My 0 Price Dicriminion. Direc rice dicriminion. Direc Price Dicriminion uing wo r ricing 3. Indirec Price Dicriminion wih wo r ricing 4. Oiml indirec rice dicriminion 5. Key Inigh ge
. Direc Price Dicriminion Conider firm elling roduc for which demnd i no erfecly elic. If he roduc cn be eily reold, i i difficul for he firm o chrge differen eole differen rice. The firm hu exloi i monooly ower by eing ingle rice. Thi i deiced below. The rofi of he firm i mximized he oin where mrginl revenue nd mrginl co re eul. * ( ) MC MR( ) * Figure : Demnd rice nd mrginl revenue How much monooly chrge bove mrginl co deend on he demnd eliciy. d d d MR = ( ) = ( ) + = ( + ) = ( ). d d d ε Since he monooly eue MR nd MC, MC ( ) = MC, hence =. E ε Difference in rice eliciy led o differen rice. The bigger he rice eliciy of demnd funcion he lower i he rice.
. Direc Price Dicriminion wih wo r ricing For wide rnge of ervice, rele i rohibiively exenive. A monooly cn furher exloi i monooly ower by doing wo r ricing cheme. The hone comny, for exmle, chrge monhly cce fee nd chrge er minue on mny ln. Le he fixed cce fee (er monh, for exmle) be K. In ddiion o he cce fee here i ue fee er uni. Then if cuomer urche uni her ol ymen i R = K +. A ye cuomer h demnd ( ). In he liner ce wih demnd rice funcion = b, we cn olve o obin = ( ) / b. A long hi i he cuomer demnd. For higher rice demnd i zero. Therefore ( ) = MAX(0,( )/ b. Profi Revenue from he ue fee i ( ). We ume h uni co i conn. Therefore he rofi o he firm from he ue fee ( ) c ( ). Thi i he doed re in he figure. In ddiion he monooly chrge n cce fee K The ol benefi o he conumer B ( ) i he re under he demnd rice funcion. Subrcing off he ue co yield he ne gin or conumer urlu CS ( ) Thi i he doed ink ringle. CS( ) In he ce of liner demnd rice funcion he conumer urlu i hlf he recngle, h i c ( ) c( ) ( ) CS ( ) = ( ) ( ) ( )
If he conumer y n cce fee of K her ne gin i u (, K ) = CS ( ) K (.) Thi i enered in Solver hown below in cell H0 (ee he formul br.) A long hi i ricly oiive, he conumer i ricly beer off urching hn ying ou of he mrke. In he limiing ce where hi i zero, he conumer cn do no beer hn urche he ( ) uni. We will ume h he doe o. Then, for ny rice, he monooly mximize rofi by exrcing ll conumer urlu from he conumer. Th i U(, K ) = 0 Thi i enered conrin in Solver. The oluion i hown below. 3
Conumer urlu i hlf he re of he o recngle CS = ( )* ( ) To mximize revenue, he cce fee i e eul o conumer urlu. Then he buyer yoff i zero. CS( ) Noe h in he red hee he cce fee h been e oo high o h ye cuomer i beer off urching nohing. Noe lo h c ( ) c( ) ( ) he ln for he wo ye of buyer differ only in he cce fee. ( ) A long > c i i rofible o lower he ue fee nd o rie conumer urlu. Thi exr urlu i hen roried by he monooli by chrging higher cce fee. Noe h he oiml ln differ only in he cce fee. Therefore if he monooly know h ome buyer re ye nd oher re ye bu doe no know which, i cnno imly offer hee wo ln ince no one will ign u for ln. Moreover even if he monooly doe know who re he high demnder i my be illegl o exclude high demnder from urching ln wih lower cce fee. 4
3. Indirec rice dicriminion wih wo r ricing The reul of he reviou ecion hinge on hree umion (i) rele i rohibiively coly (ii) he monooly cn idenify he differen ye of buyer nd (iii) here re no legl conrin o excluding differen cle of cuomer from offer. Here we uoe h eiher umion (ii) or umion (iii) i no ified. Suoe here re T differen ye of cuomer. We lbel hem o h higher ye hve higher demnd. Formlly, Aumion : Higher ye hve higher demnd rice funcion For ll nd > if ( ) 0 hen ( ) > ( ) For imliciy we will focu on liner demnd rice funcion. = b, =,..., T Then Aumion hold, for exmle, if here re wo ye nd he demnd rice funcion re ( ) = 0 nd ( ) = 40 3. Suoe h he monooly offer e of lernive ln {(, K),...,( n, K n)}. Ech conumer ick one of hee lernive or urche nohing. Noe h he lernive (, ) = (,0) i euivlen o urching nohing. For here i no cce fee nd, he rice 0 K0 T T, no conumer wihe o mke urche. I i helful o dd hi o he e of ln o h he ugmened e i {(, K ),(, K ),...,(, K )} 0 0 n n Nex define (, K ) o be he choice of ye from hi e of ln. We will wrie he e of choice {(, K ),...,(, K )}. T Since (, K ) i he choice of ye buyer, T u (, K ) u (, K ), forll (, K ) in he e of choice. where u (, K ) = CS ( ) K We now derive wo imle bu imorn reul. 5
Princile : Higher ye will never chooe ln wih higher ue fee To ee h emen i rue, le (, K ) be he choice of ye nd le (, K ) be ome oher ln wih higher ue fee nd lower cce fee. Since ye refer (, K ) he lo in conumer urlu in wiching o he oher ln mu be eiher eul or greer hn he reducion in he cce fee K K. In he figure he lo in conumer urlu for ye buyer i he hevily hded re. For ny higher ye The lo in conumer urlu (he hded nd doed re) i greer. Thu ny higher ye i ricly wore off wiching o he ln wih he higher ue fee. ( ) ( ) Princile : If ye i indifferen beween ln (, K) nd ( b, K b) nd > b hen (i) ll higher ye ricly refer (, K ) nd (ii) ll lower ye ricly refer (, K ). b b We hve lredy eblihed (i) in he dicuion of Princile. To ee h (ii) i lo rue we roceed in eenilly he me fhion. Since ye i indifferen beween he wo ln he gin in conumer urlu in wiching from ln o ln b mu be eul o he incree in he cce fee Kb K. In he figure he gin in conumer urlu for ye buyer i he um of b ( ) ( ) 6
he hded nd doed re. For ny lower ye he gin in conumer urlu (he hded re) i mller. Thu ny lower ye i ricly wore off wiching o he ln wih he lower ue fee. Secil ce: Two ye The conumer urlu of ech ye if hey boh chooe ln i deiced below. ( ) ( ) Figure 3.: Conumer urlu if boh ye chooe ln The hded region i he conumer urlu for ye nd he um of he doed nd hded region i he conumer urlu of ye. Suoe h he monooli offer e of ln nd ye buyer chooe (, K) while ye buyer chooe (, K ). Suoe h ye buyer yoff i ricly oiive. Then we cn rie he cce fee by n eul moun Δ K on boh ln unil ye buyer hve yoff of (lmo) zero. Since he co of boh ln h gone u by he me moun, ye buyer will no wich ln. Hence we hve he following reul. Princile 3: The rofi mximizing monooli chooe K = CS( ) o h he yoff of ye i zero. Noe nex h if ye buyer chooe ln her yoff i u (, K ) = CS ( ) K If he chooe ln her yoff i u (, F ) = CS ( ) K. 7
Therefore he cce fee for ye cuomer cn be ried unil he i (lmo) indifferen beween he wo ln. Princile 4: The monooli chooe K o h ye cuomer i indifferen beween ln nd ln, h i u(, K) = CS( ) K = u(, K) Given hee rincile we cn ue Solver o olve for he rofi mximizing ln. Column A hrough H re excly wih direc rice dicriminion. The nex block of cell doe he me comuion if buyer ye chooe he ln inended for he ye ju below her. For ye he ln below i he oion of no buying nyhing. We mke hi ln wih very high ue fee nd no cce fee o h he conumer urche nohing. Cell J = B0 nd cell K = C0. Column L, M nd N coy he formul in Column F,G nd H. There re wo ye of conrin. The fir i he reuiremen h ue fee be lower for higher ye (Princile ). The econd i he reuiremen h he locl downwrd conrin be ified wih euliy. Th i, he blue cell in column N = he blue cell in column H. The red-hee bove deic he oluion. Noe h ye cuomer h ue fee eul o mrginl co nd herefore conume excly he me wih direc rice dicriminion. However ye cuomer h ue fee bove mrginl co. Thu demnd i lower hn i would be wih efficien ricing. 8
Below i he oluion when he rio of ye o ye cuomer i : rher hn :. Noe h fll. Thi incree he conumer urlu of ye buyer nd hi llow he monooly o exrc more rofi by riing i cce fee K. Since he ye buyer re indifferen, ye buyer re ricly beer off wiching. Thu he monooly h o lower K o rovide he incenive no o wich. To undernd hee reul conider he figure below. ( ) ( ) ( ) ( ) c c ( ) ( ) 9
In he lef digrm he drk hded region i revenue from he ue fee le he co of roducion. The doed ringle i he conumer urlu of ye cuomer. Since he i chrged n cce fee eul o hi conumer urlu he rofi of he firm on he ye cuomer i he um of he hded nd doed region. If ye cuomer chooe ln hi conumer urlu i he re under hi demnd rice funcion nd bove he line =. Since he re of he doed ringle i he cce fee, he reidul ried re i u(, K ), he yoff o ye cuomer if he chooe ln. If ye cuomer chooe ln he hded re in he righ hnd figure i he revenue from ln le he co of roducion. Hi conumer urlu i he um of he doed nd ried re. Bu he ried re i he minimum yoff h he mu ge or he will wich o ln. Thu he cce fee i he doed re. Conider he righ hnd digrm. A long > c he monooly cn lower nd o incree he um of doed nd hded re, h hi, he ol rofi on ye cuomer. Then i i oiml for he monooly o e = c. The revied figure i hown below. ( ) ( ) ( ) ( ) c = c ( ) ( ) 0
The finl e i o k wh hen decline o ˆ. Thi i deiced below ( ) ( ) ( ) ( ) ˆ c = c ( ) ( ) The incree in rofi, +Δ S, on ech ye cuomer rie by he re in he lef digrm bounded by he hevy line. Thi i he incree in ocil urlu for ye buyer. The monooly rorie hi by riing he cce fee. The reducion in rofi, Δ r, on ech ye cuomer fll by he re in he righ digrm bounded by he hevy line. The ne incree in rofi i hen ΔΠ = nδs nδ r (3.) Noe h ge cloe o c he incree in rofi roche zero while he lo i bounded wy from zero. Thu o mximize rofi > c=. Noe lo h ny chnge in rmeer h incree he fir erm or decree he econd erm incree he yoff o lowering he rice. Thu, for exmle, if n rie or n fll, he rofi mximizing ue fee,, mu rie.
Solving nlyiclly (for hoe o inclined) Since = b nd ˆ = ˆ b i follow h ˆ ˆ = b( ). Then if eh chnge in rice i mll, Δ Δ =. b To fir roximion, nd c ΔΠ = c Δ = Δ. ( ) ( ) b ΔΠ = ( ( ) ( )) Δ. Subiuing hee exreion ino (3.) c ΔΠ = [ n ( ) n ( ( ) ( ))] Δ b Dividing by Δ nd king he limi, Π c = n ( ) n ( ( ) ( )). b Thi i negive if c i ufficienly mll. The monooly rie unil eiher he mrginl rofi from increing i zero of ye buyer re excluded from he mrke.
Three or more ye of cuomer The nlyi of he wo ye ce cn be exended direcly if here re hree or more ye of buyer. In Solver imly dd row for ech ye. Ju wih ye ech locl downwrd conrin i binding. Bu wh of ll he oher conrin. Conider he following ble where uj i he yoff o ye cuomer if he chooe ln j. ye ln 0 3 u0 = u? u? u3 u0? u = u u3 3 u30? u3? u3 = u33 To check ll he incenive conrin nd riciion conrin we need o relce ech of he ueion mrk. Princile 4 ell u h we cn do h. The rice i higher on ln 0 hn ln nd ye i indifferen beween he wo ln. Thu ll he higher ye wekly refer he ln wih he lower rice. Nex noe h, ince ye i indifferen beween ln nd ln, he lower ye refer ln nd he higher ye ln 3. Finlly, ince ye i indifferen beween ln nd ln 3, boh lower ye refer ln. Thi yield he following ble of ineuliie. ye ln 0 3 u0 = u u u3 u0 u = u u3 3 u30 u3 u3 = u33 3
Noe h none of he hree ye gin by wiching o noher ln. Thu if he locl downwrd conrin re ll binding, hen ll he incenive nd riciion conrin mu hold. Princile 5: If he locl downwrd conrin re ll binding, h i U (, K ) = U (, K ), hen no ye cn gin by wiching o noher ln. How mny ln will be offered? Wih hree ye i my be oiml o ell o ll hree ye bu only hve wo ln. For exmle conider he d below. Noe h ln nd re idenicl. Thu here re only wo differen ln offered. Why i hi? Exercie 3.: Prmeric chnge Conider he wo ye ce wih liner demnd rice funcion. () Wh i he effec on he rofi mximizing ue fee if (i) rie (ii) n rie (iii) b fll. (b) In which ce die i follow h F rie o ye cuomer re wore off. 4
Technicl Noe (limiion of Solver) Conider he following exmle. Sr wih ech rice eul o 0 nd Solver will likely give you he following nwer. Exmle However you mu lo check o ee if i i beer o exclude ye. Chnge o ome lrge number nd run Solver gin. Here i he new oluion. Thu in hi ce i i oiml o hve only wo ln. 5
4. Oiml indirec rice dicriminion We now rgue h he monooly cn do beer by offering cell hone ln rher hn r ricing ln. A cell hone ln offer fixed number of minue (or d) for ol fee r. For ye cuomer he benefi from he uni i he re under her demnd curve B ( ). Thu her yoff (or uiliy) i u (, r) = B ( ) r. The benefi i he um of he doed nd hded re below, h i B ( ) = ( ) + CS ( ). In he liner ce he conumer urlu i eily clculed ince i i hlf he re of he recngle. CS = ( ( ))* ( ) CS ( ) ( ) ( ) Figure 4.: Demnd rice nd ol benefi To ee why hi i beer for he monooli conider he lowe ln wih r ricing. The monooly exrc ll he conumer urlu from ye by chrging n cce fee K eul o CS( ). Now conider ye cuomer who wiche urche ln. Her conumer urlu i he um of he doed nd ried re. She lo h o y he cce fee K. However, we 6
hve ju rgued, he doed re i eul o K. Then he yoff o ye buyer if he wiche i he ried re. U(, K ) K = CS ( ) ( ) ( ) Fig. 4.: Pyoff o ye buyer wih r ricing Noe h he ol ymen by ye i he um of he hded nd doed re. r = * ( ) + K. Suoe h he monooly relce ln wih cell hone ln h offer = ( ) uni for monhly ymen of r. Thi i excly he me oucome for ye buyer nd he monooly. Bu wih he new cell hone ln ye buyer only ge uni o hi conumer urlu i he um of he green ried nd doed re. Since he mu y he um of he doed nd hded re hi yoff i he green ried re. Noe h hi i mller hn he ried re under r ricing. So wiching i le rcive nd i i oible o ueeze ye buyer by chrging her more. 7
U (, r ) K = CS ( ) ( ) ( ) Figure 4.3: Pyoff o wiching decline Rher hn offering ln in which he conumer chooe he number of uni, uoe h he firm offer fixed number of uni nd ol ymen er monh. Thi i he wy cell hone re old. So hink of he ln cell hone ln. The firm offer e of lernive ln {(, r),...,( n, r n)} A cuomer h he oion of no riciing. In hi ce he oucome i ( 0, r 0) = (0,0). We will cll hi ln 0. Then he e of lernive vilble o cuomer i {(, r ),(, r ),...,(, r )} 0 0 n n Le (, r ) be he choice of ye cuomer nd define {(, r),...,(, r )} o be he e of choice of ll he differen ye. A in ecion 3 we hve wo imle bu imorn reul. T T 8
Princile : Higher ye will never chooe ln wih lower uniy To ee h emen i rue, le (, r ) be he choice of ye nd le ( r, ) be ome oher ln wih lower uniy nd ymen. Since ye refer (, r ) he lo in conumer urlu in wiching o he oher ln mu be eiher eul or greer hn he reducion in he cce fee r r. In he figure he lo in conumer urlu for ye buyer i he doed re. For ny higher ye he lo in conumer urlu (he doed nd hded re) i greer. Thu ny higher ye i ricly wore off wiching o he ln wih he higher ue fee. B ( ) B ( ) ( x ) ( x ) Prooiion : If ye i indifferen beween ln (, r) nd ( b, r b) nd < b hen (i) ll higherye ricly refer (, r ) nd ll lower ye ricly refer (, r ) b b We hve lredy eblihed (i) in he dicuion of Princile. To ee h (ii) i lo rue we roceed in eenilly he me fhion. 9
Since ye i indifferen beween he wo ln he gin in conumer urlu in wiching from ln o ln b mu be eul o he incree in he ymen r b r. In he figure he gin in conumer urlu for ye buyer i he um of he doed nd hded re. For ny lower ye he gin in conumer urlu (he doed re) i mller. Thu ny lower ye i ricly wore off wiching o he ln wih he lower ue fee. b B ( ) B ( ) b ( x ) ( x ) The rucure of he red hee i lmo idenicl o h for wo r ricing. Thi i illured below. 0
For he me d here i he oluion wih r ricing. Noe h wih cell hone ricing he monooly i ble o exrc higher ymen from he buyer wih high demnd nd o chieve higher rofi. 5. Key Inigh. Of ll he riciion nd incenive conrin, he only one h we need o focu on re he locl downwrd conrin. Thi i becue higher ye re willing o y more for higher uniy.. The monooly e he ol ymen o h he locl downwrd conrin re binding. For ye hi i he conrin h he mu be willing o ricie. 3. By increing he uniy old o ye (nd o increing he benefi o ye, he monooli cn incree rofi unil he locl downwrd conrin i gin binding. The higher ye uniy rie he incenive for ye o wich. Thu r mu be reduced unil ye i gin indifferen. The ne rofi i herefore ΔΠ = n( ΔB cδ) nδ r. The monooly e he uniy where mrginl rofi i zero. 4. Wih more hn wo ye he monooly mu lower he ol ymen on ll higher uniy ln o minin he locl downwrd conrin. Thu mrginl rofi i ΔΠ = nδs ( n +... + n ) Δ r T
5. If he rio n : n i ufficienly lrge, ye cuomer will be ueezed ou of he mrke. 6. For he highe ye, chnging he uniy doe no ffec incenive conrin. Therefore if i i oiml o mximize ocil urlu nd hu e ( ) = c T T