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Price dispersion in oligopoly
I
PRMX DISPERSION
IN OLIGOPOLY
Chaim FERSHTMAN Tb Hebrew Unhmsiry of .IeruSaleqt, Jerusalem, Israel RC4dVtd
July f982, final version received January
lti the +&Id of perfect markets consumers are assumed to respond instantly to every small price ch&nge. However. in the rcaS world it is not dear that any small price change will have a great impact on cuasumea’ dq&ioqs and that, rcaardless of their habit, they ~$9 shift from one -&and to tk atiier. ‘l$e purpose of this paper is to examine oligopolistic price competition under the a%tiption rhat consumers ottt non-responsive to small prke differences. The paper proves the ctistencc of’equitibriurn in whit’ ilrms do not necessaGly charge the same price; howl:ver some of the firtq charge their monopolistic price and others charge prices close to that price.
Wnomjsts have always been suspicious about any irrationality in the consumcr’a behavior. Therefore rational behavior, which is, according to Becker fX96?JX ‘. . . a consistent maximization of a well Grdered function such as utility function . . .’ is assumed in every traditioinal model of competition among firp?s. The perfect frictionless markets derived from such behavior, are far. away from the vonomic realities we encounter every day. In the world of perfect markets a. small change of price by a 5rm will immediately have a great imp+ on tfie quantity demanded. However, in economic reality the purchasing pre is WucL more complicated than is usually assumed in tkc perfect market models, and it is not clear that any small price change will have, a grrent impact on the consumers’ decisions and that, regardless 01’their habits, th?y wiU,$hi$ from,.one brand to ihe other. $iqce purch@ng and consuming involve habits th::!t consumers acquire OV+X@IQ~~’there ,is -a tendiepcy. to consume the same: brands and not LO change &em awxding to stig&$ price changes. Morso;_Gr as Leibenstem (iW& pn.: 196) points out gny purchasc~ are made by agents on behalf of tk S~Z@UIJ~,(for qampie by other.. members of the hou*holds), theref&e itnpl;v:n&;:* . . the. existenq of instances in which individuals are ao’nrmponsiy? tQ. ~nzp c$~~r,e;, in p&e’. The s@ne critic&n about perfect m&++,was,.ma@ by @@ps and Winter (1970): In the world as it & 8~prim WT. 6’ a p.lny will not instantly attract a large crowd of buye=’ .: ,‘l __I”’ \ , 1w
m&Is of I&it &x&ion
are investigated in Pollak (&O, 1976).
Diamond {WI) agrees in his pawr with the pnerai need to adju4ment @X processes which are designed to reflect Some in w&h the cornmcpdity is such that the consumer p it only CXXG.Therefore habiti do not afEet the constmxr’s ions and the share of consumers going to each store is independent
‘ tht
-,a
the dammi racy be rep&&ted grqhkaiiy by a
.=.
,
L.
. .
C. Fershtmun, Price dispersion in oligopoly
391
convarge to a unique equilibrium point in which at least one firm charges its monopolistic price and others charge prices close to that price. Unlike IXawm~ firms do not have to charge identical prices and the equri%ium prioe~ do not necessarily maximize the j&t profits. 1x1section 2 we describe the consumers behavior assuming non-response tc, small price differences. This assumption implies that the consumer preferences cannot be represented by a web ordered function and this result contradicts Becker’s definition of rational behavior. Section 3 presents the demand function firm face as a. conscquenee of the consurne:rs behavior described in section 2. Assuming constant returns to scale we prove in this section that prices will converge to a unique equilibrium poin.. Since in the constant returns to scale case v*e prove that prices arr, not s:Gdve to thF: number of existing firms in the industry, in section 4 WN(:: a:lalyze the competition assr.uning increasing mar-8 inal cost. In this case prices are sensitive to changes in the number of firms and we explon: the relation between the two variables. We conclude by discussing the implications of this analysis on price and non-price comnetiiioir.
2, SmdJ price dispersion Consider an industry consisting of n firms. Let f(t) -(pi(t), . . .I% P,(t)) be the prices that the firms charge at time t and let aj denote the proportion of the total number of consumers buying from firm j. We assume that the number of consumers is large enough so that aj can be treat.ed as a continuous variable. According to Pbelps and Winter (19X)), it is possible that hi a given time, each firm rmty charge a different price. Consumers are not assumed to rr;act immediately to this ,rice difference; however, over time Custolil!:rs shift from firms charging higher prices to those charging lower prizes. MoruoXier if q&P) is the total quantity demanded from the industry when each firm posiA the price P, then it is assumed by Phelps and Winier that ttre denand function the jth firm faces at time t is
Since firms may charge different prices “‘j IS not the j*h firm rmaket share:. If as tlte relative market share we rewrite (1) as u,~=I lj /&Pi), UJ may be d c:ccribed .-. which is the jth firm market share if all !isJls charge thie same price. According to (13, at each point in time the firms h:ive complete monopoly power ivith respect to their customers. Thus it follows I hat the price dispersion at any @{en time can be very great arid does no. influence the firm’s current relative market share, -- “j. The prick c’lispersion in Phelps and
A
ptim L
Within this twinge the Grxns have comF?ete monopoly power to their customers,ii3 if the firmschanget$r;it prices within the
does not satisfy the transitivity condition. Therefore it is impossible to represent such behavior in a well ordered utility fun&on3
Traditional oligopoly theory has no gent ral sclutiori. Although there are several different solutions for different sets of assur:lptions, the main di~fficulty in oligopolistic theory is to describe the interdependence among firms. Each firm has to make assumptions about the paficy of other iirrms in oix5x to find Its own optimal policy, irnd moreover the firm has tc anticipate the reaction of other firms to its cwn policy. This iIlterdtatr,=nden*:eyields enorr;lous tech:&al Sifflculties in any oligopolistic model which is very se&&e to behavioral assumptions. In the Bertrand price competition firms ketp cutting pric,s until price: is equal to marginal cost. But let us consider a game tbat takes place over time (willrout an end point). If the firms realize: that auy price xtting will be followed immediately by similar price reductions by the enti.,e industry, the firms do n6t have any incentive to cut. price (unless price is above their monopolistic price). The lequiWxium prices der’ved fwn th% behavior are not unique; in fact any price between a Bertrand equilibrium price and Pareto optimal prices might be an equilibrium rrxe4 fn t’t-3 model we assume that the firm believes that any reduction of its own price below the E-range of prices, i.e., belo, r B (t) -E, will tx followed by a similar price reduction by other firms. Therefore the 5x1 canno\ increase its relative market share by price reduction al~d the demand f:;nctlon that firm j faces at time t is 4jtpj)
=O,
Pj
>
Pj
s
c’(t) + E, j=l,...,n,whets
G aJipjh ~j=Klin(P1,..
.,Pj_r,Pj+1,
cj(t)
t
.
.
. .
(2)
E,
Pa).
Figure. 1 describes the three kinds of demand ftincticns w have menboned. (,a)The demand fllnction when the firm assumes that any price reduction will be followed by the same price reduction across the market. (b) The demand 3J?hisresultholdscvcnif the cmsumerpreferet:wi include indifferencebetween bundies whose priobs;and qua&tics .‘Z#erby epsilon. Let yc den&! a point in the q Yp pl~e. For e~ly given po@t$ y*
3% fixqm$tkW J., &de~ ~mutn#ms I and 2, the prices charged by the firns C,QW+F~~TV ~~rutiq~reequUi&nicmpoint in which at least one firm charges the p&~,~ id dl the ather pt‘icGs lie in the range [~,~“es). ( / PJW@i Thd pmd Of this pro:Wsition will be carried out in two ereps. It will firsC bt &own *hat them is m equilibrium pokt in which all prices are
within the range [p”’ +--I, ancl at least one of the firms charges the price r_ Then ‘@edescribe the converg~xce to this equilibrium point. Let, ris assume witkmt loss ot’ generality that P!: = p”. Since fl is the price that maximizes the prc43 functton of the first firm, the first firm charges this price if it is possible to do so, and the firm does not have any incentive to chat+rgethis I&e. If the first firm charges the price PT all the othx firms have to charg:e Pj 6 Pi’+ 6, Sux pT = cm we get flz fl. Moreover from the concavity of the profit functions Lf(P$, j=2,. . ., ra, we h!?.ow that the profit function Js an increasing function with respect to price fol- all the prices below the monopolistic price, i.e., ZIi(Pj)> 0 for P,j< e, j = 1,. _., n. ? heli-efore the firms will not charge a price lower than PT if it is possible to charge a price higher than pT. If y ~[e,fl +sj the fa will charge its monopolistic price. If y > -pT+ei we know from the concavity of the profit function that the firm will charlIe the highest possible price, i.e., pT +E. ‘I’herefixe the point P* =(Pf, . . ., P,*) that>is defined b? Pt = q, and PJ”=Py i =fl-i-6
if p;“, [PT,fl-r
s], j=&...,n
(“)
if Py>c-l-s,
is an equ.ilibrium point. In :order to prove convergence it is sufftcixt to describe a process at the end, of which firm 1will charge thi=price fl. If all the prices are above fl it is worthwhile and pcissib!!: for firm 1 to reduce its price immediately to fl, as at this, price the firm maximizes its profit. Now assume that the prices are below Py so that lirm 1 cannot charge fi, i.e., fl> e, + e. Since the profit function is 9 concave function of the @ice that obtains its maximum value at Pi’, we know that for any price belo~V e we get W#$)>O. Thus firm 1 should charge the highest possible price, i.e., e’ +E. Assume that the price pi is charged by firm k, i.e., P,= El.” Since Pi =e” we get fl &Pi’ and therefore Pk K 17. From the concavity 4 n&) we know that the 4th firm can increase its 1wofit by increasing its p&e. Since the kth firm charges’ the lowest price (Pk =e’ and PI = Pt6+E) iz is possible for firm k tu increase its p&e. The increase implies that pi .* . ’ < 3.: -;,? 1 2’. y2
Fig. 2
4.
!¶wsy&al cost
’ Tn ,tbe txmsknt matginai cost case the price that maximizes the profit ~I,IW&MIdoes I@ depend,. crtl the number of firms, i.e., tk price that maxi.i@eq tb p&t fp&on is always the monopolistic price. However in the gkneral case &en we apply increqing marginal cost this result does got hofd. For convenikce let’s assume an industry in which all firms get the sq2 ~ela_t@ mar&# sharqs (i e.; ui= l/n; j = 1,. . ., fit). The &?I firm proi functiaab t& qse is I
(7 Ld~rentiating
(7) with
respectto
n yields
399
The reslrlts of the mode? above do not depend on the size of 8. Changes in E will only a&t the ran,z:c of prices at the equilibrium and the speed in which prices converge to t?;Esequilibrium point. The size of E depends Krst of ail on the product itself. If th: ;Jroduct is completely homogeneous and consumers car, not different iate between different bnmds we expect E to equal zero. me:efore we need some kind of product differentiation in :)rder to assume a positive epsilon. The size of E depends also on the price of the product. It is clear that a one dollar +rease in the price of cars will not have the same effect as a one dollar incr :z,;e in the prim of cigarettes. Tht: existence of E> 0 can also be explained by consumers’ habits. ‘Tlni;sthe history of the market and the consumption ldabkts in the previous ~riods play an importano role in the determin;ticn of E. For example if for many yelirs a consumer has bought t’he same brand It is unlikely that he; will cflangc his habits when he faces solmesmall ~::.ce changes. On the other hand, if the history of the markst ic such tltaat consumers often switch from one brand to the other we expect E to b very small. The size of E dot.; not d.epend solely on the consuming habits but also ori the purchasicg habits. We expect a higher E when purchases are made by agents on behalf of the consumers than when the consumers purchase the product themselves. The existence of a positive E can also be e’xplained by a search model. In this case the state of information in the ma;- c.et, the cost of obtaining information c.bout prices, the cost of searching fo * a new brand and the cost of getting used to a new brand (habits), will deterniine the size of epsilon. Finally, we claim that the purchasing process is much more complicated than is usually assumed in the *perfect market E
According to Assumption 1 if prices are within the E-range of one another, prices cannot change the relative market shares. Therefore in a model where price is :the only competition variable, the relative market shares are exogenous variables. Thus by extending the discussion to competition view price and non-price variables, it can be argued that the relative market shares ;tre determined solely by the non-price variables. AssuGng constant returns to scale, the prices at the equilibrium point are independent of the relative market shares. When the industry demand
function is gi\'en and unchanged there is a complete dichotomy between pnce competition and non-price competition. 8 The prices at the equilibrium POint are determined according to the discussion above, and non-price competition can be described separately [see Schmalensee (1976)] as competition about the market shares. In the Increasing marginal cost case, non-price competition cannot be discuncd separately. The prices at the equilibrium point depend on the vector of relative market shares (as discussed in section 4). 7. Sumnury The Impact of small price changes on the consumer's behavior is a c;untrovcnial question. In the perfect market models consumers are assumed to know and always seek the lowest price. Thus a small change in price shifts Ihe c;onsumers from firms charging a higher price to the firm charging the 10"01 pm,'C. Yet in the world as it is, it is not so clear Ihat for any price dnpenion consumers will bother to change their consumption habits and to ,hlft from one brand 10 the other, This assumption implies that at any Inslant firms ha\'e monopolistic power with respect to their customers. flowe\'cr Ihis monopolistic power is restricted to small price differences. Assuming Ihls rigidity in consumers' behavior we have explored price compc:lition in oligopolistic markets. We have proved that prices converge to a unique equilibrium point in which at least one firm charges its monopolistic price and the other prices are within a e-range of that price. Assuming conslant returns to scale the equilibrium prices are not affected by Ihe number of firms. Moreover when all the firms are identical they will charge Ihe monopolislic price and enlry or exit from the industry will not change that price. Assuming increasing marginal cost, the prices at Ihe equilibrium tend to decline as the number of fi~s. incre~ses. but nevertheless "ill always be higher than the perfect competatlve pnces. In addition to proving the uislence of such an equilibrium point, we described the way praces converge to thaI equilibri~m. . . Fifl311y further research at thiS tOPIC may ~ead In several directions. For el3mple, Ihe lame kind ~f. argu~ent descnbed here can be applied 10 1fl3Iy7Jng non-price competltaon: Sance we. may presume that consumers will be as unresponsive to small differences In products (such as quality and durablhty) as they arc 10 small differences in price. e., 1M noD'pnce ~ariAbla dun&\! the total demand runction the monopolistiC pncn or eadl ~ .ill N and therelor. the equllibnum priCCt will be thanKed.
,w,cd
Rrrtrrnc~
IJ«hr. 0 S. t962. tffallonal behaVIor and economic Iheory. Journal or Political Economy 10, f'eb... 1,11,
I: A,, 1971, A mh.11 of pric& ;rdjustments, Journal di Economic Theory 3, June,
New York).. is (Nonh-M&and, Ainsterdam). %prdUniveririry Press, Cambridge, MA). A micro-behavioral analysis, American 373. licy under atomistic competiiicq m: ES. yment knd Sation theory @krtora, New
, Rk, 1970, H&it formation and dynamic demand hnction. Journal of Poiiticai $osmonsy 78, Juiy/Aug., 745-763. PO&&, R.&i i976, &&it formation aud long run utility functions, JournaS of Economic Theory 13, ckt., 272-297. S&m&n$ee, &., 1976, A model of promotilonal competition m c4igopoly, Review of Economic studies 43, Oct., 4%507.
, .
.
I
PRMX DISPERSION
IN OLIGOPOLY
Chaim FERSHTMAN Tb Hebrew Unhmsiry of .IeruSaleqt, Jerusalem, Israel RC4dVtd
July f982, final version received January
lti the +&Id of perfect markets consumers are assumed to respond instantly to every small price ch&nge. However. in the rcaS world it is not dear that any small price change will have a great impact on cuasumea’ dq&ioqs and that, rcaardless of their habit, they ~$9 shift from one -&and to tk atiier. ‘l$e purpose of this paper is to examine oligopolistic price competition under the a%tiption rhat consumers ottt non-responsive to small prke differences. The paper proves the ctistencc of’equitibriurn in whit’ ilrms do not necessaGly charge the same price; howl:ver some of the firtq charge their monopolistic price and others charge prices close to that price.
Wnomjsts have always been suspicious about any irrationality in the consumcr’a behavior. Therefore rational behavior, which is, according to Becker fX96?JX ‘. . . a consistent maximization of a well Grdered function such as utility function . . .’ is assumed in every traditioinal model of competition among firp?s. The perfect frictionless markets derived from such behavior, are far. away from the vonomic realities we encounter every day. In the world of perfect markets a. small change of price by a 5rm will immediately have a great imp+ on tfie quantity demanded. However, in economic reality the purchasing pre is WucL more complicated than is usually assumed in tkc perfect market models, and it is not clear that any small price change will have, a grrent impact on the consumers’ decisions and that, regardless 01’their habits, th?y wiU,$hi$ from,.one brand to ihe other. $iqce purch@ng and consuming involve habits th::!t consumers acquire OV+X@IQ~~’there ,is -a tendiepcy. to consume the same: brands and not LO change &em awxding to stig&$ price changes. Morso;_Gr as Leibenstem (iW& pn.: 196) points out gny purchasc~ are made by agents on behalf of tk S~Z@UIJ~,(for qampie by other.. members of the hou*holds), theref&e itnpl;v:n&;:* . . the. existenq of instances in which individuals are ao’nrmponsiy? tQ. ~nzp c$~~r,e;, in p&e’. The s@ne critic&n about perfect m&++,was,.ma@ by @@ps and Winter (1970): In the world as it & 8~prim WT. 6’ a p.lny will not instantly attract a large crowd of buye=’ .: ,‘l __I”’ \ , 1w
m&Is of I&it &x&ion
are investigated in Pollak (&O, 1976).
Diamond {WI) agrees in his pawr with the pnerai need to adju4ment @X processes which are designed to reflect Some in w&h the cornmcpdity is such that the consumer p it only CXXG.Therefore habiti do not afEet the constmxr’s ions and the share of consumers going to each store is independent
‘ tht
-,a
the dammi racy be rep&&ted grqhkaiiy by a
.=.
,
L.
. .
C. Fershtmun, Price dispersion in oligopoly
391
convarge to a unique equilibrium point in which at least one firm charges its monopolistic price and others charge prices close to that price. Unlike IXawm~ firms do not have to charge identical prices and the equri%ium prioe~ do not necessarily maximize the j&t profits. 1x1section 2 we describe the consumers behavior assuming non-response tc, small price differences. This assumption implies that the consumer preferences cannot be represented by a web ordered function and this result contradicts Becker’s definition of rational behavior. Section 3 presents the demand function firm face as a. conscquenee of the consurne:rs behavior described in section 2. Assuming constant returns to scale we prove in this section that prices will converge to a unique equilibrium poin.. Since in the constant returns to scale case v*e prove that prices arr, not s:Gdve to thF: number of existing firms in the industry, in section 4 WN(:: a:lalyze the competition assr.uning increasing mar-8 inal cost. In this case prices are sensitive to changes in the number of firms and we explon: the relation between the two variables. We conclude by discussing the implications of this analysis on price and non-price comnetiiioir.
2, SmdJ price dispersion Consider an industry consisting of n firms. Let f(t) -(pi(t), . . .I% P,(t)) be the prices that the firms charge at time t and let aj denote the proportion of the total number of consumers buying from firm j. We assume that the number of consumers is large enough so that aj can be treat.ed as a continuous variable. According to Pbelps and Winter (19X)), it is possible that hi a given time, each firm rmty charge a different price. Consumers are not assumed to rr;act immediately to this ,rice difference; however, over time Custolil!:rs shift from firms charging higher prices to those charging lower prizes. MoruoXier if q&P) is the total quantity demanded from the industry when each firm posiA the price P, then it is assumed by Phelps and Winier that ttre denand function the jth firm faces at time t is
Since firms may charge different prices “‘j IS not the j*h firm rmaket share:. If as tlte relative market share we rewrite (1) as u,~=I lj /&Pi), UJ may be d c:ccribed .-. which is the jth firm market share if all !isJls charge thie same price. According to (13, at each point in time the firms h:ive complete monopoly power ivith respect to their customers. Thus it follows I hat the price dispersion at any @{en time can be very great arid does no. influence the firm’s current relative market share, -- “j. The prick c’lispersion in Phelps and
A
ptim L
Within this twinge the Grxns have comF?ete monopoly power to their customers,ii3 if the firmschanget$r;it prices within the
does not satisfy the transitivity condition. Therefore it is impossible to represent such behavior in a well ordered utility fun&on3
Traditional oligopoly theory has no gent ral sclutiori. Although there are several different solutions for different sets of assur:lptions, the main di~fficulty in oligopolistic theory is to describe the interdependence among firms. Each firm has to make assumptions about the paficy of other iirrms in oix5x to find Its own optimal policy, irnd moreover the firm has tc anticipate the reaction of other firms to its cwn policy. This iIlterdtatr,=nden*:eyields enorr;lous tech:&al Sifflculties in any oligopolistic model which is very se&&e to behavioral assumptions. In the Bertrand price competition firms ketp cutting pric,s until price: is equal to marginal cost. But let us consider a game tbat takes place over time (willrout an end point). If the firms realize: that auy price xtting will be followed immediately by similar price reductions by the enti.,e industry, the firms do n6t have any incentive to cut. price (unless price is above their monopolistic price). The lequiWxium prices der’ved fwn th% behavior are not unique; in fact any price between a Bertrand equilibrium price and Pareto optimal prices might be an equilibrium rrxe4 fn t’t-3 model we assume that the firm believes that any reduction of its own price below the E-range of prices, i.e., belo, r B (t) -E, will tx followed by a similar price reduction by other firms. Therefore the 5x1 canno\ increase its relative market share by price reduction al~d the demand f:;nctlon that firm j faces at time t is 4jtpj)
=O,
Pj
>
Pj
s
c’(t) + E, j=l,...,n,whets
G aJipjh ~j=Klin(P1,..
.,Pj_r,Pj+1,
cj(t)
t
.
.
. .
(2)
E,
Pa).
Figure. 1 describes the three kinds of demand ftincticns w have menboned. (,a)The demand fllnction when the firm assumes that any price reduction will be followed by the same price reduction across the market. (b) The demand 3J?hisresultholdscvcnif the cmsumerpreferet:wi include indifferencebetween bundies whose priobs;and qua&tics .‘Z#erby epsilon. Let yc den&! a point in the q Yp pl~e. For e~ly given po@t$ y*
3% fixqm$tkW J., &de~ ~mutn#ms I and 2, the prices charged by the firns C,QW+F~~TV ~~rutiq~reequUi&nicmpoint in which at least one firm charges the p&~,~ id dl the ather pt‘icGs lie in the range [~,~“es). ( / PJW@i Thd pmd Of this pro:Wsition will be carried out in two ereps. It will firsC bt &own *hat them is m equilibrium pokt in which all prices are
within the range [p”’ +--I, ancl at least one of the firms charges the price r_ Then ‘@edescribe the converg~xce to this equilibrium point. Let, ris assume witkmt loss ot’ generality that P!: = p”. Since fl is the price that maximizes the prc43 functton of the first firm, the first firm charges this price if it is possible to do so, and the firm does not have any incentive to chat+rgethis I&e. If the first firm charges the price PT all the othx firms have to charg:e Pj 6 Pi’+ 6, Sux pT = cm we get flz fl. Moreover from the concavity of the profit functions Lf(P$, j=2,. . ., ra, we h!?.ow that the profit function Js an increasing function with respect to price fol- all the prices below the monopolistic price, i.e., ZIi(Pj)> 0 for P,j< e, j = 1,. _., n. ? heli-efore the firms will not charge a price lower than PT if it is possible to charge a price higher than pT. If y ~[e,fl +sj the fa will charge its monopolistic price. If y > -pT+ei we know from the concavity of the profit function that the firm will charlIe the highest possible price, i.e., pT +E. ‘I’herefixe the point P* =(Pf, . . ., P,*) that>is defined b? Pt = q, and PJ”=Py i =fl-i-6
if p;“, [PT,fl-r
s], j=&...,n
(“)
if Py>c-l-s,
is an equ.ilibrium point. In :order to prove convergence it is sufftcixt to describe a process at the end, of which firm 1will charge thi=price fl. If all the prices are above fl it is worthwhile and pcissib!!: for firm 1 to reduce its price immediately to fl, as at this, price the firm maximizes its profit. Now assume that the prices are below Py so that lirm 1 cannot charge fi, i.e., fl> e, + e. Since the profit function is 9 concave function of the @ice that obtains its maximum value at Pi’, we know that for any price belo~V e we get W#$)>O. Thus firm 1 should charge the highest possible price, i.e., e’ +E. Assume that the price pi is charged by firm k, i.e., P,= El.” Since Pi =e” we get fl &Pi’ and therefore Pk K 17. From the concavity 4 n&) we know that the 4th firm can increase its 1wofit by increasing its p&e. Since the kth firm charges’ the lowest price (Pk =e’ and PI = Pt6+E) iz is possible for firm k tu increase its p&e. The increase implies that pi .* . ’ < 3.: -;,? 1 2’. y2
Fig. 2
4.
!¶wsy&al cost
’ Tn ,tbe txmsknt matginai cost case the price that maximizes the profit ~I,IW&MIdoes I@ depend,. crtl the number of firms, i.e., tk price that maxi.i@eq tb p&t fp&on is always the monopolistic price. However in the gkneral case &en we apply increqing marginal cost this result does got hofd. For convenikce let’s assume an industry in which all firms get the sq2 ~ela_t@ mar&# sharqs (i e.; ui= l/n; j = 1,. . ., fit). The &?I firm proi functiaab t& qse is I
(7 Ld~rentiating
(7) with
respectto
n yields
399
The reslrlts of the mode? above do not depend on the size of 8. Changes in E will only a&t the ran,z:c of prices at the equilibrium and the speed in which prices converge to t?;Esequilibrium point. The size of E depends Krst of ail on the product itself. If th: ;Jroduct is completely homogeneous and consumers car, not different iate between different bnmds we expect E to equal zero. me:efore we need some kind of product differentiation in :)rder to assume a positive epsilon. The size of E depends also on the price of the product. It is clear that a one dollar +rease in the price of cars will not have the same effect as a one dollar incr :z,;e in the prim of cigarettes. Tht: existence of E> 0 can also be explained by consumers’ habits. ‘Tlni;sthe history of the market and the consumption ldabkts in the previous ~riods play an importano role in the determin;ticn of E. For example if for many yelirs a consumer has bought t’he same brand It is unlikely that he; will cflangc his habits when he faces solmesmall ~::.ce changes. On the other hand, if the history of the markst ic such tltaat consumers often switch from one brand to the other we expect E to b very small. The size of E dot.; not d.epend solely on the consuming habits but also ori the purchasicg habits. We expect a higher E when purchases are made by agents on behalf of the consumers than when the consumers purchase the product themselves. The existence of a positive E can also be e’xplained by a search model. In this case the state of information in the ma;- c.et, the cost of obtaining information c.bout prices, the cost of searching fo * a new brand and the cost of getting used to a new brand (habits), will deterniine the size of epsilon. Finally, we claim that the purchasing process is much more complicated than is usually assumed in the *perfect market E
According to Assumption 1 if prices are within the E-range of one another, prices cannot change the relative market shares. Therefore in a model where price is :the only competition variable, the relative market shares are exogenous variables. Thus by extending the discussion to competition view price and non-price variables, it can be argued that the relative market shares ;tre determined solely by the non-price variables. AssuGng constant returns to scale, the prices at the equilibrium point are independent of the relative market shares. When the industry demand
function is gi\'en and unchanged there is a complete dichotomy between pnce competition and non-price competition. 8 The prices at the equilibrium POint are determined according to the discussion above, and non-price competition can be described separately [see Schmalensee (1976)] as competition about the market shares. In the Increasing marginal cost case, non-price competition cannot be discuncd separately. The prices at the equilibrium point depend on the vector of relative market shares (as discussed in section 4). 7. Sumnury The Impact of small price changes on the consumer's behavior is a c;untrovcnial question. In the perfect market models consumers are assumed to know and always seek the lowest price. Thus a small change in price shifts Ihe c;onsumers from firms charging a higher price to the firm charging the 10"01 pm,'C. Yet in the world as it is, it is not so clear Ihat for any price dnpenion consumers will bother to change their consumption habits and to ,hlft from one brand 10 the other, This assumption implies that at any Inslant firms ha\'e monopolistic power with respect to their customers. flowe\'cr Ihis monopolistic power is restricted to small price differences. Assuming Ihls rigidity in consumers' behavior we have explored price compc:lition in oligopolistic markets. We have proved that prices converge to a unique equilibrium point in which at least one firm charges its monopolistic price and the other prices are within a e-range of that price. Assuming conslant returns to scale the equilibrium prices are not affected by Ihe number of firms. Moreover when all the firms are identical they will charge Ihe monopolislic price and enlry or exit from the industry will not change that price. Assuming increasing marginal cost, the prices at Ihe equilibrium tend to decline as the number of fi~s. incre~ses. but nevertheless "ill always be higher than the perfect competatlve pnces. In addition to proving the uislence of such an equilibrium point, we described the way praces converge to thaI equilibri~m. . . Fifl311y further research at thiS tOPIC may ~ead In several directions. For el3mple, Ihe lame kind ~f. argu~ent descnbed here can be applied 10 1fl3Iy7Jng non-price competltaon: Sance we. may presume that consumers will be as unresponsive to small differences In products (such as quality and durablhty) as they arc 10 small differences in price. e., 1M noD'pnce ~ariAbla dun&\! the total demand runction the monopolistiC pncn or eadl ~ .ill N and therelor. the equllibnum priCCt will be thanKed.
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IJ«hr. 0 S. t962. tffallonal behaVIor and economic Iheory. Journal or Political Economy 10, f'eb... 1,11,
I: A,, 1971, A mh.11 of pric& ;rdjustments, Journal di Economic Theory 3, June,
New York).. is (Nonh-M&and, Ainsterdam). %prdUniveririry Press, Cambridge, MA). A micro-behavioral analysis, American 373. licy under atomistic competiiicq m: ES. yment knd Sation theory @krtora, New
, Rk, 1970, H&it formation and dynamic demand hnction. Journal of Poiiticai $osmonsy 78, Juiy/Aug., 745-763. PO&&, R.&i i976, &&it formation aud long run utility functions, JournaS of Economic Theory 13, ckt., 272-297. S&m&n$ee, &., 1976, A model of promotilonal competition m c4igopoly, Review of Economic studies 43, Oct., 4%507.
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