Mi = Mi(Si, S) is the firm's share of type I consumers.
qi=Q(cv.Mi+~--~)
(2)
Profit Maximization Given the structure developed above, we can specify the firm's optimization problem as equation (3). Maxs, 1-Ii=(P-
(
lo /
Si)Q (xMi'Jt"7
- F- C
(3)
The profit-maximizing stamp intensity is restricted to the interval (0, p).~8
msThis is an ad hoc assumption about consumer behavior, but can be motivated in several ways. As stated in Salop and Stiglitz (1977), it may be that only a proportion of the consumers spend the resources necessary to inform themselves about the location of stamp-issuing stores. Alternatively, one might suppose that there is a distribution of the propensity to collect among consumers. Finally, as stated in Salop and Stiglitz (1982), it may be that consumers choose stores randomly but purchase more in a stamp store than a nonstamp store. 16 An alternative would be to allow a(S) = 0 for all ~ _< S for some ~ _> 0. Although this specification considerably complicates the analysis, variations on the main results can be proved. See note 28. m7Fixed N is not crucial, but it considerably simplifies the analysis. At any rate, it seems sensible to assume that stamping policies will not affect entry and exit decisions, especially since Q is fixed and stamp policies only determine market shares. ~s Given these assumptions, it should be clear that S can be thought of as the shadow price of stamps, and P - S can be interpreted as a discounted price. The variable C becomes a fixed cost to the low-price strategy, and again can be justified by the need to advertise the lower price.
6
William Hallagan and Wayne Joerding F u r t h e r m o r e , in o r d e r to e x a m i n e p o l y m o r p h i s m , we have p u r p o s e l y assumed that all firms are identical so that p r o f i t - m a x i m i z i n g stamp intensity satisfying equation (3) will be the same for all stores adopting the stamp strategy. As a consequence, Si is the same for all stamp-issuing stores, and ~¢ = Si so that the subscript and b a r notation is superfluous and is hereafter dropped. It is a s s u m e d that in solving this p r o b l e m , firms c o n s i d e r only the effect that stamp intensity has on their costs and market share. Although ~ is a function o f the average stamp intensity o f the market, each firm is assumed to act as i f its own choice o f stamp intensity has no effect on the m a r k e t stamp intensity (i.e., a s m a l l - f i r m assumption), so that firms treat a as a m a r k e t parameter. Solution o f equation (3) implies that:
[
-
s ~(s) -~ M(A-N J - ( P
- sO~(s0E=0
(4)
where E ~ ( d M / d S ) ( S / M ) is the f i r m ' s expectation o f the elasticity o f its share o f type I consumers with respect to its own stamp use. This elasticity contains the direct effect o f its own stamp use as well as the indirect effect on the stamp intensity o f other firms, and is assumed to be a constant. 19 Condition (4) can be used to derive a function o f S which is useful in our analysis. Let Zm(S), for Se(0, P ) , be the p r o p o r t i o n o f firms issuing stamps that is consistent with the equilibrium condition (4) and with identical stamp intensities o f all stampissuing firms. Identical stamp intensities implies that M = I / N s , and (MAr) = l = N s / N ~ Z where N s is the number o f stamp-issuing firms. A f t e r substituting these restrictions into equation (4), Zr~(S) must satisfy the resulting condition identically on a set o f S such that Ze(0, I). W e can derive 2° undefined if S e A l = [Se(0, P ) : a ( S ) > S / ( P -
Zm(S)~
(P-
S)a(S)E-offS)S [l
-
o~(s)]s
S)E]
if SeA - - [Se(0, P):S not in A~ or A0]
undefined if SeA0-----[S~(0, P ) : S > P E / ( E
+ 1)]
(5)
a real valued function defined on (0, P ) . 21 Concentrating for a m o m e n t on the interior solution for Zm, we can m a k e some rough calculations which will p r o v i d e some insight into the relation b e t w e e n Zm and E . F o r g r o c e r y store operations, the gross m a r g i n ( P ) is about 20% o f sales, and the most frequently o b s e r v e d stamp intensity was two cents p e r d o l l a r o f sales. 22 F r o m our discussion o f the history o f stamp use, it appears that the proportion o f consumers that are stamp seekers (o0 is s o m e w h e r e between 0.10 and 0.45, say, c~ = 0.30 as an intermediate value for the calculations. Substituting for the interior solution o f ~9Since M(., •) is a function with a range restricted to the unit interval, it is not possible in general for E to be a constant. Thus, more accurately, we assume that/14(., .) is a hyperbola for relevant values of S and ~. 20 This is easily done by substituting Z = MN-1 into equation (4) and solving for Z. Then use the resulting condition to define sets of S for which Z > 1 and Z < 0. Thus, Zm(') is continuous, but its derivative is not. 2J A , A0 and At are sets on the real line. If sup At < I, then sup A t < inf Ao, since Pc~E/t~E + 1 <
PE/E + 1. 2~ In 1982 the gross profit divided by sales was 18.5% for grocery stores and 15.3% for gasoline stations, as reported in Industry Norms and Key Business Ratios, Dun and Bradstreet (1982).
Polymorphism in Competitive Strategies
7
equation (5) we get Z = 3.14E - 0.36. Thus, setting the roughness of the calculations as well as the equal profit condition aside, polymorphism (0 > Z > 1) is consistent with expected stamp elasticities between 0.11 and 0.43. Although the range is narrow, the values are plausible. The earlier observation that 55 % of all grocery stores issued stamps at the peak of the trading stamp boom (Z = 0.55) suggests that profit-maximizing grocery stores expected the stamp elasticity to be about 0.29 at that point in time. These calculations apply only to the profitmaximizing decision and are not about an equilibrium. A plausible configuration for Zm(S) is shown in Figure 1. Of course, without further restrictions, there are many different possible shapes for Zm(S). The example shown in Figure 1 would apply if the slope of eft. ) at S = 0, e ' (0) is not too large and declines monotonically with increasing S. In the example, AI = (0, Sin) and A0 = [PE/(E + 1), P]. In this example, an exogeneous increase in either the price cost margin or the expected stamp elasticity shifts the Zm function out and, for any interior values of Z, increases the profit-maximizing stamp intensity. These results hold only for the firm's profit-maximizing decision. To analyze the market equilibrium, we must also consider the equal profit condition, to which we now turn.
Z
z=(s}
Se Figure 1.
Sm
S*
P'E. E+I
P
S
$
William Hallagan and Wayne Joerding
Equal Profits Imposing the equilibrium condition that M = I / N s for all firms adopting the stamp strategy, the profit functions for representative stamp-issuing firms (IIs) and nonstamp firms (HN) are given by equations (6) and (7). I-Is = (P -
S)Q
-~
~
rlu=PQ[ ('-N
c0]-F
- F- C
(6) (7)
At an equilibrium, firms will adopt and abandon strategies until the profits for stamp and nonstamp firms are equal, l-Is = l-IN, which implies S
~
P-
S=(1
-
ot)Z
(P-
NC S)Q(I -
or)
(8)
The equal profit condition can be rearranged to calculate how much the sales of stamp-issuing firms must exceed the sales of the nonstamp firms in order for each strategy to generate equal profits. For the case in which C = 0, the equal profit condition is satisfied when the ratio of the sales of the stamp-issuing firms to the nonstamp-issuing firms equals P / ( P - S). Roughly, using the market norms for grocery stores, the equal profit condition is satisfied if stamp-issuing stores have about 10% higher sales than non-stamp-issuing stores. Again, this result is within the plausible realm. However, equation (8) need not hold for single-strategy equilibria since profits for one of the strategies is irrelevant. Let Z~(S), for S~(0, P ) , be the proportion of firms issuing stamps, which is consistent with the equilibrium condition that all firms earn the same profits and all stamp-issuing firms have the same stamp intensity. 23 Then Z = Ze(S) satisfies equation (8) as an identity for 0 < Z < 1, which implies that Z~(S) can be written as 1 if S e B t ~ [ S e O , P):ot(S)>(SQ + N C ) / P Q ] Ze(S)=
(P - S)Qa(S) Qs[1 - . ( s ) ] + NC
if SeB=[Se(O, P):S not in Bz or B0]
0 if SeBo~[Se(O, P):A(S)<0]
(9)
a real valued function on (0, P), where A(S) is the excess profits of stamp firms, 7rs 24 --
~'N.
A possible configuration for Z e ( ' ) is shown in Figure 1; many other shapes are possible depending on the restrictions placed on a ( ' ) . As before, the illustrated Ze(" ) would result from an ct(.) with or'(0)small and declining with increases in S. Note that Z , ( S ) is not a function of E. The set Bi = (0, Se) and Ze(.) reaches a minimum of zero at S = P; thus, the set B0 = {P}. 23This condition applies to single- as well as multiple-strategyequilibria. 24B, Bo, and B, are sets on the real line. Bj is definedby noting that (P - S)Qa(S)/{QS[I - ot(S)l + NC} ~ I as a(S) f (SQ + NC)/PQ; consequently,Z,(S) is continuous, but its derivative is not. B0 is the set of S for which the stamp strategy is less profitable than the nonstamp strategy.
Polymorphism in Competitive Strategies
9
Polymorphic Equilibrium An equilibrium is defined by the condition that all stamp-issuing firms are at the profit-maximizing stamp intensity, and all firms are equally profitable. That is, S* will be an equilibrium level of stamp intensity by stamp-issuing stores if S* satisfies Zm(S*) ~ Ze(S*). The equilibrium proportion of firms choosing the stamp strategy can then be determined by either equation (5) or equation (9). Such a solution is illustrated in Figure I. Since a closed-form solution for S* is not possible, we first examine a special case in which c~ is a positive constant and C = 0; then propositions characterizing the general case of ~ = c~(S) are proved. C o n s t a n t ~ a n d C = 0:
An equilibrium Z which is interior to the unit interval can be obtained directly by solving the maximum profit condition (4) and the equal profits condition (8) simultaneously for Z*. This results in 1
1 if E < 1
Z~ ~
and S*_
--
ot
Ot
(1
-
ot)(g
0 if S*>So
-
1)
1
if E > 1
-
and S*_< So ot
(lo)
where So ~ a P ( N - 1)/[1 + c¢(N - 1)]. 25 The point So is easily seen to be less than 1, and is the inf Bo. That is, So is the minimum stamp level for which the profits of the nonstamp strategy exceed or equal the profits of the stamp strategy, A(S) < 0 for all S > So. Intuitive insight into the specification and implications of equation (10) can be obtained from Figure 1. Also, since the figure is not dependent on the specific simplifying assumptions of this section, the discussion is useful in understanding the results of the general model. Remember, a decrease in E causes Zm(') to shift down, while Z e ( ' ) is unaffected. Consider that E > 1/(1 - ct), so that the equilibrium Z is as shown in Figure 1. As E declines, Z* increases and S* decreases until Sm coincides with Se [at E = 1/(1 - ot)] and Z* just equals 1. For all values of E less than or equal to 1/(1 - a), but such that S* < So, the profit of stamp firms is greater than the profit of nonstamp firms, and so Z* = 1. Whenever S* > So, the profits of nonstamp firms are greater than those of stamp firms, and a single-strategy equilibrium occurs at Z* = 0. The intuition behind these results is that an increase in E causes profitmaximizing stamp intensity to increase. This means that the costs of stamp-issuing stores are higher and implies that fewer such stores can survive (if they are to profitable as non-stamp-issuing stores). Consequently, Z* must fall. In order for equation (10) to generate a polymorphic equilibrium (0 < Z* < 1), highly implausible conditions, E > 1, are required. We turn to analysis of the general model. 26
25 N o t i c e that Z* in equation (10) is not a function of P or Q, w h i c h is a result of setting C = 0. 26 T r e a t i n g S as a price cut, even this highly restrictive model can generate s i n g l e - p r i c e e q u i l i b r i a (SPE) at both the low and high prices ( Z = 1 or 0). This contrasts to the w o r k of Salop and Stiglitz (1982), w h o had an SPE only at the high price (Z* = 1).
10
William Hallagan and Wayne Joerding
The General c a s e In this section, w e are concerned with the distinction between two types o f equilibrium, a single-strategy equilibrium (SSE) and a polymorphic or two-strategy equilibrium (TSE). We present conditions relating to the existence o f each o f these equilibria. An SSE occurs when Z* = 0 or 1. Note that an SSE trivially satisfies the equal profits condition (9) but may not satisfy (5), the profit-maximizing condition. A TSE occurs when Z*e(O, 1). An SSE will occur at Z* = 0 if the profits of a single stamp-issuing store, Z = 1/ N, are less than the profits of nonstamp stores. Conversely, an SSE will not occur at Z* = 0 if a single stamp-issuing store can earn more profits by switching to the stamp strategy. This occurs if and only if A(S*m)-~(P -
S*)Qot(S*)- S'Q[1 -
e t ( S * ) l / N - C>_O,
(11)
where S* is the profit-maximizing stamp intensity when Z = 1/N, and thus is a function o f parameters. 27 Condition (1 1) is difficult to interpret, but some insight into the plausability o f an SSE can be gained from the following proposition. Proposition 1 : If up _> C / Q ( P - S*), a necessary condition for the existence of an S S E a t Z * = 0 i s E > 1. Proof: If an SSE exists at Z* = 0, then A(S*) -< O. Since a ( S * ) _< a ( P ) ~ ap, observe that A(S*) _< 0 only if
(P -
S*)Qotp-S*Q(1 -
Ctp)N-~-C<_O.
(12)
Rearrange equation (12) to obtain
olpN i
CN
ap
-
Q(P-
S*)(1
S* -
otu)
P
-
(13) S*
Setting M = 1 (since there is only one stamp firm) in the maximum profit condition (4), one can derive
S* P-
Not(S*)E S*=Not(S*)+[1 - e~(S*)]
(14)
The right side o f equation (14) is an increasing function o f e~, which means that
ap I -
CN Otp Q ( P -
s*)(1
agE
-
_< Up) N a p + ( 1
-
Up)
(15)
is a necessary condition for equation (13) to hold. This condition can be rearranged to obtain
E>ap(N - 1)+1 1 - otp
C[(n -
Q(P-
1)otp+l]
S*)(I -
--- T(aA etp)eLp
(16)
27 Equation (9) is obtained from 17s - lIjv where lqs and l-INare evaluated at Z = I/N and the profitmaximizing stamp intensity S*.
Polymorphism in Competitive Strategies
11
Rearrange T(ap) to see that T(t~p) _> 1 as ~(Otp)~NQ(P -
S*)ot2p-C(N-
1)ap-C_>0
The quadratic ~'(. ) is a parabola which opens upward and has a positive root at & < 1 if Q ( P - S*) > C (since ~'(0) = - C and ~'(1) = N [ Q ( P - S* - C]), which must hold if the stamp strategy is feasible. The value ~'(. ) is not positive semidefinite, but from the quadratic equation C(N-
&=
1) + ~/{C2(N - 1) + 4 N Q ( P - S * ) C } 2 N Q ( P - S *)
Since x/{x + y} < x/x + ~/y and C / N Q ( P - S*) < 1, it can be shown that 6~ < C~ Q(P S*). But ap >_ C / Q ( P - S*), by assumption, which implies that ~'(C~p) _> 0 and E _> 1 is a necessary condition for equation (12). [] Proposition 1 is quite counterintuitive, since it seems to say that as the expected marginal impact on the market share of the stamp strategy increases, then the likelihood of an SSE in which nobody issues stamps increases. This result can be shown for the case represented in Figure 1. Recall that for any interior value of Z, an increase in the stamp elasticity causes an increase in the profit-maximizing stamp intensity (shifts the Zm function to the right). The stamp elasticity does not enter the equal profit condition. Thus, for sufficiently high levels of E, the profit-maximizing stamp intensity approaches the profit margin, and it becomes impossible for a stampissuing store to earn profits equal to those of a store that doesn't issue stamps. It should be remembered that this proposition is about an equilibrium. It is entirely possible that for a large enough E the SSE at Z* = 0 is very unstable; nevertheless, it is an SSE. Proposition 1 is useful because Otp >_ C / Q ( P - S * ) is plausible, while E > 1 is implausible. For example, if C is considered an advertising cost, then the condition Olp >~ C / Q ( P - S*) requires that the maximum proportion of type I customers, c~(P), be greater than or equal to the advertising sales ratio. In general, one would expect C to be very small; thus, this condition is easy to satisfy. If C = 0, then any positive Otp will do. The other condition is implausible, E > 1 implies that agents expect their share of type I customers to increase proportionally more than an increase in stamp intensity. Certainly, this condition cannot be satisfied by all agents in equilibrium. Furthermore, if C is small, then the right side of equation (16) will be considerably larger than 1, and the necessary condition of proposition 1 could be considerably strengthened. I r E > 1 or equation (16) is implausible, then proposition 1 implies that one should not expect to see an SSE at Z* = 0, a result consistent with the empirical observations reviewed in the second section. Consequently, for the rest of this section, we will restrict our attention to the case of no SSE at Z* = 0. It is also possible to have an SSE at Z* = 1. Informally, an SSE at Z* = 1 could occur if, for all combinations of Z and S which satisfy profit maximization, all stampissuing stores make greater profits than nonstamp stores. Unlike the potential SSE at Z* = 0, there are no plausible reasons to reject the possibility of a n S S E at Z* = 1. This is because many plausible specifications of c~(.) can satisfy the conditions of the following lemma. -
Lemma: Given no SSE at Z* = 0, an SSE exists at Z* = 1 if and only ifPQot(Sm) QSm >__ N C , where Sm =- inf A.
12
William Hallagan and Wayne Joerding Proof: The condition E < 1 implies that we only need to consider possible SSEs at Z*~(0, 1). Note that Sm is the smallest S which just satisfies ot(S,n) = Sm/(P - S,,)E. (See the definition of Al.) Also, Zm(Sm) = 1. Consequently, if Ze(Sm) = 1, then Z,n(S,n) ~ Ze(S=) and an SSE will occur at Z* = 1 and S* = Sin. But Ze(Sr,) = 1 if and only if Sr~Bl or, equivalently, PQot(S,n) - QSm >- N C . [] There clearly exist a large number of plausible specifications for a ( ' ) which could satisfy the lemma. However, the condition of the lemma does constrain the permissible set somewhat, as shown in the next proposition, which is analogous to proposition 1. Proposition 2: Given no SSE at Z* = 0, a necessary condition for the existence of an SSE at Z* = 1 is either ao >- N C / P Q or ot'(O) ~ doddS(O) > l / P . Proof: Define O(S) = p Q a ( S ) - QS. Then the condition for the lemma requires either 0(0) = PQoto > N C (which is the first condition above) or, if O(0) < N C , 0 (.) must have a positive slope at S = 0. This latter case will hold if ct'(0) > I / P . D The intuition behind the lemma and proposition 2 is that an SSE can exist at Z* = 1 only if stamp-issuing stores are more profitable than nonstamp stores. But this will occur only if the proportion of the consumers who collect stamps is large at low stamp intensities. The large ot can occur either with a large ct0 or a high rate of increase in ~x for small increases in the stamp intensity. As one would suppose, the necessary size of a0 or or' (0) depends on the fixed cost associated with adopting the stamp strategy. I f C = 0, then any positive a0 may be sufficient to allow an SSE at Z* = 1 to exist. We next consider the main result of this article: conditions for the existence of a TSE or polymorphic equilibrium. Proposition 3: Given no SSE at Z* = 0, a sufficient condition for the existence of a TSE is PQct(S,,) - QSm < N C . 28 Proof: An equilibrium S* is defined by the condition F(S*) =-- Ze(S*) - Zm(S*) ~ O. Let # = P E / ( E + I) < P. From equation (5), it is easy to see that l i m s ~ _ Zm(#) = 0.29 Also, it is easy to see that Ze(l~) > 0, since/~ < P. Consequently, we have F(/~) > 0. Note Zm(Sm) = 1 by definition of the set A. Next, Ze(Sm) < 1 since
o~(s.,) <
QSm + N C
PQ
(17)
or
PQa(Sm) - QSm < N C
(18)
2s A similar proposition holds for the case a ( S ) = 0 for all S < ~ > 0. The restrictive conditions are that ff~(0, ,if'), where if' is the positive root of $(x) = - E Q x 2 + I ( P Q - N C ) E - ( N C + P Q ) l x + P E N C = 0, and 0 < E < ( P Q + N C ) / ( P Q - N C ) > 1. It can be shown that there exists a TSE at
S*~(g, p). 20 lims.~_ Z,n(S) is the left-hand limit of Z,,,(.) at #.
Polymorphism in Competitive Strategies
13
Thus, under the conditions o f the theorem, F(Sm) < 0. Since F ( ' ) is continuous on the connected set (Sin, #), the intermediate-value theorem can be used to conclude that there exists an S* such that F(S*) =- O. [] The condition o f the proposition 3 is simply the negative o f the condition for the lemma preceding proposition 2. The reason P Q a ( S ~ ) - QSn < N C is not a sufficient condition is that not enough restrictions have been placed on or(.). For example, if o~(.) has a monotonically decreasing slope, then one can easily see from Figure 1 that P Q a ( S ~ ) - QS~ < N C becomes a sufficient condition as well. It amounts to a condition in which Sm > So. 30 A number of interpretations are possible. An increase in the number o f firms or the cost of the stamp strategy makes a TSE more likely to occur by increasing the right side of equation (18). In the case o f N, the rationale for this result is that an increase in N raises the maximum profit stamp intensity making it more difficult for the market to support a large number of stamp-issuing firms. Thus, a high enough N would make the stamp strategy less profitable than the rronstamp strategy. In the case o f C, an increase in the cost of the stamp strategy makes the non-stamp strategy more attractive. Since the possibility of an SSE at Z* = 0 is eliminated by the condition E < 1, this means that the likelihood of a TSE at Z* < 1 increases as C increases. Finally, an increase in P raises the likelihood of observing a TSE by increasing the potential profits of the nonstamp strategy while increasing the profit-maximizing stamp intensity. Thus, an increase in P benefits the nonstamp firms more than stamp firms.
A Digression on P r i c e D i s c o u n t s Our model needs no adjustment to serve as a model o f sales. The stamp intensity S can be interpreted as a price discount, and the variable C would then represent the fixed costs associated with implementing a sale (e.g., remarking prices, advertising). While we offer no answer, it is interesting to speculate on why the trading stamp strategy would ever dominate the price discount strategy. Since the stamp strategy restricts consumer choice to those goods offered in the stamp catalog, an equivalent price discount would appear to be preferred by consumers. Yet, after the Tennessee legislature banned the use o f trading stamps in 1957, it was flooded with over 40,000 letters of protest, and a group o f housewives picketed the capitol. 31 While some argue that the stamp strategy is a price discrimination mechanism, this argument does not apply to the national stamp plan described in the second section. 32 Under the usual trading stamp practices, retail stores purchased stamps from national firms and gave these stamps to all customers. Thus, the retail store issuing stamps earned the same net profit ( P - S) from each consumer, regardless of whether or not the consumer
30 A necessary and sufficient condition could be stated as inf A > sup B~, but yields little insight. 31 Fox (1968), p. 16. 32 The usual treatment of consumer behavior predicts that consumers would prefer an equally valued price discount, since it does not constrain their budget allocation, whereas trading stamps are redeemable only for the limited group of goods offered by trading stamp redemption centers. Most existing economic analyses of this puzzle model trading stamp use as a practice adopted to facilitate price discrimination between consumers, with differing values of time (Clarkson and Miller 1982, pp. 269-271; Alchian and Klein 1976; Sherman 1968; and Davis 1959). However, there is still some shadow price which measures the value of stamps as a price cut.
14
William Hallagan and Wayne Joerding eventually redeemed the stamps from the national firm's catalog. Since the retail outlet receives the same net price from each consumer, the stamp strategy does not represent price discrimination from the store's viewpoint. It has been suggested that trading stamps may be more effective in attracting consumers than equivalent price discounts because trading stamps are more visible and credible than a claim that all prices have been cut by 2%. Although this explanation of trading stamp use may operate to some extent, empirical studies suggest that, when explicitly offered trading stamps and an equivalent amount of cash, many consumers chose the stamps. For example, in 1963 the Burgoyne Study of Supermarket Shoppers found that a majority of consumers "prefer to shop in a store giving trading stamps rather than an otherwise identical store that is 2 % cheaper on all g o o d s . " 33 Another study found that "when the offer was one dollar cash or four hundred stamps (presumably equivalent value), 37% chose cash and 62% chose stamps. 34 Thus, by itself, the visibility argument cannot explain the use of stamps rather than price discounts. Market researchers have put forth other explanations for the popularity of trading stamps. Two hypotheses dominate. The first is that trading stamps, due to their shortrun illiquidity (until a sufficient number of books are filled), force consumers to save. Since, for some, forced saving is a good, they are willing to restrict themselves to the types of semi-durable goods offered in the stamp catalog. The second hypothesis is that trading stamps were useful allies for housewives in the battle over the household budget. According to this argument, housewives sought out stamp-issuing stores so that they could redeem the stamps " o n goods for which their husbands may be unwilling to incur a financial expenditure." 35 Consistent with this argument is the fact that the popular premiums were automatic coffee makers, toasters, steam irons, household lamps, and bedspreads. 36 This argument presumes that housewives had less control over the money saved from equivalent price discounts. Since we have not set out to explain why trading stamps are sometimes preferred to price discounts, let us presume that the alternative strategies are roughly equivalent. We now turn to a brief comparison of our model and an existing model of price discounts by Salop and Stiglitz (1982). Our model (HJ) differs from the Salop and Stiglitz model (SS), since in our model it is possible to have a single-price equilibrium (SPE) at the low price (Z* = 1). Indeed, in our model, the SPE at the high price (Z* = 0) is possible but not plausible, so the most likely SPE to occur is at the low price. This contrasts to the SS model, in which a low-price SPE cannot occur. The difference in models occurs despite the presence of C > 0, which should make the low-price SPE less tenable compared to C = 0, as implicitly assumed in SS. The source of this difference in the models is the treatment of excess sales by low-price firms. To compare the two models, let the number of firms and total industry sales be the same in HJ and SS, 37 and set C = 0 and M = 1/Ns. Using HJ notation, the excess sales of the low-price firm is Q/2N and Q~/ZN in the SS and HJ models, respectively. First, note that the extra sales is a
33 Study cited in Fox (1968), p. 25. 34Ibid., p. 26. 3s Ibid., p. 22. 36Ibid., p. 98. 37 Using HJ (SS) notation on the left (right) o f the equal sign, this means that N = n and Q = 2L.
Polymorphism in Competitive Strategies
15
constant in the SS model. Now consider the effect on extra sales in the HJ model of an increase in S, a price cut. Both the zero profit and the maximum profit conditions imply that Z will decline, while or(.) is assumed to be an increasing function of S. Thus, a price cut in the HJ model will increase extra sales of the low-price firms, in contrast to the SS model. In this sense, price cutting is more profitable to low-price stores in the HJ model than in the SS model, which leads to the greater plausibility in the HJ model than in the SS model of an SPE with all low-price firms.
Cyclical Stamp Use As discussed in the second section, the proportion of firms issuing stamps has experienced cycles. It is well recognized in the retailing literature that most promotional strategies are characterized by cycles, a phenomenon called "the wheel of retailing." 3s In applying the wheel of retailing to trading stamp use, one writer has enumerated four stages of the wheel. 39 In the first stage (labeled the "stimulation stage"), early users of stamps find that store sales are relatively elastic with respect to stamp use despite an underlying inelastic demand for its products. In the second stage ("stabilization"), rival stores imitate the stamp strategy, and the stamping strategy's "power to amass a disproportionate increase is sales is neutralized." In the third stage ("saturation"), stamp-issuing firms begin to experience lower profits. In the saturation stage, " i f all major competitors have them and if they no longer attract additional customers, trading stamps have many disadvantages." In the fourth stage ("separation"), firms abandon the stamp strategy. After the separation stage, the cycle is complete, and begins again as the wheel of retailing continues to turn. While the primary concern of this article is the existence of a polymorphic equilibrium in nonprice promotional strategies, the wheel of retailing story, with its implication that firms overshoot the equilibrium point, suggests an ad hoc explanation for the observed cycles in stamp intensity. Assume that the firm considers the stamp elasticity to be a function of the expected proportion of firms issuing stamps. Let Et = f ( Z ~ ) where Et and Z~ is the firm's belief about the elasticity ( d M / d S ) ( S / M ) and expectation of Z in period t, respectively. Further assume that, optimizing over information costs, it turns out that at the margin it just pays to collect information on the previous period's value of Z; that is, firms use the previous period's Z only to estimate the current Z. Then, assuming that firms have rational expectations of Z, Z~ = E ( Z t : Z t _ l ) ~- g ( Z t _ l ) . From the previous section we know that, barring a stochastic shock, the equilibrium Z for period t is given by Z * = h(Et) for some function h(. ). Pulling these ideas together and adding a stochastic element to h(. ), we have Z* = h{f[g(Zt*_ 0]} + ~ t ~ ( Z *- ~)+/~t
(19)
a nonlinear first-order stochastic difference equation. Linearizing ~(. ) with a Taylor expansion yields a linear first-order stochastic difference equation which is quite capable of modeling a wide range of cyclical processes (Sargent 1979). In particular, such a first-order stochastic difference equation can exhibit overshooting as Z moves about its steady state. 3~McNair (1958) and Hollander (1960). 39Fox (1968), pp. 30-50.
16
William Hallagan and Wayne Joerding The resulting cyclical behavior of Z can be seen by referring to Figure 1 and remembering that the effect of changing E is to shift the Zm(') locus up and down. This will cause the equilibrium Z* to rise and fall as E changes over time. If the actual level of Z moves with Z*, then the result will be cyclical behavior in the proportion firms following the stamp strategy.
Conclusion Market structures, like biological systems, are sometimes characterized by a dualism in which individuals follow widely divergent strategies. Biologists have called this " p o l y m o r p h i s m . " In economics it is commonly argued that the persistent existence of strategic dualism (polymorphism) is a symptom of mobility barriers inhibiting the adoption of the best strategy by all firms. In this paper we have argued that, in some cases, polymorphism can arise in a competitive market in which no mobility barriers exist. Using trading stamp strategies as an example, we have provided an explanation for the persistent coexistence of markets composed of stamp-issuing firms and nonstamp firms. Our model implies that polymorphism can represent an equilibrium which is consistent with profit maximization and equal profits for each strategy. The second result of interest in this paper is that a simple ad hoc extension of our model can account for the observed cycles in trading stamp use. Further research in this field should place a high priority on a better theoretical understanding of the cyclical nature of promotional strategies.
The authors are grateful to the editor and anonymous referees for helpful suggestions.
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