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Center
for
Economic Research
No. 9412
COUPONS AND OLIGOPOLISTIC
PRICS DISCRININATION
by Helmut Bester and
Emmanuel Petrakis
January 1994
COUPONS AND OLIGOPOLISTIC PRICE
DISCRIMINATION
Helmut Bester and Emmanuel Petrakis't
January 1994
Abstract
This paperstudies sales promotionthroughcouponsin anoligopolistic market. Sending out coupons allows the sellers to separate market segments with different degrees ofconsumer btandloyalty. This kind ofprice discrimination is profitable for the individual seller when the cost of couponing is sufficientlylow. In equilibrium, however, couponing increases competition and reducesprofits. The paper provides
comparative statics results and studies welfare implications.
Keywords: Advertising, Coupons, Oligopoly, Price Discrimination; JF,L
Classifi-cation No.: D43, D83
'CentER, Tilburg University, and Universidad Carlos I11 de Madrid; respectively.
tmailing address: Nelmut Bester, CentER, Tilburg University, P.O. Box 90153, 5000 LE Tilburg, The Nethedands
1
1
Introduction
This paper studies sales promotion through coupons and rebates in an oligopolistic indus-try. Sales promotion is a complementary, and in cases even more ímportant marketing strategy than media advertising for a firm Lo increase its market share. Coupons and rebates count for the bulk of billion of dollars spent each year on t.hese promotional activities. Manufacturers in the US distributed 310 billion coupons in 1992, representing an 6alo increase over the year before (see Hume (1993)). Rebates are virtually equivalent to coupons except that they impose some additional redemption cost on the customer, namely the cost of mailing and waiting for the cash refund. For our purposes coupons and rebates play the same role and we will treat them interchangeably in our analysis.
We investigate the role of coupons in a market that is segmented due to location, brand loyalty, or access to product information. By offering a rebate, a manufacturer is able to attract some consumers who are otherwise more inclined Lo buy a competing brand. Consumc~rs differ in Lheir degree of brand loyalt.y. In the locat.ion interpretation o( our modcl, they have different transportation costs. Sending out coupons is costly for the seller, and this cost increases as he targets to reach a higher percentage of the competing brand's consumers. By price discriminating between his own customers and the clientele of competing manufacturers, the seller can increase his own market share. We study the optimal marketing behaviour of the oligopolists in a game where prices, coupon values, and the levels of coupon distribution are chosen simultaneously. We re-strict our analysis to the case where consumers are fully informed about all prices.
An empirical illustration of how firms use coupons to expand their market share is the classic struggle between I'Br.C-Folger and the General Foods Corporation. Before Fulger was acquiredby PRrC in 19G.3, it ope~rated urainly west of tho Mississippi where it had a Icading share in the local cofíee industry. ln 197'l aud 1973 holge~r rnoved iutu Cleveland and Philadelphia with an advertising campaign that included sending 25 cents off coupons to more than a million households. The Maxwell House Division of General
2
Foods react.ed by mailing 50-cent coupons to households in Clcveland. I,ati~r, siniilar counter-attacks were launched when PBtG entered other markets in the East. Overall, General Foods succeeded in defending its market, but at a high cost.
With couponing the manuCacturer has at least partial control of the type of household reached. T'hus he is able to offer a reduced price to a specific segment of the market. Coupons can serve as a price discrimination device. Only those consumers who have received a coupon are able to benefit from the rebate. Since this enables the manu-facturer to separate market segments, coupons are more than just a low price offered to all consumers. An alternative explanation of coupons has been proposed by Cremer (1984) and Caminal and Matutes (1990). In these models, coupons create a lock-in effect. because the consumer is o(fered a rebate on future purchases of the product. The seller can stabilize his market share by creating ari artificial cost of switching suppliers. This is in direct contrast with our model, where coupons tend to make switching more attrac-tive. Yet another explanation (Gerstner and Hess (1991)) is that the seller can motivate retailer participation in the sales promotion by offering rebates to the consumers.
The literature on the price discrimination aspects of coupons is rather scarce. Narasimhan (1984) studies the consumers' decision to use a coupon. Only consumers with a sufficiently low opportunity cost of time take advantage oí coupons. Thus price discrimination can be achieved through self-selection. Caminal and Matutes (1990) is, to our knowledge, the only study of oligopolistic behaviour, but in a repeat purchase context.. In t.heir model Lhe consurner receives a rebate only after purchasing a second unit from the same seller. The role of coupons is to create a switching cost in the second period. If the firms precommit to a discount, competition in that period is decreased and the equilibrium profits inrrcasc.
In contrast, in our model coupons increase competition between firrns. Each
individ-ual seller has an incentive to reduce the brand loyalty of the other firms' clientele in order
to increase his market share. But, offering a rebate amounts to reducing the consumers'
3
switching cost and so competition is intensified. In equilibrium, each seller's profits are lower than if coupons or price discrimination were not allowed. Price discrimination in combination with oligopolistic competition leads to lower prices; the consumers as a whole are better off when the sellers compete by using coupons.
The finding that firm profits may be lower as a result of price discrimination is similar to the findings of Thisse and Vives (1988). They show that discrimination is a dominant strategy for each firm. Yet, it leads to lower profits than a uniform price. Levy and Gerlowski (1991) show that "meeting competition clausesn, whereby a seller announces to meet the competitor's price, may reduce equilibrium profits. One may view such clauses as a special type of coupon; they allow the seller to discriminate between those consumers who only receive his own ad and those who receive ads also from other firms. In our model the manufacturers use coupons as long as the marginal cost of distribut-ing coupons are iiot too hígh. Under certain assumptions on distribution costs, there is a unique symmetric equilibrium. In this equilibrium, as couponing becomes more ex-pensive, the firms' profits increase, while consumer welfare decreases. This is so because a higher cost of price discrimination reduces competition between the firms. The sellers' profits increase and consumer welfare decreases also when the consumers become more heterogeneous. The intuition is that a higher degree of brand loyalty reduces the firms' incentive to use coupons as a discrimination device, which in turn leads to higher average
prices for the consumers.
The following Section presents a simplified example, where we abstract from the
costs of issuing coupons and from differences in the consumers' degree of brand loyalty.
Section 3 describes the general model. We study the equilibrium marketing behaviour of
the firms in Section 4. Section 5 provides some comparative statics and welfare tesults.
All proofs are relegated to an Appendix.
4
2
An Example
T'o illustrate the main features of our analysis, we first present a simplified example of the more general model studied in the following sections. We study a duopoly market char-acterized by some segmentation according to location, brand loyalty, or access to product information. Using the language of the locational application, the model considers the following situation: There are two sellers, A and B, located in different neighbourhoods of some geographical market. The sellers' production costs are normalized to zero. In each localit.y, there is a unit mass of consumers with a totally inelastic demand for one unit of the good for all prices between zero and v 1 0. When a consumer purchases the good from the seller in the distant location, he has to pay a transportation cost s c v. Shilony (1977) studies the equilibrium of this market when the sellers compete by setting prices in a standard Bertrand fashion. If v C 2s, both sellers will post the price p" - v in equilibrium. Undercutting the competitor's price is not profitable since this would yield a profit of at most 2(v - s) G v. The price setting game between the duopolists fails to have a pure strategy equilibrium if v 7 2s. Indeed, any combination of prices (pA, pB )would allow one of the two sellers to gain either by undercutting the other seller or by increasing his price by some small amount. Shilony (1977) shows that that there is a unique symmetric mixed strategy solution where each seller gains an expected profit higher than s.
~Ve now introduce oligopolistic price discrimination. Even though the seller is un-able Lo identify thc origin of the customers at his store, he can separate t,hem through coupons. 't:ach seller is able to offer the good at different prices in the two regions by mailing coupons to the other region. Coupons enta:l a legally binding promise by the seller to offer a rebate upon presentation. In our example, we abstract from mailing costs and assume that seller i can costlessly send coupons to all consumers in region j. The coupon entitles its owner to buy the good from seller i at the price p; - r;, while buyers without a coupon have topay p;.
In this sirnple example, competition betwc~en the duopolists, A and f3, results in the following equilibrium outcome:
PA-Pa-P~-s~ ra-ra-r'-s.
(1)
Each consumer is indifferent between buying the good from seller A or seller B. If he buys at the neighbouring store, has to pay the price p' - s; otherwise, has to pay p' - r' - 0 but incurs the transportation cost s. In equilibrium, it must be the case that each consumer buys at the local store. If not, the local seller would have an incentive to lower his price slightly below s. Clearly, the outcorne described by (1) constitutes an equilibrium: By charging a price above s a seller would lose all his customers. Charging a price below s can never be optimal, since each seller enjoys a local monopoly position for all prices up to s. Actually, the equilibrium is uniyue: If some seller i would charge a price p; ~ s, then the opponent could induce all consumers in region i to switch by offering p~ - r~ sl~ghtly below p, - s. Accordingly, a price p, 1 s cannot. be part of eyui-librium behaviou-. The same argument shows that both sellers must offer the rebate r' - s. Obviousl~, given p' this is the highest rebate a seller is willing to offer. If seller i sets r; C s, then seller j would optimally charge his local customers some price p~ ~ s, which was already shown to be inconsistent with equilibrium.
The example demonstrates that price discrimination increases competition. The
con-sumers have to pay lower prices and the firms' profits are reduced. Indeed, each seller
carns a prufiL of .v, which is luwcr Lhan Lhc proÍil. hc, gcts in thc abscncc of
discrirnina-tory pricing. In thc following, we will verify this observation in a rnore geueral model.
In fact, the above example has some unappealing features because the consumers'
pur-chasing decisions have to be based on a tie-breaking rule. Even though the equilibrium
requires both
sellers
to use coupons, this marketing instrument is actually ineffective
since no consumer is induced to switch. If the sellers had to pay a mailing cost, the
above equilibrium would therefore collapse. The subsequent model will overcome these
difficulties by introducing some consumer heterogeneity.
G
3
The Model
Following Shilony (1977), we study a tnarket where. conswners can purchase costlessly from a neighbourhood store, but incur some visiting cost if they venture a more distant
store. Thereare two firms located in different regions, A and B, thatproduce a homo-geneous good at zero cost. Each region is inhabited by a unit mass of consumers who have a common reservation utility v for the good. F,ach consumer can costlessly visit the store at his home location, while he has to pay a transportation cost s to go to the other seller. It is assumed that s is uniformly distributed on [0, s] across the population of consumers in each region. This assumption differs from Shilony's (1977) model oí
mixed pricing in oligopoly, where all consumers have the same visiting cost. It generates continuous demand functions, which are a prerequisite for the existence of a pure price setting equilibrium(see Bester (1992)).
The firms offer the good at the price pA and pB, respectively. Without loss of gen-erality, let 0 C p; C v, i- A, B. The seller cannot distinguish buyers from different regions once they enter the store. Similarly, he is uniformed about the visiting cost of the individual buyer. This means, he cannot make his price offer contingent on such information. But, seller i may promote purchases by offering the consumers at location j~ i a rebate r;, with 0 C r; C p;. The seller may offer such rebates by mailing coupons to the other region. When firm i sends coupons to a fraction a; E[0, 1] of the consumers at location j, it has to pay the mailing cost k(a;). Each consumer is equally likely to rc~ci~iw~ Lhe rc~óat~~. 'I'his means, with probabilit.y a, a consumcr in rcgion j has to pay ouly p, - r, for Lhe giwd availabh~ at location i. "1'h~~ buyer who has not rcci~ived a coupon is noL t~ntilh~d 1.o a rcbate and has Lu pay p,. Wc assunu, LhaL constun~~rs du not iuteracl. with each other so that trading coupons is not possible.
The marketing strategy of firm i may be described by
x; -(p;, r;, ~;).
The cost
function k(.) is assumed to satisfy the following restrictions:
7
The marginal cost of couponing is increasing in the number of households reached.
Thisassumption is standard in the advertising literature(see Butters (1977) and Gross-man and Shapiro (1989)). The underlying idea is that thP manufacturer can distributr`
coupons via mail and by placing ads in a set of magazines or newspapers. The probabil-ity that a given consumer receives a coupon through one of these media is independent of receiving a coupon through the other media. In this case, the cost of making sure that a fraction a of consumers receives at least one coupon becomes a convex function oí a. The additional assumptions on k'(.) guarantee that each manufacturer will choose some advertising intensity0 C 1; G I. In addition, we restrict the analysis to the case v~ s. This ensures that competition is sufficiently strong so that setting p; - v cannot be optimal for seller i.
4
Equilibrium
We assume that each consumer is aware of the availability of the good in both regions. In addition he knows Lhe price charged by each of the sellers. In this situation, advertising conveys no information about the existence or the price of a good. Distributing coupons only serves to price discriminate between buyers from different regions. By offering a rebate the seller increases the attractiveness of his product for those consumers who have to spend the cost s in order to visit his store.
To compute each seller's demand, one has to distinguish four groups of consumers: ln eacli of the two mgions thc~ purchasing decision of thc consurners who have received a coupon diffen from those without a coupon. A consumer at locatiou A who has not received a coupon frorn firm B purchases the good at his home location as long as
pA C pB -~ s. If, however, he gets a coupon of value r~, he will buy from seller A only ifpq C pB - rgf s. When consumer s at locationB is not offered a rebate, he will be attracted by firm A's price offer ifpA f s C pB. Otherwise, with a coupon rA, he will purchase from firmA if pA - rA ~- sCpB.
Given the consumers' demand decisions, firm A's profit HA(xA, xB) depends on both firms' sales strategies and is given by
HA(xA, xB) - PA(1 - aB)D(S- PA t pe) ~ PAaBD(s - PA f PB - rB) fPA(1 -~A)D(PB - PA) f(PA - rA)~AD(PB - PAf rA) - k(~A), where
(3)
D(z) - z~s for 0 C z C s, D(z) - 0 for z C 0, and D(z) - 1, for z 1 s. (4) By symmetry, firrn B's profit equals HB(xA,xB) - lIA(xB,xA). To study equilibrium advertising in this market, we consider the Nash equilibrium in the sellers' game. Thus a pair (xA,xB) of marketing strategies constitutes an equilibrium if IIA(xA,xB) ~ IIA(xA,xB) for all xA, and IIB(xA,xA) ~ IIB(xA,xB) for all xB.
Our first result provides a characterizationof the symmetric pure strategy equilibrium.
Proposition 1: Let (xA, x~) 6ean equiliórium such that xA - xR -(P, r", a'). Then
(p',r',a') is given by the unique solution to
P- S~(1 -h 0.5a'), r~ - 0-5P, k~(~~) - P~(4-F 2a').
Figure 1 illustrates that the equilibrium is indeed unique: The P- Pschedule de-picts all p- a combinations such that p- s~(1 f 0.5a); theK- K schedule describes the function p- k'(~)(4 f 2a). Assumption (1) ensures that the two functions intersect at some point (p',a`) within the area (O,s) x(0,1). This intersection determines the equilibrium values p' and a'.
With the help of Figure 1 we can easily establish some comparative statics propertie~s of the equilibrium outcome (li ,r',a'). For instanee, an increase iu Lhe rnarginal cost. k'(.) leads to a higher equilibrium price p' and a lower level of advertising a' because the K- K schedule is shifted upwards. How does the equilibrium react to a change in the consumers' transportation cost? Increasing the parameter s is equivalent to increasing
9
Flgute 1: Equilibrium(p', ~')
each consumer's switching cost by the same íactor. In Figure 1 this leads to an upward
shift of the P- P schedule. Conse.quently, both p' and ~' are increased.
Not all consumers who are couponed will makc~ use of the rebate. 1{irdeeming Lhc coupon is worthwhile only when s C r' - p' - r'. Accordiugly, the redernption rate is
(p' - r')~s - 1~(2 t a'). That is, more than one half of the coupons is not returned to the manufacturer. Using the above comparative statics results, it follows that the equi-librium redemption rate is increasing in the marginal cost of couponing. A cost increase reduces the number of households that are couponed, but the fraction of households that redeem the coupon is increased. An increase in the consumers' transportation cost has the opposite effect: More households are couponed, but a lower fraction returns the coupon. Note, however, that the total number of coupons actually redeemed is given by a`~(2f ~'), which is increasing in a'. This number is, therefore, negatively related to the niargiual cozl. of coupuning anii posit.ivcly relal.cd lo Lhc Icvcl of cousuini~r swilching cost.s.
Of course,
the result that the coupon oífers
a 50"~u price
rebate is specific to the setting
of our model. In particular, the uniform distribution of switching costs is important in
this context. One can show that the sellers would optimally offer a lower rebate if the
distribution of s puts more weight on low switching costs. The intuition is that attracting
high cost consumers through a rebate is not profitable when these consumers represent
only a small fraction of the total population.
10
Proposition 1 only provides an implicit characterization of the equilibrium values p', r', and a'. To obtain an explicit solution of the equilibrium, one has to look at a parametric example of the cost function k(.). The simplest example is k(a) - ca3. This
function satisfies restriction ('l) if c 1 s~27. Theequilibrium described by Yroposition I Is Lhen gIVCII by
p' - 2[3cs -} s(3cs)1~~]1~~ - 2(3cs)1~2, a' -[1 f s(3cs)-1~~]1~~ - l.
(5)
The equilibrium characterization by Proposition 1 is derived from the first order con-ditions that the profit maximizing marketing strategies xA and xB necessarily have to satisfy. Unfortunately, the first order conditions are not sufficient for profit maximiza-tion. The reason is that the firms' profit functions are not jointly concave in (p;, r;, ~; ).
Because of this non-concavity, one has to be careful that no seller can increase his profit. by, for instance, simultaneously lowering his price and his advertising intensity. The following assumption guarantees that such deviations from (p', r', a') are not profitable.
Assumption 1: For all 0 G al C aZ G 1, k(.) satisfies k'(a2)~k'(al) ~(I t at - al)2.
This assumption requires the cost function k(.) to be suíficiently convex, which allows
us to establish existence of a pure strategy equilibrium.
Proposition 2:
Let k(.) satàsjy Assumption 1. Then, there
às an
equiliórium (xA,xB)
S7lftt til(ti TA - TH - (p ,r',.~').
Within the family of cost functions k(a) - cad, Assumption I is satisfied as long as a~ 3. Therefore, the above example satisfies this assumption and the solution described by equation (5) constitutes an equilibrium outcome. Assumption 1 guarantees that the seller's optimal advertising int,ensity is not very sensitive to price changes. In general, the optimal distribution rate .~; is positively related to seller i's price p;. Under Assump-tion 1, however, a given reducAssump-tion in p; has only a small impact on the optimal ~;. This ensures that seller i cannot gain from setting p; c p' together with a; c a'.
11
Even though the manufacturer's profit function is not jointly concave in (p;, r, a, ),
it. is continuous. This implies Lhat there is a mixed strategy Nash equilibrium when Assumption 1 is not satisfied (sa~ llasgupta and Maskin (1986)). In such au eyuilibriurn the sellers choose a random marketing policy. Since for a given a„ seller i's profit is con-cave in (p;,r;), it follows from the first order conditions that the optimal p; is positively related with a,. Similarly, the optimal r; increases with p;. ln the mixed strategy equi-librium, therefore, each seller chooses a; stochastically and he combines higher values of a; with higher prices p; and higher coupon values r;.
5
Welfare
ln this section we investigate how the market participants' welfare depends on the ad-vertising cost k(.) and the transportation cost parameter s. As a measure of consumer welfare, wc, consider aggregate consumer surplus, C, which depends on (p',r', a') ac-cording to
C(p', r', a')- 2(1 -.1')(v- p') ~- 2a' f max[v - p', v - p' ~ r' - s]~s ds. (6) lo,a]
The first term represents the utility of the consumers who do not receive a coupon. The other consumers' purchasing decision depends on the rebate r' and the switching cost s. Consumer s will make use of the coupon only if v- p' ~ v- p' ~ r' - s. Finally, we define social welfare as the sum of producer profits and consumer surplus. We begin by studying the relation between welfare and s.
Proposition 3: The sellers'equilibrium profitisincreasingin s. Consumersurplus and socia! wel~are are decreasing in s.
Not surprisingly, producer profits are positively related to the level of brand loyalty or geographical differentiation. When the sellers are able to choose products, they will seek to maximize the degree of market segmentation. The principle of `maximal differ-entiation', as described by D'Aspremont et al. (1979), remains valid when couponing is introduced. An increase in s has a two-fold impact on consumer surplus: First, the
12
consumer has to pay higher prices. Second, he ís more likely to receive a coupon. The above result demonstrates that the first negative efiect outweighs the second positive effect. Indeed, as was shown before, the coupon redemption rate decreases with s. Fi-nally, social welfare is negatively related to s: The higher s, the higher is the coupon distribution intensity, a', and the number of coupons actually redeemed, a'~(2 f a'). This means that the resources spend on couponing and the consumers' aggregate travel costs increase with s.
To study the impact of the advertising cost on equilibrium payoffs, we consider cost functionsof the type k(a) - cao-. An increase in the parameter c then shifts the cost function upwards so that also the marginal cost of advertising is increased.
Proposition 4: Letk(a) - cav, with o 1 3. Then equilibrium profits and social surplus are increasing in c. Consumer surplusis decreasing in c.
Surprisingly, an increase in the cost of couponing makes the sellers better off. Of course, seller i gains by a reduction in c when the rnarketing strategy x~ of his opponent is kept fixed. In addition to this direct effect, however, there is an indirect effect. Lower advertising costs make competition more aggressive. This reduces the sellers' prices and their equilibrium profits. A similar observation has been made in models of informative advertising (see Bester and Petrakis (1992), and Peters (1984)), where a tax on advertis-ing may increase profits. Since higher couponadvertis-ing costs increase prices and reduce coupon distribution, the consumers become worse off. Aggregate welfare, however, is increased. '1'h~~ intuiliun is t.hal. Ih~~ s~,lh~i:v wasle- h~ss resuuni~s un ~unpuning cuusutni~rs and Lhr consurners save on switching costs.
6
Conclusions
Thís paper has studied couponing as a price discrimination device in oligopolistic com-petition. By couponing those consumers who have some preference for a competing brand, a seller can increase his market share. Coupons may compensate the consumer
13
for a costly movement to another brand. In contrast with the repeat-purchasc explana-tion, coupons reduce consumer switching costs in our model. The price discrimination model predicts that couponing intensifies competition between the sellers, leading to lower prices and profits.
Our simple model may be extended in several interesting directions. We assumed that all the consumers have the same valuation for the good. As a result, total demand was fixed. Introducing some dispersion of consumer valuations would make aggregate demand elastic. As couponing increases competition, it would raise aggregate output. Couponing may have a positive effect on social welfare if the elasticity of demand is sufficiently high. Also, the value of the coupon relative to the príce of the product would presumably depend on the distribution of consumer valuations.
Removing the symmet,ric st,ructure of the modcl would be another interest,ing ex-tension. In our n odel, the sellers were identical and so they used the same marketing strategy. This wc;uld no longer be the case when different production technologies are considered. An e:ctension along these lines could provide insights into the relation be-tween a firm's efficiency and its marketing policy. Similarly, one could study the role of a firm's size whet~ the number of consumers differs across market segments. One might expect that smaller firms have a higher incentive to distribute coupons because they can gain more by attracting consumers from other market segments.
19
7
Appendix
Lemma 1: Let r, 6e an optimalmarketing strategy for firm i given that firm j chooses x~. Then the foRowing must hold: (i) p~ - s Gp; - r;G p;; (iiJ p~ - sG p; G p~ f s; and
(iii) p, ~ min~p~ - r~, 0.5p~].
Proof: If p;- r;7 p„ no consumer will switch from locationjto firm í. Therefore, firm i could increase its profit by not mailing coupons. If p; - r; G p~ - s, all consumers at location j who receive a coupon from firm i strictly prefer to switch. Therefore, firm i could increase its profits by slightly decreasing r;. This proves (i). To provc (ii), not~ that all consumers atj strictly prefer to switch to firm i ifp; G p~ - s. 'I'herefore, firm i could increase its profit by slightly increasing p;. If p; ~ p~t s, no consumer at location
i buys from firm i. But then firm i could get higher profits by settingp; slightly below p~ f s, while keepingp; - r;constant. Finally, (iii) must hold because keeping p; - r;
fixed and increasing p; increases II; if p; G min[p~ - r„ 0.5p~]. Q.E.D.
Lemma 2: Let a, be an optimal marketing strategy for firm i given that firm j chooses z~. Then p; - r, - 0.5p~.
Proof: By the definition of I7;(.), p; - r; must maximize (p; - r;)D[p~ -(p; - r;)]. By lemma 1, the solution must satisfy the first order condition p~ - 2(p; - r;) - 0. Q.E.D.
Proof of Proposition 1: Lemma 2 implies r' - 0.5p'. This together with the first order condition for the opt,imal choice of a; implies k'(a') - (p'~2]2~s. We show firm A cannul gain by scl.l.ing p~ 1 p' only if p' ~ s~(1 t0.5a'). By Icrnma'l, firm A will optimally set pA - rA - 0.5p'. Therefore, the profit from choosing pA E(p`,p' f s) together with aA -a' equals
1In -~Pa(I - a')(s - pn -F P
) f pna'(s
- Pn f 0.5p')
t a'(P
~2)~]~s - k(a~).
(7)
This impli~sí3ll ale3pn - I~' - 2pn f P(1 - 0.5~'))Js. If p' G s~(1 t O.~ia'), then for pA
close enough to p' one gets r3fl A~ópA ~ 0 so that firm A could gain by charging a price slightly abovc p'. As a result, onc must havc p' ? s~(1 f 0.5a').
15
Finally, we show that firmA cannot gain by settingpA C p' only if p' C s~(1 f0.5a'). The profit from choosing pA E(p' - r', p')together with aA - a' and pA - rA - 0.5p'
equals
nA - PA(l - a') ~[PA~~(s -PA f 0.5p') f pA(I -~')(P -PA)]~~5 (g)
fa`(P ~2)2~s - k(a').
Therefore, anA~apA -[s - 2pA f p'(1 - 0.5a')]~s. If p' ) s~(1 -~ 0.5a'), then for pA
close enough to p' one getsallA~apA G 0 so that firm A could gain by charging a price slightly below p'. As a result, one must have p' c s~(1 f 0.5a'). Q.E.D. Proof ofProposition 2: Let firmBuse the strategy aB -(p', r', a`)such that p', r',
and a' satisfy the conditions of Proposition 1. We will show that adopting the same strategy is a best response for firrn A.
First we show that choosing some pA 1 p' is not profitable for firm A. By lemma 2, firmA will optimally set pA - rA - 0.5p'. Therefore, the profit frorn choosing pA E (p',p` f s) together with some aA equals
IIA-ÍPA(1-í~~)(S-PAfP
)fPAa~(s-PAf0.5p')f~A[P'~2121~s-k(~A).
(9)
From the first order condition it follows that firm A will optimally set aA - a'. This
implies
allA~aPA -[(1 - a')(s- 2pA f P') f~'(s - 2pA f 0.5P )]~s c 0, (10) where the in~yuality follows frompA ~ p'- s~(I f 0.5a') Thus, firm A cannot gain by adopting sorne strategy xA ~ (li ,r',~')such thai. pq ~ p'.
'1'o complcte thc argumcul., wc show that choosi~igsomc pA C)i is not pro(itabl~~ for firm A. As firm A will optimally set pA - rA - 0.5p`, its profit írom choosing
pA E(p' - r', p') together withsomeaA equals
nA - PA(I - a') f
[pAa'(s - pA
-6 0.5p') f pA(I
- aA)(P - PA)]~s
(11)
16
Forany pA c p' the optima] choice of aA has to satisfy the first order conditionpA(p' -pA) - s[k'(a') - k'(~A)], so that aA G a' by convexity of k(.) This equation can be
rewritten as
PÁ - pnPf s[k'(a') - k'(an)1 - 0. (12)
Note that p'2 - 9s(k'(a') - k'(aA)] - 4sk'(aA) and that, by lemma 1, pA ~ 0.5n .
Therefore, the above equation yields the solution
pA - 0.5p'f [sk'(an)1'~Z. (13)
As 2pA - p' - 2[sk'(aA)]r~~and s- 0.5~'p' - p' - 2[sk'(a')]r~2, differentiation of fIA yields
BIIA~aP.r
-
[s - 0.5a'p'
- (1 t a' - an)(2Pa -
P )]Is
(14)
-
[2[sk'(~')1'~'
- 2(1 f ~' - aA)[sk'(~A)]'~21~s.
Since aA c a',A~sumption 1 implies 8IIA~8pA ~ 0. Thus firm A cannot gainby
adopt-ing some strategy xA ~(p',r',a') such thatpA G p'. Q.E.D. Proof ofProposition 3: Equilibrium profits are given by
n~ - F~ - a'p'2~4s - k(~~) - 4k'(~~)(1 -f- 0.5a') - a'k'(a') - k(~")
(15)
- 4k'(a') f a'k'(a') - k(a').
By convexity of k(.}, II; is increasing in a'. As a result, s and II; are positively related bccause a' is an incrcasing (rmct,ion o( s.
Consumc~r surplus cquals
C - 2v - 2p'
f a'p'2~4s -
2v - 8k'(a')(1
t 0.5a') f a'k'(a'j
(I6)
- 2v - 8k'(a') - 3a'k'(a')
As an increase in s raises a', this proves that consumer surplus is decreasing in d.
Social welfare equals
W- 2II; f C-
2v - 2k(a')
-
a'k'(a"). An increase in s leads
to a higher value of a' so that the social welfare is reduced.
Q.E.D.
17
Proof of Proposition 4: For k(a) - caaone has ak'(J~)Ia -k(a). Therefore
n~ - P - a'p'~I4s - a'k'(a')Io - P~~1 - (1 -f a)a'p'I(4sn)].
(17)
But a'p' - 2(s - p'). Since an increase in k'(1) raises p', a'p' falls whcn c is increased. This proves that !l, and c are positively related.
By the proof of Propositíon 3, social welfare equals
W- 2v - 2a'k'(a')I~ - a~k~(~~) - 2v -(o f 2)a~P~I(4~s) (18) - 2v -(o f 2)(s - P) P I(2as).
Since 2sI3 C p' c s, W is increasing in p'. Therefore, an increase in c will raise W. Finally, as an increase in c raises p' and lowers a', each consumer's equilibrium utility will de.crease with c. Accordingly, consumer surplus is a decreasing function of c. Q.E.D.
18
References
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19
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Discussion Paper Series,CentER, Tilburg University,The Netherlaods:
(For previous papers please consult previous discussion papers.)
No. Author(s) 9303 D. Karotkin and S. Nitzan 9304 A. Lusardi 9305 W. Giith 9306 B. Peleg and S. Tijs 9307 G. Imbens and A. Lancaster 9308 T.Ellingsen and K. Warneryd 9309 H. Bester 9310 T.Callan and A. van Scest 9311 M. Pradhan and A. van Scest 9312 Th. Nijman and E. Sentana 9313 K. Warneryd 9314 O.P.Attanasio and M. Browning 9315 F.C. Drost and B. J. M. Wcrkcr 9316 H. Hamers, P. Borm and S. Tijs 9317 W. Guth 9318 M.J.G. van Eijs 9319 S. Hurkens 9320 J.J.G. Lemmen and S.C.W. Eijffinger Title
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Euler Equations in MicroData: MergingData from TwoSamples A Simplelustificationof QuantityCompetition and the Cournot-OligopolySolution
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9321 A.L. Bovenberg and S. Smulders 9322 K.-E. Wíimeryd 9323 D. Talman, Y. Yamamoto and Z. Yang 9324 H. Huizinga 9325 S.C.W.Eijffinger and E. Schaling 9326 T.C. To 9327 1.P.J.F. Scheepens 9328 T.C. To 9329 F. de 1ong, T. Nijman and A. RSell 9330 H. Huizinga 9331 H. Huizinga 9332 V. Feltkamp, A. Koster, A. van den Nouweland, P. Borm and S. Tijs
9333 B. Lauterbach and U. Ben-Zion
9334 B. Melenberg and A. van Scest
9335 A.L. Bovenberg and F. van der Plceg
9336 E. Schaling
9337 G -J. Otten
9338 M. Gradstein
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EnvironmentalQuality andPollution-saving Technological
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The (2"'m" - 2}Ray Algorithm: ANew Variable Dimension Simplicial Algorithm For Computing Economic Equilibria on S"xR~
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9342 G.J. Almekinders
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9344 H.Carlsson and S. Dasgupta
9345 F. van der Plceg and A.L. Bovenberg
9346 J.P.C. Blanc and R.D. van der Mei
9347 J.P.C. Blanc
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9356 E. van Damme and S. Hurkens
9357 W. Gtith and B. Peleg 9358 V. Bhaskar
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Aspects of GoodsMarket Integration. A Two-Country-Two -Sector Analysis
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1ob Search, Search Intensity and Labour Market Transitions: An Empirical Exercise
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A Game Theoretic Approach to Problems in Telecommunication
Consequences of Environmental Tax Refortn for Involuntary Unemployment and Welfare
Amoldi Type Methodsfor Eigenvalue Calculalion: Theory and
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Non-additive Beliefs and Game Theory
A Non-cooperative View of ConsistentBankruptcy Rules
