A Cournot-Stackelberg Advertising Duopoly Derived From A Cobb-Douglas Utility Function

Transcription

1 MDEF Workshop 01, Urbno, 0- September A Cournot-Stackelberg Advertsng Duopoly Derved From A Cobb-Douglas Utlty Functon Alna Ghrvu * and Tönu Puu** *Faculty of Economc Studes and Busness Admnstraton, Babeş- Bolya Unversty, Clu-Napoca, Româna, alna.ghrvu@ubbclu.ro **Center for Regonal Scence, Umeå Unversty, Umeå, Sweden, tonu.puu@cerum.se

2 What s ths paper about? The present paper presents an advertsng quantty game where two frms are engaged n a competton were they try to obtan mamal proft by adustng ther advertsng volumes. By ncludng Stackelberg leadershp acton, the adustment of advertsng volumes leads to a shft between Cournot and Stackelberg equlbrum ponts. The compettors look to mamze ther profts by usng a strategy based on profts epected from both Cournot and Stackelberg actons when choosng the best net move possble. MDEF Workshop 01, Urbno, 0- September

3 In ths presentaton I.The Model II. The Dynamc Setup III. The Numercal Results IV. Conclusons MDEF Workshop 01, Urbno, 0- September

4 I.THE MODEL 1.1 Demand and utlty mamzaton Assume consumer utlty functons of the Cobb Douglas type: U = q q (1) Budget constrant y = p1q1 + pq y =, = 1, + p Cobb-Douglas demand functons (3) q () We can aggregate the so-elastc" demand functons over N consumers: = N y = N = 1 q = = = 1 1+ p, 1, (4)

5 Revenue for the frm sellng becomes: q pq =, = 1,, + (5) 1. Mamzng Profts = c, = 1,, + (6) Mamzng we obtan the condtons: = 0, 1,, c = = ( + ) (7)

6 ' 1 = φ1( ) = 1.3 Reacton Functons (8) = φ ( ) = ' c 1.4 Cournot Equlbrum and the Cournot equlbrum coordnates c 1 = = ' 1 1 ' Stablty Dfferentatng the reacton functons = c ( c1 + c) = c 1 ( c1 + c) 1 1 Dφ1( ) = 1 c 1 1 Dφ( 1) = 1 c 1 (9) (10) Substtutng (9) n (10) we obtan 1 (11) Dφ ( ) = 1 c c c 1 c c Dφ( 1) = c 1

7 The Jacoban n the Cournot pont s thus: 0 1 I= 1c c c 1c c1 c 0 (1) Wth the characterstc equaton: 1( c c) 4 cc I I = = λ λ 0 (13) The roots of ths equaton, the Egenvalues, are: λ 1, = ± 1 c c cc (14)

8 II. THE DYNAMIC SCENARIO: PERIOD DOUBLING AND CHAOS Preamble What happens when the fed pont (Cournot, 1838) loses stablty? Stackelberg Acton In equlbra t s assumed that one compettor could learn the other compettor's reacton functon, and so establsh a leader-follower par. In dsequlbrum stuatons of an evolvng dynamc process, Cournot acton s always the best reply.

9 .1 Stackelberg Proft = φ( ) = ' 1 c1 Substtutng n = φ ( ) = ' c = c, 1,, ' = c c = + = c, = 1,, + we get (16) Take the total dervatves, and equate to zero: whch solves for 1 c = c = 0, = 1,, c ɶ, 1,, = = 4c (17) (18)

10 . The Advantage of Stackelberg Equlbrum Compared to Cournot Stackelberg leadershp profts c ɶ = cɶ cɶ =, = 1,, 4c (19) The Cournot profts Calculate = c, 1,, c + = ( c+ c) = c c c ( c c) ɶ = = 0 4 c ( c c) 4 ( ) + c c + c (0) (1)

11 .3 A Parado: Short Term Advantage of Cournot Acton Consder the general epresson for profts: = c, = 1,, () Substtutng from the reacton functon of the frm tself we obtan: + ( ) = c = 1 c + c = 1 c + In Stackelberg equlbrum the leader chooses acton accordng to c ɶ, 1,, = = 4c (3) (4) and the follower responds c 1 c c c wth Cournot acton: = = = (5) c 4c c 4c 4c ˆ 1 c = 1 Usng (5) n (3) c (6) ˆ 1 c 1c 1 c = ɶ 1 = 1 0 Now, the dfference c 4c c (7)

12 .4 The Map and Agenda An endogenous crteron n the map θ ˆ where takes < ɶ postve values. We compare epected Cournot proft = c = 1 c + c = 1 c + seasoned by the choce parameters to Stackelberg leadershp profts. c 4c The crteron for choosng Cournot acton: ( 1 c ) θ ( ) θ (8) ( 1 c ) θ > c 4c (9)

13 The teratve map: ' 1 1 1c, < 1 c c θ c = 1c 1 1 1c, 1 4c c θ c = 1,, MDEF Workshop 01, Urbno, 0- September

14 III. NUMERICAL RESULTS We check the facts at a parameter pont c = 1.5, c = 1 and θ = 1, θ = The equlbrum Cournot and Stackelberg profts and the average proft are: The phases of the cycle are:

15 A: In the prevous move, the second frm has chosen a Cournot acton, so the frst frm reacts by also choosng a Cournot acton B: The second frm reacts by choosng agan a Cournot acton whch gves hm the mamal proft n ths case C: The frst compettor reverts to Stackelberg acton D: The second frm chooses agan the Cournot acton and the cycle starts all over agan. The profts per perod for the two frms are: Fg.1: Reacton functons when c1 = 1.5, c = 1and θ 1 = 1, θ =

16 IV. Conclusons The numercal results show that t may be better for a compettor to swtch between Cournot and Stackelberg acton than to be a successful Stackelberg leader, and also that t may be proftable for a frm to let the other become a leader than to be one tself. MDEF Workshop 01, Urbno, 0- September