Speed of price adjustment with price conjectures

Transcription

1 Seed of rce adustment wh rce conectures Mchael Olve Macquare Unversy, Sydney, Australa Emal: Abstract We derve a measure of frm seed of rce adustment that s drectly nversely related to market ower and comare ths to the measure derved by Martn (1993). However, both measures are ncorrect when frms have non-zero rce conectural varatons and treat cometng rce levels as exogenous. Ths s because Taylor seres exansons of the demand functon mlcly assume that frms nfluence the level of cometng rces n a way that s consstent wh ther conectures. JEL: D43, L11, L13, L16 Keywords: Prce adustment; market ower; conectural varatons 1. Introducton One cause suggested for the stable nflatonary envronment that has exsted n many countres snce the late eghtes and early nnetes s a lower seed of rce adustment by frms (Dwyer and Leong, 2001). Emrcal dynamc ndustry studes often suggest that rce adustment s nfluenced by the level of ndustry cometon, although not all agree on the drecton of that nfluence (comare Kraft, 1995; and Shaanan and Fenberg, 1995). As a means of ntroducng rce rgdes nto the theoretcal model, Rotemberg (1982) ncludes a quadratc rce adustment cost and mnmses loss n

2 rof due to the ncomlete adustment to the statc equlbrum rce. Other studes that derve dynamc rcng equatons usng ths method nclude Yetman (2003) and Martn (1993). Alternatvely, Kasa (1998) and Worthngton (1989) maxmse the rof functon wh the ncluson of a quadratc quanty adustment cost. By emloyng a Taylor seres aroxmaton to actual rof, Martn (1993) shows that the seed of rce adustment s a functon of the second dervatve of rof wh resect to rce at the statc equlbrum rce and s negatvely related to market ower n the cases of monoolstc cometon and olgooly wh quanty conectures. Maxmsng the rof functon, we show that an alternatve seed of rce adustment aroxmaton s drectly a negatve functon of market ower when frms are assumed to have quadratc rce adustment costs. However, neher method correctly derves the seed of rce adustment when frms have rce conectural varatons as usually aled. Ths dscreancy s resolved f frms beleve that they nfluence the level of cometng rces n ways that are consstent wh ther conectures. 2. Seed of rce adustment and market ower Let the th frm have the followng rof functon: π (1) ( ) ( ) ( ) 2 = mc q α 1 where and t are frm and tme subscrts, resectvely, and, q, mc, and α ndcate the frm rce, outut, constant margnal cost (excludng adustment costs) and the rce adustment cost arameter, resectvely. In the absence of adustment costs, the frm charges the statc rof maxmsng rce and roduces outut q. 2

3 In ths case, the frst order condon s q ( mc )( dq d ) = 0, where ( dq d + ) s the sloe of the demand functon n statc equlbrum. Takng a frst-order Taylor seres aroxmaton of outut around the statc rof maxmsng rce gves: q q + ( dq d )( ) (2) The followng artal adustment model results after substutng (2) nto (1) and rof s maxmsed wh resect to rce: = λ ( ) 1 (3) λ ( dq d ) /[( dq d ) +α] = where = 1 and λ s the seed of rce adustment. Note (3) s derved wh the use of the statc frst-order condon. Gven that the elastcy of demand n statc equlbrum [ η = ( dq d )( q ) ] s a measure of market ower, (3) mles that as market ower ncreases, the seed of rcng adustment decreases for a gven and q. The seed of rce adustment derved by Martn (1993) s as follows: λ ( π "( ) / 2) [( π "( ) / 2) + α] (4) = 3

4 where π "( ) s the second dervatve of the rof functon at the statc equlbrum rce when there are no adustment costs. Generally, however, the seeds of rce adustment n (3) and (4) wll not be equal. The exceton to ths s when the demand functon s lnear and the Taylor seres exansons are exact.e. ( dq d ) = ( π "( ) / 2). Next we show that even when frms have lnear demand functons, neher of the above seed of adustment measures s correct when frms have non-zero rce conectures but treat cometng rce levels as exogenous. 3. Seed of rce adustment when frms have rce conectures. Prce conectural varatons are frequently used to model frm conduct n heterogenous roduct ndustres (for examle, Bloch and Olve, 2003; Allen, 1998; and Dornbusch, 1987). They reresent the exected rcng resonses of other frms n the ndustry to a change n the frm s own rce. The larger the frm s rce conectures the less cometve s behavour. Let the th frm have the followng lnear demand functon: q = A a + b (5) t where A s a demand shft varable, t s the rce of the th frm, and a and b are osve arameters. Takng the dervatve of (5) wh resect to the th frm s rce gves: dq d = a + b θ (6) 4

5 where d t d = θ s the frm s rce conectural varaton wh resect to the rce of the th frm. 1 After substutng (6) nto (3), the resultant seed of rce adustment s λ = ( a b θ ) /[( a b θ ) + α]. It can be seen that the seed of rce adustment decreases as the weghted sum of the rce conectural varatons ncreases. Ths seed of rcng adustment s the same as (4) when demand s lnear and the rce conectural varatons are constant wh resect to rce. An alternatve method of estmatng the seed of rce adustment s to maxmse the rof functon from (1) when the th frm has the lnear demand functon from (5). The artal adustment model that results s as follows: = δ ( ) 1 (7) δ = ( 2a b θ ) /[(2a b θ ) + 2α ] = [ A + b t + mc ( a b θ )]/(2a b θ ) where δ s the seed of rce adustment and s the equlbrum rce n the absence of adustment costs. Whle the qualatve mact of the rce conectural varatons s the same as above, s clear thatδ λ. Ths dscreancy s resolved f the frm beleves that a art of cometng rces s endogenous n a way that s consstent wh the frm s conectures. Let the th frm s belef about the th frm s rce be reresented by the followng lnear relatonsh: 1 Prce conectural varatons are most commonly resented as elastces. However, resentng rce conectures as the conectured rate of change n a cometng frm s rce for a margnal change n the frm s own rce s analogous to the usual resentaton of quanty conectures. 5

6 t = c + θ (8) where c reresents a comonent of the cometng rce that the th frm beleves cannot nfluence. 2 Then the artal adustment model becomes: = λ ( ) 1 (9) λ = ( a b θ ) /[( a b θ ) + α] = [ A + b c + mc ( a b θ )]/ 2( a b θ ) Comarng (7) and (9), can be seen that the makeu of the statc equlbrum rce and the seed of rce adustment are nterrelated. Now all three methods for calculatng the seed of rce adustment lead to the same result when frms have lnear demand functons. The dscreancy n seed of rce adustment measures arses because Taylor seres exansons of the demand functon mlcly assume that a comonent of cometng rces s endogenous n a way that s consstent wh frm conectures. To see ths, take the followng frst-order Taylor seres aroxmaton of outut around the statc equlbrum own rce and cometng rces: q q + ( q )( ) + ( q )( ) t t t (10) 2 Note that ths s dfferent to consstent conectures that equate the sloe of the cometng frm reacton functons wh s conectures, although consstent conectures could be encomassed n ths formulaton. 6

7 where t s the th frm s statc equlbrum rce, q s the own-rce artal dervatve of demand and q t s the cross-rce artal dervatve of demand. If the rces of all cometng frms are exogenous, as n the conventonal conectural varatons model, then the last term n (10) s zero. Substutng (8) nto (10) gves: q q + [ q + ( q ) θ ]( ) (11) t (11) and (2) are equvalent, as the total dervatve of the th frm s outut s dq d = q + ( q t ) θ. 4. Concluson Ths aer derves frm seed of rce adustment as a functon of the sloe of the demand functon when frms have quadratc rce adustment costs. Ths drectly nversely relates market ower to the seed of rce adustment and rovdes an alternatve to the measure derved by Martn (1993). However, both measures are ncorrect when frms have rce conectural varatons as they are usually aled. Ths s because Taylor seres exansons of the demand functon mlcly suggest that frms dvde cometng rces nto exogenous and endogenous comonents n a way that s consstent wh ther conectures, whle the standard method does not make ths dvson. In whch of these ways frms behave s an emrcal queston that s worthy of further consderaton. 7

8 References Allen, C., An emrcal model of rcng, market share and market conduct: an alcaton to mort cometon n US manufacturng. Manchester School 66, Bloch, H., Olve, M., Influences on rcng and marku n segmented manufacturng markets. Journal of Industry Cometon and Trade 3, Dornbusch, R., Exchange rates and rces. Amercan Economc Revew 67, Dwyer, J., Leong, K., Changes n the determnaton of nflaton n Australa. Reserve Bank of Australa Research Dscusson Paer Kasa, K., Identfyng the source of dynamcs n dsaggregated mort data. Journal of Aled Econometrcs 13, Kraft, K., Determnants of rce adustment. Aled Economcs 27, Martn, C., Prce adustment and market structure. Economcs Letters 41, Rotemberg, J.J., Stcky rces n the Uned States. Journal of Polcal Economy 90, Shaanan, J., Fenberg, R.M., Dynamc cometon and rce adustments. Southern Economc Journal 62, Worthngton, P.R., On the dstncton between structure and conduct: adustment costs, concentraton, and rce behavor. Journal of Industral Economcs 38, Yetman, J., Fxed rces versus redetermned rces and the equlbrum robably of rce adustment. Economc Letters 80,