Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors Yuanzhu Lu Chna Economcs and Management Academy, Central Unversty o Fnance and Economcs Abstract We nvestgate endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors and compare the results wth those n Pal (998) to see the eect o the natonalty o prvate rms on the endogenous role o the publc rm. We nd that the results are the same n two cases: () there are only two tme perods or quantty choce, and () there are more than two tme perods or quantty choce and there are more than two prvate rms; but qute derent when there are more than two tme perods or quantty choce and there are only one or two prvate rms. Ctaton: Lu, Yuanzhu, (7) "Endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors." Economcs Bulletn, Vol., o. pp. -7 Submtted: December 8, 6. Accepted: January 8, 7. URL: http://economcsbulletn.vanderblt.edu/7/volume/eb-6l7a.pd
. Introducton Endogenous order o moves s an mportant ssue n a pure prvate olgopoly and n a mxed olgopoly as well. In the lterature on mxed olgopoly, Pal (998) analyzed endogenous order o moves n quantty choce n a mxed olgopoly consstng o a sngle publc rm and domestc prvate rms. Matsumura (3) consdered endogenous roles o rms n a mxed duopoly consstng o a state-owned publc rm and a oregn prvate rm. Lu (6) dscussed endogenous tmng n a mxed olgopoly wth both domestc and oregn prvate rms n the lnear demand case. Gven the results n Pal (998), Jacques (4) and Lu (7), the last two o whch slghtly correct Proposton 4. n the rst paper, t s nterestng to nvestgate endogenous tmng n a mxed olgopoly consstng o a sngle publc rm and oregn compettors. What s the eect o the natonalty o prvate rms on the endogenous role o the sngle publc rm? Ths s exactly what we do n ths paper by adoptng the observable delay game o Hamlton and Slutsky (99) n the context o a quantty settng mxed olgopoly where the rms rst choose the tmng o choosng ther quanttes. Usng a general demand uncton, Matsumura (3) dscussed a mxed duopoly case n whch there are only two possble tme perods or quantty choce. The derences between ths paper and Matsumura (3) are: () the number o oregn prvate rms can be more than one; () the number o possble tme perods can be more than two; (3) we use a lnear demand uncton n order to compare the results wth those n Pal (998), Jacques (4) and Lu (7). We nd that the results are the same n two cases: () there are only two tme perods or quantty choce, and () there are more than two tme perods or quantty choce and there are more than two prvate rms; but qute derent when there are more than two tme perods or quantty choce and there are only one or two prvate rms. The organzaton o the paper s as ollows. In Secton, we descrbe the model. Secton 3 presents the results when there are only two possble tme perods or quantty choce. The SPEs are presented n Secton 4 when there are more than two possble tme perods to be chosen. Secton 5 closes the paper.. The model Consder a mxed olgopoly model consstng o one sngle publc rm and ( ) oregn prvate rms, all producng a sngle homogenous product. Let q and q (,,, ) be the quanttes o the publc rm and the oregn prvate rms, respectvely. Let Q= q+ q denote the aggregate quantty. The market prce s determned by the nverse demand uncton p= a Q. To make the results n ths paper drectly comparable to those o Pal (998), Jacques (4) and Lu (7), we make the same assumptons except that the natonalty o the prvate rms s derent. Speccally, the ollowng assumptons are made: () a s sucently large; () All oregn prvate rms have constant and dentcal margnal costs o producton, whch are normalzed to ; (3) The publc rm has a postve, constant margnal cost o producton, c > ; (4) Fxed costs are zero or all rms; (5) The publc rm s objectve s to maxmze domestc socal surplus dened as the sum o consumer
surplus and ts prot, whereas each oregn prvate rm s objectve s to maxmze ts own prot. We consder the observable delay game o Hamlton and Slutsky (99) n the context o a quantty settng mxed olgopoly where rms rst announce at whch tme they wll choose ther quanttes and are commtted to ths choce beore they actually choose ther quanttes. There are M possble tme perods or quantty choce and each rm may choose ts quantty n only one o those M perods. The objectve unctons o the publc rm and oregn prvate rm are respectvely gven by SS= a( q+ q ) ( q+ q ) ( a q q ) q cq and π = pq = ( a q q ) q. Our objectve s to solve the SPEs o ths extended quantty settng mxed olgopoly game. We restrct our attenton to symmetrc equlbra n whch all rms o the same type choose to produce n the same perod. Frst, we derve the results or two tme perods (M=). ext, we present the results or more than two tme perods. 3. Results or two tme perods ( M = ) Frst, we prove that the publc rm wll not produce smultaneously wth all oregn prvate rms. Ths s stated n the ollowng proposton. Proposton 3.: All rms producng smultaneously n the same tme perod cannot be sustaned as a SPE outcome. Ths proposton s the same as Proposton 3. n Pal (998) except that the prvate rms n Pal s model are domestc and also the same as Lemma 3. n Lu (6) except that there s no domestc prvate rm n our work. It mples that ths result s robust regardless o the type o prvate rms n the market. Gven Proposton 3. and that we restrct our attenton to symmetrc equlbra, there are two possble equlbra when M=: one nvolves all prvate rms producng smultaneously n perod and the publc rm producng n perod, whle n the other possble equlbrum, the publc rm produces n perod and all prvate rms produce smultaneously n perod. We show that the ormer possble equlbrum s really a SPE or any whle the latter one s a SPE only when. Proposton 3.: I 3, there s a unque SPE, at whch the prvate rms produce n perod and the publc rm produces n perod. I, then there s a second SPE n whch the publc rm produces n perod and all prvate rms produce n perod. One mght wonder why Proposton 3 n Matsumura (3) states there exsts a unque SPE n whch the publc rm produces n perod and all prvate rms produce n perod whle we denty two SPEs or the same mxed duopoly. The reason s that Matsumura restrcts hs attenton to the equlbra whch are not supported by weakly All proos are n the appendx.
domnated strateges. We can check that or a mxed duopoly case (=), the addtonal SPE dented n Proposton 3. s ndeed supported by a weakly domnated strategy. Comparng the results o ths secton wth those n Pal (998), we nd that the endogenous order o moves s actually the same. It seems that the natonalty o prvate rms does not aect the endogenous tmng. However, ths s not completely true when there are more than two tme perods or quantty choce. 4. Man Results or more than two perods ( M > ) Proposton 4.: I M >, then () when 3, there s a unque SPE, at whch all prvate rms produce smultaneously n perod and the publc rm produces n a subsequent perod. () when =, there s a second SPE, at whch the publc rm produces n any perod except the last one and the two prvate rms produce n the subsequent perod. (3) when =, there are two SPEs. In one SPE, the prvate rm produces n perod and the publc rm produces n the last perod; n the other SPE, the publc rm produces n any perod except the last one and the prvate rm produces n a subsequent perod. Comparng the results o ths secton wth those n Pal (998), Jacques (4) and Lu (7), we nd that the endogenous order o moves s actually the same when 3 but qute derent when. When =, we stll have the same SPE as n Pal (998), but we also have a second SPE at whch the publc rm produces n any perod except the last one and the two prvate rms produce n the subsequent perod. When =, we stll have two SPEs but they are totally derent rom Jacques (4) and Lu (7). The reason s smple. That s because the publc rm preers to be a leader when prvate rms are oregn whle t preers to be a ollower when competng wth domestc prvate rms. 5. Concludng Remarks In ths paper, we nvestgate endogenous tmng n a mxed olgopoly consstng o one sngle publc rm and ( ) oregn prvate rms by consderng the observable delay game o Hamlton and Slutsky (99) n the context o a quantty settng mxed olgopoly. We nd that the results are the same when there are only two tme perods or quantty choce and when there are more than two tme perods or quantty choce and there are more than two prvate rms but qute derent when there are more than two tme perods or quantty choce and there are one or two prvate rms. Ths derence s the result o the publc rm s derent desred role when competng wth prvate rms o derent natonalty. Reerences Hamlton, J. and S. Slutsky (99) Endogenous Tmng n Duopoly Games: Stackelberg or Cournot Equlbra Games and Economc Behavor, 9 46. Jacques, A. (4) Endogenous Tmng n a Mxed Olgopoly: a Forgotten Equlbrum Economcs Letters 83, 47-48. 3
Lu, Y. (6) Endogenous Tmng n a Mxed Olgopoly wth Foregn Prvate Compettors: the Lnear Demand Case Journal o Economcs 88, 49-68. Lu, Y. (7) Endogenous Tmng n a Mxed Olgopoly: another Forgotten Equlbrum Economcs Letters, orthcomng. Matsumura, T. (3) Stackelberg Mxed Duopoly wth a Foregn Compettor Bulletn o Economc Research 55, 75-87. Pal, D. (998) Endogenous Tmng n a Mxed Olgopoly Economcs Letters 6, 8-85. Appendx In the ollowng proos, we let q, Q and p respectvely denote the publc rm s quantty, the total quantty and the prce n equlbrum or any gven tmng, and q denote a oregn prvate rm s quantty or any gven tmng n whch all oregn prvate rms produce n the same perod. When we consder whether a oregn prvate rm has the ncentve to devate rom any gven tmng, we always choose oregn prvate rm to be the deector. I oregn prvate rm devates, we let q denote the deector s quantty, and q (, 3,, ) denote the quantty o those oregn prvate rms who do not deect. I all rms produce smultaneously n perod t (=, ), then every rm s payo maxmzaton problem gves us the ollowng rst-order condtons: SS = a ( q + q ) + q c= a q c= π q, (A) = a q qk q =, or,,..., q k=, k =. (A) Solvng these equatons gves us q = a c and q = c /( + ). It ollows that Q = a c /( + ), p = c /( + ), SS = a / ac+ ( + + ) c / ( + ), and π = c /( + ). Proo o Proposton 3. We can show that ether the publc rm or a oregn prvate rm has the ncentve to devate all rms produce smultaneously n the same perod, that s, devate rom the ollowng two cases. Case.: All rms produce smultaneously n perod. Consder oregn prvate rm devatng to be a ollower. Then n perod, t wll choose q to maxmze π= a q q q q and the rst order condton (A.) ( ) mples q = a q q. It ollows that p= a q q and thus n perod, oregn prvate rm s (,..., ) prot uncton s π = q a q q 4
and the publc rm s objectve uncton s SS= a a+ q+ q a q q a q q a q q cq + + +. The rst order condtons mply q = a 4c ( + ) and q = 4 c /( + ) (,..., ). It ollows that q = c /(+ ) Q = a c /(+ ), p = c /( + ), and π = 4 c /( + ) > c /( + ). Thereore, oregn prvate rm has the ncentve to devate. Case.: All rms produce smultaneously n perod. Consder the publc rm devatng to be a leader. Then n perod, (A.) mples q = a q +. It ollows that n perod, the publc rms objectve uncton s ( ) ( ) SS= a( a+ q) ( + ) ( a+ q) ( + ) ( a q) ( + ) cq. The rst order condton mples q a ( ) c ( ) = ( + ) /( + ), ( ) q c π = + /( + ), c ( ) [ ] ( ) = + +. It ollows that SS = a / ac+ + c / (+ ) > a / ac+ + + c / ( + ). Thereore, the publc rm has the ncentve to devate. Proo o Proposton 3. () We prove that the possble equlbrum n whch all prvate rms produce smultaneously n perod and the publc rm produces n perod s really a SPE or any by showng that no rm has the ncentve to devate. Frst we obtan the equlbrum quanttes, prce and each rm s payo n ths possble equlbrum. In perod, (A.) mples q = a c. It ollows that p= c q and n perod, oregn prvate rm s prot uncton s π c q q. The rst order condtons mply q = c /( + ). It ollows that ( ) SS = a / ac+ + + c / ( + ), and π = c /( + ). Clearly the publc rm has no ncentve to devate snce the socal surplus would be the same t devated to produce smultaneously wth all the oregn prvate rms n perod. ow consder oregn prvate rm devatng to produce n perod. (A.) and (A.) ( ) mply q = a c and q = c q. In perod, oregn prvate rm s (,..., ) prot uncton s π = q c q and the rst order condtons mply q = c /. It ollows that q = c /( ), p = c /( ), and π = c c + (equal and only / 4 /( ) to devate. = ). So no oregn prvate rm wants 5
() We prove that the possble equlbrum n whch the publc rm produces n perod and all prvate rms produce smultaneously n perod s a SPE only when. The equlbrum quanttes, prce and each rm s payo n ths possble equlbrum have been obtaned n the proo o proposton 3. (case.), q ( ) a c ( ) q = ( + ) c/(+ ), ( ) π = + /( + ), SS a / ac ( ) c /[ ( ) ] = + +, c = + + +. Clearly, the publc rm has no ncentve to devate. ow consder oregn prvate rm devates to produce n perod. (A.) (,..., ) mply ( ) q ( a q q ) q = a q q. In perod, oregn prvate rm s prot uncton s / / π = and the publc rm s objectve uncton s ( ) ( ) SS a q a q q cq a+ q + q a+ q + q a q q = + ( ). q = a c 3 and q = c ( 3 ). It The rst order condtons mply ( ) ollows that q = c ( 3 ) (,..., ), p c ( 3 ) 3 π = c ( ) whch s lower than ( ) c /( ) / 3 hgher when 3. =, and + + when = or but Proo o Proposton 4.: Frstly, clearly, smultaneous play cannot be sustaned as a SPE outcome. Secondly, prvate rms producng n perod t(>) whle the publc rm producng as a ollower cannot be sustaned as a SPE outcome. To prove ths, we lst domestc socal surplus n three derent cases: () when the publc rm produces smultaneously wth all prvate rms, SS = a / ac+ ( + + ) c / ( + ) ; () when the publc rm produces as a leader o all prvate rms, SS = a / ac+ ( + ) c / ( + ) ; (3) when the publc rm produces as a ollower o all prvate rms, SS = a / ac+ ( + + ) c / ( + ). So the publc rm preers to be a leader. I prvate rms produce n perod t(>), the publc rm wll choose to produce n perod. Thrdly, prvate rms produce n perod and the publc rm produces n perod t ( t< T ), then we can show a oregn prvate rm has the ncentve to devate to be a ollower o the publc rm when = but no ncentve to do so when. We can also show a oregn prvate rm has no ncentve to devate to produce n perod s ( s< t ) when t 3. 3 I prvate rms produce n perod and the publc rm produces Straghtorward calculaton yelds the deector s prot s π ( = 4 c / 9 ) π = /( + ) when 3 3 c, equal when =, but hgher when =. Straghtorward calculaton yelds the deector s prot s π ( = c / 4 ) π = /( + ) when c, equal when =., whch s lower than, whch s lower than 6
n perod T, then clearly no rm has the ncentve to devate. So ar we have proved that prvate rms want to be leaders o the publc rm, they have to produce n perod. When they do so, the publc rm producng n any subsequent perod when can be sustaned as SPE, whle the publc rm has to choose to produce n the last perod when =. Fourthly, by Proposton 3., the publc rm producng as a leader o all prvate rms cannot be sustaned as SPE when 3. Fthly, the publc rm produces n perod t(<t) and prvate rms produce n a subsequent perod, then clearly the publc rm has no ncentve to devate, and we can show that a prvate rm has no ncentve to devate to be a leader o the publc rm when t>, that a prvate rm has the ncentve to devate to be a leader o the other prvate rm when = except that prvate rms produce n the subsequent perod, and that a prvate rm has no ncentve to devate =. 7