Imperfect Competition

Transcription

1 Iperfect Copetton ISBN: Iñak Agurre 07-09

2 Notes on Iperfect Copetton Iñak Agurre Departaento de Fundaentos del Análss Econóco I Unversdad del País Vasco

3 Inde Chapter. Monopoly Introducton.. Proft azaton by a onopolstc fr... Lnear deand and constant elastcty deand..3. Coparatve statcs..4. Welfare and output..5. Prce dscrnaton..6. Frst-degree prce dscrnaton..7. Second-degree prce dscrnaton..8. Thrd-degree prce dscrnaton. Chapter. Non-Cooperatve Gae Theory Introducton... Basc notons... Etensve for gaes.... Strategc for gaes... Soluton concepts for non-cooperatve gae theory.... Donance crteron.... Backward nducton crteron...3. Nash equlbru...4. Probles and refneents of Nash equlbru..3. Repeated gaes.

4 .3.. Fnte teporal horzon..3.. Infnte teporal horzon..4. Conclusons. Chapter 3. Olgopoly Introducton 3.. The Cournot odel Duopoly Olgopoly (n frs) Welfare analyss. 3.. The Bertrand odel Hoogeneous product Heterogeneous product Leadershp n the choce of output. The Stackelberg odel Colluson and the stablty of agreeents Short-ter colluson The stablty of agreeents under a fnte teporal horzon and under an nfnte teporal horzon.

5 Chapter. Monopoly Introducton We say that a fr s a onopoly f t s the only seller of a good (or goods) n a arket. Proble: t s not easy to defne good and arket. A fr ay becoe a onopoly by varous reasons: - Control over raw aterals. - Acquston of the eclusve sellng rghts (by a patent, by a publc aucton etc.). - Better access to the captal arket. - Increasng returns of scale etc. In contrast wth a perfectly copettve fr whch faces a perfectly elastc deand (takng prce as gven), a onopolst faces the arket deand. Therefore, a fr wth onopolstc power n a arket t s aware of the aount of output that t s be able to sell t s a contnuous functon of the prce charged. Put dfferently, the onopolstc fr takes nto account that a reducton n output wll ncrease the prce that can be charged. In consequence, a onopolst has the power to set the arket prce. Whle we can consder a copettve fr as a prce taker, a onopolst s prce decson-aker or prce setter... Proft azaton () The proble of proft azaton n prces and n quanttes. Frst order condtons. Second order condtons. A graphcal nterpretaton of the proft azaton proble. () Interpretaton of argnal revenue. 3

6 () Margnal revenue equals argnal cost condton. (v) Output and deand elastcty. (v) Lerner Inde of onopolstc power. (v) Graphcal analyss. (v) Second order condtons. () The proble of proft azaton n prces and n quanttes There are two types of constrant that restrct the behavour of a onopolst: a) Technologcal constrants suarzed n the cost functon C(). b) Deand constrants: (p). We can wrte the proft functon of the onopolst n two alternatve ways: - Π ( p) = p( p) C( ( p)) by usng the deand functon. - Π ( ) = p ( ) C ( ) by usng the nverse deand functon. The deand, (p), and the nverse deand, p(), represent the sae relatonshp between prce and deanded quantty fro dfferent ponts of vew. The deand functon s a coplete descrpton of deanded quantty at each prce whereas the nverse deand gves us the au prce at whch a gven output ay be sold n the arket. a Π( p) a Π( ) p p 0 = ( p ) p = p( ) The proble of proft azaton as a functon of prce 4

7 a Π( p) a p( p) C( ( p)) p p ( p) ( p) p( p) C ( ( p)) ( p) 0 Π = + = Π ( p) = ( p) + p ( p) C ( ( p)) ( p) C ( ( p)) ( p) < 0 The proble of proft azaton as a functon of the output a Π( ) a p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p( ) C ( ) = 0 Π ( ) = 0 Frst order condton. Π ( ) = p( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). Π Π ( ) = 0 () Interpretaton of argnal revenue Margnal revenue, Π (0) > 0 r ( ), s: Π( ) r ( ) = p( ) + p( ) () Addtonal revenue fro sellng an addtonal unt. Loss of revenue fro sellng unts already produced at a lower prce. 5

8 () Margnal revenue equals argnal cost condton The proft-azng output level (nteror soluton) satsfes: Π ( ) = r ( ) C ( ) = p( ) + p( ) C ( ) = 0 () At the onopolstc optal output the argnal proft s zero, Π ( ) = 0; that s, an nfntesal change n the level of output antans proft unchanged. An output level such that Π (.) > 0 does not aze profts: an (nfntesal) ncrease n output would ncrease profts. In a slar way, a level of output such that profts: a (nfntesal) decrease n output would ncrease profts. Π (.) < 0 does not aze At the proft-azng level of output argnal revenue equals argnal cost, r ( ) C ( ); = that s, an nfntesal change n the level of output changes revenue and cost equally. (In other words, an nfntesal ncrease n the level of output ncreases revenue and cost by the sae aount and an nfntesal decrease n the level of output reduces revenue and cost by the sae aount). An output level such that r (.) > C (.) does not aze profts: an (nfntesal) ncrease n output would ncrease revenue ore than cost (therefore ncreasng profts). Lkewse, a level of output such that r (.) C (.) < does not aze profts: a (nfntesal) decrease n output would reduce cost ore than revenue (therefore ncreasng profts). (v) Output and elastcty: ε ( ) We seek to show that at the onopoly output the prce-elastcty of deand s or ore. Frst, we defne the prce-elastcty of deand n absolute value: 6

9 - as a functon of prce: p p ( p ) ε ( p) = ( ), (3) - as a functon of output : ε p( ) ( ) =. p ( ) (4) We net represent argnal revenue as a functon of the prce-elastcty of deand: r p p ( ) = ( ) + ( ) (5) p ( ) r ( ) p( ) = + p ( ) (6) r ( ) = p( ) ε ( ) (7) In the onopoly output argnal revenue and argnal cost are equal: ( ) = ( ) = ( ). ε ( ) r p C (8) Gven that the argnal cost s by defnton non-negatve (zero or ore) then the argnal revenue ust be non-negatve. Ths occurs when the prce-elastcty of deand n absolute value s or ore. That s: ( ) 0 ( ) 0 ( ). ε ( ) p C p ε ( ) 0 (v) Lerner nde of arket power Now we obtan the Lerner nde of onopoly power (or arket power) also called the relatve prce-argnal cost argn. Fro condton (8) we obtan: p ( ) p C ε ( ) = ( ) ( ). 7

10 By rearrangng we get: p ( ) C( ) =. (9) p( ) ε ( ) Therefore, the Lerner nde s a decreasng functon of the prce-elastcty of deand n absolute value. In partcular, when ε ( ) = 0 onopoly power would be p p ( ) ( ) C( ) = and when ε ( ) = (as t would occur f the fr behaved as a perfectly copettve fr) arket power would be zero, ( ) ( ) p C p ( ) = 0.. (v) Graphcal representaton p C ( ) p Π r ( ) p ( ) Margnal revenue, r ( ) = p( ) + p ( ), s located below nverse deand gven that the nverse deand functon s downward slopng, p ( ) < 0. That s, r ( ) < p( ) for > 0, but both functons have the sae ntercept, r (0) = p(0). The proft of the onopolst (when there s no fed cost) s gven by: 8

11 C ( ) ( ) p C( ) p C ( z) dz p 0 Π =Π = = = (v) Second order condtons Interpretaton We assue for the sake of splcty that the proft functon s strctly concave. Π ( ) = r ( ) C ( ) = p( ) + p ( ) C ( ) < 0 (0) Condton (0) s equvalent to sayng that the slope of the argnal revenue has to be lower than the slope of the argnal cost: dr d ( ( )) dc ( ( )) In other words, the argnal revenue curve ust cross argnal cost fro above. < d r, C dr d ( ( )) dc ( ( )) > d dr d ( ( )) dc ( ( )) < d C ( ) r ( ) 9

12 Cases. Strctly conve cost or lnear cost: C ( ) 0 (ncreasng or constant argnal cost) a) Strctly concave deand or lner deand: p ( ) 0 Π ( ) = p( ) + p ( ) C ( ) < 0 < b) Strctly conve deand: p ( ) > 0 r ( ) = p ( ) + p ( ). We need to check < 0 > 0 r < C ( ) ( ).. Strctly concave cost: C ( ) < 0 (decreasng argnal cost) We always have to check whether r < C ( ) ( )... Lnear deand, constant elastcty deand and constant argnal cost () Lnear deand and constant argnal cost Inverse deand: p( ) = a b ( a> 0, b> 0 ). Producton cost: C ( ) Margnal revenue: = c ( c 0 ). ( a > c) r ( ) = a b. Slope of nverse deand: p ( ) = b Slope of argnal revenue: dr d ( ( )) = b Strctly concave proft functon: Π ( ) = r ( ) = b< 0. Margnal proft at zero: Π = p C = a c> (0) (0) (0) 0. Proft azaton: a c ( ) = ( ) = = b r C a b c 0

13 a+ c Monopoly prce: p = p( ) p = a b p = Monopoly profts: ( ) ( ) [ ( ) ] [ ] a c a c a Π =Π = p c c = p c = = b 4b () Constant elastcty deand and constant argnal cost Deand: ( p) = Ap b ( A> 0, b> ). Producton cost: C ( ) = c ( c > 0 ). Prce-elastcty of deand: ε p ( p) p Ap ( b+ ) ( p) = ( p) = bap = b. b Inverse deand: b b p( ) = A. Margnal revenue: ( b ) b b r ( ) = A. b Slope of argnal revenue: ( b ) =. b ( + b) b b r ( ) A Strctly concave proft functon: ( b ) b ( + b) b b Π ( ) = r ( ) = A < 0 b>. Margnal proft at zero: Proft azaton: Π (0) = > 0. ( b ) ( ) ( ) ( ) b ( ) b ( ) b b r = C r = A = c = A b c b ( b ) b b b b b b b b ( ) = A c = A c ( b ) ( b ) Monopoly prce:

14 b b b b b b b b p = p ( ) p = A( ) = A A c p = c ( b ) ( b ) Monopoly profts: b b c b b b ( b ) ( ) [ p( ) c] [ p c] A c A c ( b ) Π =Π = = = = b ( b ) ( b ) By solvng the proble of proft azaton as a functon of prce, we obtan the Lerner nde: p c = p ε ( p) Under constant elastcty deand the Lerner nde becoes p c = and t s p b straghtforward to obtan the onopoly prce. Then t s easy to obtan the onopoly output and the onopoly profts..3. Coparatve statcs We now study how the onopoly prce and output respond to a change n producton costs. Econoc ntuton tells us that an ncrease n argnal cost should ental a reducton n output and an ncrease n prce. We assue that the argnal cost s constant (and there s no fed cost). The cost functon s gven by C ( ) = c. a Π( ) a p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p( ) c= 0 () ( c) the onopoly output s an plct functon of the argnal cost.

15 Π ( ) = p( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). We can analyze the change n onopoly output due to a change n argnal cost n two equvalent ways: () By copletely dfferentatng condton () wth respect to and c. p( ) + p ( ) d dc= 0 We get: d = <0 dc p ( ) + p ( ) < 0 CºO () Therefore, an nfntesal ncrease n argnal cost reduces output and an nfntesal reducton n argnal cost ncreases output. () By usng the fact that the optal output for the onopolst ( c ) s an plct functon of argnal cost. Therefore, by defnton, ( c ) satsfes the frst order condton; that s, p c cp c c ( ( )) + ( ) ( ( )) = 0 By dfferentatng wth respect to argnal cost: p( ()) c () c + () c p ( ()) c () c = p( ()) c + () c p ( ()) c () c = Rearrangng: () c = 0 p( ()) c + () c p ( ()) c < Fnally the change n prce due to the change n argnal cost s: 3

16 < 0 dp ( ) = dp d = p >0 dc d dc p ( ) + p ( ) < 0 CºO (3) Eaples () Lnear deand a+ c dp p = = dc dp p ( ) = = dc p ( ) + p ( ) Under lnear deand the change n prce s half the change n argnal cost: dp = dc = 0 () Constant elastcty deand b dp b p = c b dc = b > ( + b) p ( ) = A p( ) = A p( ) = A b b ( + b) (+ b) b b b b b b dp b = = = = > (+ b) dc p ( ) ( + b) ( + b) b + b b A p ( ) b + b ( + b) b b A b Under constant elastcty deand the ncrease n the onopoly prce s greater than the ncrease n argnal cost: dp > dc. 4

17 .4. Welfare and output () The representatve consuer approach. Quas-lnear utlty. () Mau wllngness to pay. Margnal wllngness to pay. () The deand functon s ndependent of ncoe. (v) Socal welfare functon and socal welfare azng output. (v) Total surplus, consuer surplus and producer surplus. (v) Effcency condtons n the presence of several consuers or arkets. (v) A coparson between onopoly output and effcent output by usng the proft azaton proble. (v) A coparson between onopoly output and effcent output by usng the socal welfare azaton proble. () Irrecoverable effcency loss. () The representatve consuer approach. Quas-lnear utlty We wll follow the representatve consuer approach to analyze welfare and evaluate onopoly fro a socal welfare pont of vew. Under ths approach, t s assued that arket deand (p) s generated by azng the (quas-lnear) utlty of a representatve consuer. Consder an econoy wth two goods, and y. Good s produced n the onopolstc arket whle we can nterpret the good as the aount of oney to be spent on the other good by the consuer once he/she has spent the optal aount of oney on good. We assue that the representatve consuer has a Quas-lnear Utlty Functon: U y u y u u u (, ) = ( ) + ( (0) = 0; (.) > 0; (.) < 0) 5

18 () Mau wllngness to pay and argnal wllngness to pay Mau wllngness to pay, R( ) : the au aount of oney that the consuer s wllng to pay for unts of the good. He/she pays the au f he/she s ndfferent between consung unts by payng R( ) and not consung the good, thus usng all hs/her ncoe endowent to consue the other goods.that s: U(, R( )) = U(0, ) Note that the consuer ust be ndfferent and, therefore, the above condton ust be satsfed wth equalty. If for eaple U(, R ( )) > U(0, ) then the consuer would wsh to pay a greater aount to R ( ) and f hgher than hs/her au wllngness to pay. Gven that the utlty functon s quas-lnear then: Therefore, under quas-lnear utlty: U(, R ( )) < U(0, ) then R ( ) would be U(, R( )) = U(0, ) u ( ) + R ( ) = u(0) + R( ) = u( ) u ( ) Mau wllngness to pay Margnal wllngness to pay: ths s the change n au wllngness to pay due to an nfntesal change n the quantty consued. u ( ) Margnal wllngness to pay 6

19 () The deand functon s ndependent of ncoe L(, y, λ ) a u ( ) + y y, a u ( ) + y+ λ y p y,, λ sa. y+ p= [ ] L = u ( ) λ p = 0 L p= u ( ) Inverse deand funton = λ = 0 y L = y p = 0 λ The deand functon (p) s the nverse of ths functon and therefore satsfes the frst order condton: p= u ( ( p)) Deand functon Property of the quas-lnear utlty functon: the deand functon s ndependent of ncoe. By dfferentatng wth respect to p we get: = u ( ( p)) ( p) ( ) = < 0 negatve slopng deand u ( ( p)) p < 0 (v) Socal welfare functon and socal welfare azng output In ths subsecton we justfy the use of W( ) = u( ) C( ) as the socal welfare functon. We consder the proble of obtanng the allocaton that azes the utlty of the representatve consuer wth a resources constrant: we nterpret the producton cost of good as the aount of good y to whch ust be gven up n order to have the good. 7

20 a u ( ) + y y, sa. y= C( ) By replacng y n the objectve functon we get: a u ( ) + C ( ) a u ( ) C ( ) cons tante Therefore the socal welfare azng proble becoes: a W( ) a u( ) C( ) 0 0 W (0) = u (0) C (0) > 0 e W u C W ( ) = ( ) ( ) = 0 ( ) = 0 (3) Frst order condton. W u C ( ) = ( ) ( ) < 0 Strctly concave welfare functon (regular case). Therefore, the welfare azng output or effcent output satsfes e e e W ( ) = 0 u ( ) = C ( ). Under constant argnal cost the effcency condton becoes: e u ( ) = c, That s, at the effcent output argnal wllngness to pay equals argnal cost. (v) Total surplus, consuer surplus and producer surplus The functon W( ) = u( ) C( ) can be nterpreted as the total surplus; that s, the dfference between au wllngness to pay and producton cost. By defnton the followng s satsfed: u ( ) u (0) u ( zdz ) 0 = = 0 C ( ) C (0) C ( zdz ) 0 = = F = 0 8

21 Therefore azng u ( ) C ( ) s equvalent to chosng the level of output that azes the area below the nverse deand and above the argnal cost. p C ( ) p C ( ) e u ( ) e C ( ) p ( ) p ( ) e e p C ( ) e W( ) p ( ) e By addng and subtractng p we can rewrte the total surplus as: 9

22 [ ] [ ] W ( ) = u( ) C( ) = u( ) p + p c EC ( ) EP( ) The consuer surplus, CS(), easures the dfference between au wllngness to pay and the aount of oney actually pad. The producer surplus, PS(), easures the profts of the fr (when there are no fed costs). Therefore, effcent producton also azes the addton of the consuer surplus and the producer surplus. p e CS( ) C ( ) e PS( ) p ( ) e (v) Effcency condtons n the presence of several consuers or arkets We net analyze the proble of obtanng a Pareto effcent allocaton when we consder an econoy wth two consuers under quas-lnear utlty, u ( ) + y, and an endowent of, =,.. We aze the utlty of one agent (for eaple consuer ) whle antanng constant the utlty of the other (consuer ), gven a resource constrant (argnal cost, c, s assued to be constant). 0

23 a u ( ) + y, y,, y sa. u ( ) + y = u y + y = + c.( + ) By substtutng y and y n the objectve functon the proble becoes: a u ( ) + u ( ) c.( + ) + + u, Fro the frst order condtons we get: e u( ) c= 0 = = e e e u( ) u( ) c Effcency condton (4) = u ( ) c 0 (v) A coparson between onopoly output and effcent output by usng the proft azaton proble. a Π( ) a p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p( ) C ( ) = 0 Π ( ) = 0 Frst order condton. Π ( ) = p( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). ( ) 0 e Π ( )? ( ) 0 Π = Π < e e e e e e e e e Π ( ) = p( ) + p( ) C ( ) = u ( ) C ( ) + p( ) < 0 e = u ( ) = 0 < 0 By defnton of effcent output.

24 Π ( ) = 0 Π ( ) < 0 Π ( ) <Π ( ) > Π ( ) < 0 e e e dπ ( ) ( ) 0 0 ( ) d Π < < Π Π Π ( ) = 0 e Π ( ) < 0 e (v) A coparson between onopoly output and effcent output by usng the socal welfare azaton proble. a W( ) a u( ) C( ) 0 0 W (0) = u (0) C (0) > 0 p(0) > C (0) e W u C W ( ) = ( ) ( ) = 0 ( ) = 0 Frst order condton. W u C ( ) = ( ) ( ) < 0 Strctly concave welfare functon. e W ( ) = 0 W ( )? W ( ) < 0

25 u ( ) W ( ) = u ( ) C ( ) = p ( ) > 0 p( ) < 0 By defnton of onopoly output. e W ( ) = 0 W ( ) > 0 W ( ) < W ( ) > W ( ) < 0 e e dw ( ) ( ) 0 0 ( ) d W < < W W W ( ) > 0 e W ( ) = 0 e 3

26 (v) Irrecoverable effcency loss (IEL). e e ( ) ( ) e [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] 0 0 IEL= W W = u z C z dz u z C z dz = u z C z dz p e CS( ) C ( ) p CS( ) C ( ) p e PS( ) PS( ) p ( ) p ( ) e p C ( ) p IEL p ( ) e 4

27 .5. Prce dscrnaton () Defnton. () The ncentve to dscrnate prces. () Condtons. (v) Types of prce dscrnaton (Pgou, 90). (v) Eaples. (v) The odel. () Defnton There ests prce dscrnaton when dfferent unts of the sae good are sold at dfferent prces ether to the sae consuer or to dfferent consuers. Dscusson - Dfferences n qualty: passenger transport, cultural or sportng events etc. - A sngle prce ay be dscrnatory and dfferent prces not. We say that there s no prce dscrnaton when the dfference between the prces pad by two consuers for a unt of the good eactly responds to the dfference n the cost of provdng the wth the good. () The ncentve to dscrnate prces At the proft-azng level of output argnal revenue equals argnal cost, r ( ) C ( ); =.e., an nfntesal change n the level of output changes revenue and cost equally. That s: 5

28 p( ) + p( ) = C ( ) () Addtonal revenue fro sellng an addtonal unt. Loss of revenue fro sellng unts already produced at a lower prce. The onopolst would be reluctant to sell ore unts f t does not have to reduce the prce. Therefore, there are ncentves to try to capture a hgher proporton of the consuer surplus ncentves to dscrnate prces. p C ( ) p The ncentve to dscrnate prces: to capture a hgher proporton of socal surplus. Π r ( ) p ( ) () Condtons Two condtons are needed for a fr to be able to dscrnate prces: a) The fr ust be able to classfy consuers (whch depends on nforaton). b) The fr ust be capable of preventng the resell of the good (whch depends on the possbltes of arbtrage and on transacton costs). 6

29 The splest case occurs when a fr receves an eogenous sgn (age, locaton, occupaton, etc.) whch allows t to classfy consuers nto dfferent groups. It s ore dffcult to classfy accordng to an endogenous category (e.g., quantty purchased or the te of purchase). In that case the onopolst ust establsh prces n such a way that consuers classfy theselves n the correct categores. (v) Types of prce dscrnaton (Pgou, 90) ) Frst-degree prce dscrnaton or perfect dscrnaton. The seller charges a dfferent prce for each unt equal to the au wllngness to pay for that unt. Ths requres full nforaton concernng consuer preferences and no arbtrage. The onopolst succeeds n etractng the coplete consuer surplus. ) Second-degree prce dscrnaton (or nonlnear prcng). Prces dffer dependng on the nuber of unts of the good but not across consuers. Each consuer faces the sae prce catalogue but prces depend on the quantty purchased (or on another varable, e.g., product qualty). Eaples: volue dscounts. Self selecton. 3) Thrd-degree prce dscrnaton. Dfferent prces are charged to dfferent consuers but each consuer pays a constant aount (the sae prce) for each unt. The fr receves an eogenous sgn whch allows t to classfy consuers nto dfferent groups. Ths s the ost frequent type of prce dscrnaton. Eaples: dscounts for students, senor ctzens, etc. Identfcaton. 7

30 Another way of classfyng prce dscrnaton s to dstngush between drect prce dscrnaton and ndrect prce dscrnaton. Second-degree prce dscrnaton s a case of ndrect dscrnaton (consuers face a unque prce schedule and they classfy theselves by ther choces) whle frst-degree prce dscrnaton and thrd-degree prce dscrnaton would be drect dscrnaton. In the case of thrd-degree prce dscrnaton the fr gves dfferent prce enus for consuers belongng to dfferent groups or arkets. (v) Eaples It s ore dffcult to fnd real arkets where there s no prce dscrnaton than arkets where such dscrnaton ests. Although t s often not possble to dstngush clearly what type of prce dscrnaton ests t s an nterestng eercse to thnk about what type of prce dscrnaton s been practced n the followng cases. - Two-part tarffs: telephone, Internet, electrcty, cable televson, etc. - Dfferent electrcty rates for ndustral use and doestc use. - Dscounts n useus, agazne subscrptons, cultural and sportng events, for chldren, young people or senor ctzens. - Volue dscount n publc transport. - Qualty dfferences: dfferent prces dependng on the qualty of the product n cultural or sportng events, passenger transport (trans, etc.). - Dscounts for repeated buyng. -, 3, etc. n superarkets, etc. - Hoe-servce food, tele-shoppng etc. 8

31 (v) The odel We study the three types of prce dscrnaton by usng a very sple odel. Assue that there are two potental consuers wth quas-lnear utlty functons: u ( ) + y, =,. u (0) = 0, =,. u ( ) : au wllngness to pay of consuer =,. u ( ) : argnal wllngness to pay of consuer =,. We say that the consuer s a hgh-deand consuer and that the consuer s a lowdeand consuer f the followng s satsfed: u ( ) > u ( ) u ( ) > u ( ) Thus, consuer s a hgh-deand consuer and consuer s a low-deand consuer f both the au wllngness to pay and the argnal wllngness to pay are hgher for consuer than for consuer for any quantty of the good. The coparson between consuers of au wllngness to pay and argnal wllngness to pay only akes sense for the sae level of output. Moreover, the coparson has to be ade for any level of output. p u ( ) u ( ) u ( ) u ( ) u ( ) u ( ) u ( ) > u ( ) u ( ) > u ( ) 9

32 The argnal cost of the onopolst s assued to be constant (and there are no fed costs) c > 0. In an equvalent way, we can see the cost functon as: C ( ) = c. = c.( + )..6. Frst-degree prce dscrnaton or perfect prce dscrnaton () Defnton and contet. () The case of a sngle consuer. () Observatons. Is the quantty suppled by the onopolst effcent? (v) The case of two consuers. (v) Does the onopolst supply effcent outputs to consuers? The onopolst supples a hgher quantty to the hgh-deand consuer (proof). (v) What would happen f the onopolst were not able to dentfy consuers? () Defnton and contet The seller charges a dfferent prce for each unt of product and equals the au wllngness to pay for that unt. Ths requres full nforaton on consuer preferences and no arbtrage of any knd. In partcular, the onopolst needs to be able to dentfy consuers when they buy the good. (Classc eaple: a vllage doctor). 30

33 () The case of a sngle consuer The onopolst supples a prce-quantty bundle ( r *, * ) whch azes profts. The onopolst proposes a take t or leave t choce: * * ( r, ) (0,0). The consuer ether pays * r for * unts or does not receve the good. The azaton proble of the onopolst s: a r c r, sau. ( ) r () Constrant () can be equvalently wrtten as u ( ) r 0 : the consuer has to obtan a nonnegatve surplus fro good. Ths type of constrant s known as partcpaton restrcton or ndvdual ratonalty restrcton. Gven that the onopolst wshes to aze profts t wll choose the hghest possble tarff r and, therefore, condton () wll be satsfed as equalty: r = u( ). The proble thus conssts of: Π( ) a u ( ) c dπ = = = d d Π u ( ) 0 = < d * u ( ) c 0 u ( ) c Gven ths level of output the tarff wll be: r = u( ). * * () Observatons a) Is the quantty suppled by the onopolst effcent? The onopolst produces a Pareto-effcent output, * e =, gven that t supples a quantty such that the argnal wllngness to pay equals the argnal cost. (Revew the proble of 3

34 azng socal welfare and copare wth the proble we have just solved). However, the onopolst obtans the entre socal surplus. p * * * e e e ( ) u( ) c u( ) c W( ) Π =Π = = = Π c * e u ( ) = C ( ) b) The onopolst produces the sae quantty that t would produce f t behaved as a perfectly copettve fr. If t took prce as a paraeter then ts output decson would be p( ) = cbut gven that utlty s quas-lnear then p ( ) u ( ) = and consequently u ( ) = c. However, the dstrbuton of trade gans would be just the opposte. c) We ght obtan the sae results by usng a two-part tarff. p = + = + * * T( ) A p u( ) c c A * * * Π = T ( ) c = u( ) c c Π u ( ) = * e C ( ) 3

35 d) We would obtan the sae result f the onopolst sold each unt to the consuer at a dfferent prce equal to hs/her au wllngness to pay for that unt. Assue that we break producton down nto n equal portons of sze Δ so as = nδ. The au wllngness to pay for the frst unt (of sze Δ ) s gven by: u(0) + = u( Δ ) + p u(0) = u( Δ) p The au wllngness to pay for the second unt s: u( Δ ) + p = u( Δ ) + p p u( Δ ) = u( Δ) p And so on. We would obtan the followng sequence of equatons: u(0) = u( Δ) p u( Δ ) = u( Δ) p u( Δ ) = u(3 Δ) p... u(( n ) Δ ) = u( nδ) pn 3 Addng and takng nto account that u (0) = 0 we get un ( Δ ) = p. When the sze of the = unts becoes nfntesal, we obtan that proposng a take t or leave t choce to the consuer s equvalent to sellng h/her each (nfntesal) unt at a prce equal to the argnal wllngness to pay for t. n p * u ( ) = u 0 ( zdz ) * p( z) c * u ( ) C ( ) 33

36 (v) The case of two consuers The onopolst supples consuer, =,, wth a prce-output bundle * * (, ) r n order to aze profts. The onopolst gves consuer, =,, a take t or leave t choce: * * ( r, ) (0,0). Consuer, =,, ether pays r * for * unts or does not receve the good. The azaton proble of the onopolst s: a r + r c.( + ) r,, r, sa. u( ) - r 0 u ( ) - r 0 r = u ( ) r = u ( ) proft azaton Therefore, the proble becoes: a u ( ) + u ( ) c.( + ), Π = u( ) c= 0 * * u( ) = u( ) = c Π = u( ) c= 0 Gven these levels of output the tarffs are: r = u ( ) and * * r = u ( ). * * (v) Does the onopolst supply effcent outputs to consuers? The onopolst supples a hgher quantty to the hgh-deand consuer (proof) The onopolst offers effcent outputs: * e = and * e. = (Revew the proble of obtanng a Pareto-effcent allocaton and copare wth the proble we have just solved) 34

37 We net deonstrate that the onopolst offers a hgher quantty to the hgh-deand consuer: * > *. * u( ) = c * * * u( ) = u( ) < u( ) * u( ) = c Consuer s the hghdeand consuer: u ( ) > u ( ) Therefore, * * u( ) < u( ) but gven that functon u s strctly concave then du ( ( )) < 0 d and n consequence * > *. (v) What would happen f the onopolst were not able to dentfy consuers? (Ths subsecton serves to ntroduce the analyss of second-degree prce dscrnaton). Assue now that the onopolst s not able to dentfy consuers when they go to buy the good. That s, the onopolst cannot propose personalzed supples and s therefore restrcted to statng a sngle prce enu. Assue that t states a prce enu by usng the tarffs and quanttes whch are optal under perfect prce dscrnaton: ( r, ) * * * * ( r, ) (0,0) where r = u ( ) and r * = u ( * ). We can see that the hgh-deand consuer has * * ncentves to buy the bundle desgned for the low-deand consuer. 35

38 * * * * * * 0 = u( ) r < u( ) r = u( ) u( ) > 0 Incentve to engage n personal arbtrage. The surplus obtaned by consuer f he/she buys the bundle desgned for h. The surplus that consuer would obtan f he/she buys the bundle desgned for consuer..7. Second-degree prce dscrnaton (or non-lnear prcng) (Keywords: no dentfcaton, unque prce enu and self selecton). () Defnton and contet. () Partcpaton restrctons and self selecton restrctons. Interpretaton. () Deonstraton of what constrants are satsfed wth equalty. Interpretaton. (v) The proft azaton proble. (v) Observatons. Does the onopolst supply effcent quanttes? The onopolst offers a lower-than- effcent quantty to the low-deand consuer (Proof). (v) Under what condtons does the onopolst offer the good to both consuers? (v) Graphc representaton. () Defnton and contet The prces dffer dependng on the nuber of unts bought but not fro one consuer to another. We consder a contet where the onopolst knows the preferences of the consuers (t knows the preference dstrbuton functon) but s unable to dentfy consuers when they 36

39 go to buy the good. So the fr s oblged to establsh a unque prce enu and to allow consuers to self classfy or self select. In ths sense we can say that there s ndrect prce dscrnaton. The consuers face the sae prce schedule but prces depend on quantty (or soe other varable, e.g. the qualty of the good) bought. () Partcpaton restrctons and self selecton restrctons. Interpretaton The objectve of the onopolst s to optally desgn the prce enu n such a way that each consuer chooses the prce-quantty bundle desgned for h/her. ( r, ) ( r, ) (0,0) Consuer Consuer Restrctons for the onopolst - Partcpaton restrctons (or ndvdual ratonalty constrants) u ( ) r 0 () u ( ) r 0 () These restrctons guarantee that each consuer wshes to buy the good. Each consuer obtans at least as uch utlty by consung the good as by not consung. Put dfferently, each consuer obtans a non-negatve surplus by purchasng the good. - Self selecton restrctons (or ncentve copatblty constrants) u ( ) r u ( ) r (3) u ( ) r u ( ) r (4) 37

40 These restrctons guarantee that each consuer prefers the prce-quantty bundle desgned for h/her to the prce-quantty bundle desgned for the other consuer. Put dfferently, these constrants avod personal arbtrage: each consuer gets as least as great a surplus by choosng the bundle desgned for h/her as he/she does by choosng the bundle desgned for the other consuer. () Deonstraton of what constrants are satsfed wth equalty. Interpretaton We now arrange constrants accordng to each consuer. () y (3) r u ( ) () ( ) ( ) () r u u + r () y (4) r u ( ) (3) ( ) ( ) (4) r u u + r The onopolst wshes to aze profts and wll therefore choose the hghest possble r and r. As a consequence, only one of the frst two nequaltes and only one of the second two nequaltes wll be bndng (that s, they wll be satsfed wth equalty). The assupton that consuer s the hgh-deand consuer and consuer the low-deand consuer ( u ( ) > u ( ) and u ( ) > u ( ) ) s suffcent to deterne what constrants are bndng. ) Deonstraton that (4) s satsfed wth equalty and (3) wth strct nequalty. Assue that (3) s satsfed wth equalty and, therefore, that r = u ( ). Then (4) r r u ( ) + r r u ( ). Gven that consuer s the hgh-deand consuer u( ) > u( ) then r u( ) > u( ). That s, r > u ( ) whch eans that restrcton 38

41 () would not be satsfed whch s a contradcton. (The fact that the partcpaton constrant of the hgh-deand consuer s satsfed wth equalty s not copatble wth the fact that the low-deand consuer buys the good). As a concluson, (3) s not bndng and (4) s satsfed wth equalty: r = u ( ) u ( ) + r (5) ) Deonstraton that () s satsfed wth equalty and () wth strct nequalty Assue that condton () s satsfed wth equalty and, therefore, that r = u( ) u( ) + r. By substtutng r fro condton (5) we get: r = u( ) u( ) + u( ) u( ) + r = r whch ples u ( ) u ( ) = u ( ) u ( ) u t dt = () u () t dt [ u ( t) u ( t)] dt = 0 But ths contradcts the assupton that consuer s the hgh-deand consuer, u ( ) > u ( ). Therefore, () s not bndng and () s satsfed wth equalty: r = u ( ) (6) Interpretaton The onopolst charges consuer a tarff equal to hs au wllngness to pay gven that the low-deand consuer has no ncentve to engage n personal arbtrage. Gven that 39

42 the hgh deand consuer has ncentve to engage n personal arbtrage (and to c the low-deand consuer) the onopolst charges h/her the au prce that nduces h/her to choose the bundle desgned for h/her (the aount of oney that just leaves h/her ndfferent between hs/her bundle and that desgned for the low-deand consuer). We now show (n a dfferent ore ntutve way) why the onopolst ust provde a postve surplus to the hgh-deand consuer. Consder the self selecton constrant for the hgh-deand consuer: u ( ) r u ( ) r (4) Note that the rght sde of ths constrant s postve condtonal on the low-deand consuer s wshng to buy the good. That s, f we choose the au value for r condton (4) would be: u ( ) r u ( ) u ( ) > 0 gven that consuer s the hgh-deand consuer (whch ples that the partcpaton restrcton of consuer cannot be satsfed wth equalty). But gven that the onopolst ust allow the hgh-deand consuer to obtan a postve surplus, t decdes to leave the consuer wth the nu possble surplus, just that aount such that the hgh-deand consuer s ndfferent between hs/her bundle and the bundle desgned for consuer. That s, rearrangng restrcton (5): u ( ) r = u ( ) u ( ) > 0 Gven that the low-deand consuer has no ncentve to engage n personal arbtrage the onopolst charges h/her the au that he/she s wllng to pay r = u ( ). 40

43 (v) The proft azaton proble a r + r c.( + ) a r + r c.( + ) r,, r, r,, r, u( )-r 0 () r = u( ) (6) sa. sa. u( )-r 0 () r = u( ) [ u( ) r ] (5) u ( )- r u ( )- r (3) u ( )- r u ( )- r (4) By substtutng we get:, Π(, ) a u ( ) + u ( ) [ u ( ) u ( )] c.( + ) Π = u( ) c [ u( ) u( )] = 0 (7) Π = u( ) c = 0 (8) The tarffs are gven by: r = u ( ) r = u ( ) [ u ( ) u ( )] (v) Observatons ) The onopolst provdes the hgh-deand consuer wth the effcent quantty and leaves h/her wth a postve surplus. Condton (8) ples u ( ) = c and, therefore, the onopolst offers the effcent quantty e to the hgh-deand consuer = (revew Pareto-effcency condtons). Moreover, the onopolst charges h/her a prce (a tarff) lower than hs/her au wllngness to pay leavng h/her wth a postve surplus equals to that whch he/she would obtan f he/she 4

44 chose the bundle desgned for consuer. r = u( ) [ u( ) u( )] and hs/her surplus would thus be: u( ) r = [ u( ) u( )]. ) The onopolst offers the low-deand consuer a quantty lower than the effcent quantty and leaves h/her wth no surplus. Π = u( ) c [ u( ) u( )] = 0 (7) > 0 Gven that consuer s the hgh-deand consuer [ u ( ) u ( )] > 0 and then fro condton (7) we get u ( ) > c. By defnton, the effcent output satsfes u ( ) = c, and as e a consequence ( ) e u > u ( ). The au wllngness to pay s a strctly concave functon: e u( ) > u( ) < du ( ( )) < 0 d e We net look at the ntuton of ths result. We nterpret the argnal proft of and evaluate t at dfferent producton levels. Π = u ( ) c [ u ( ) u ( )] * > 0( < ) > 0 Margnal proft fro consuer : a change n the quantty suppled to ths consuer ples a change n the proft obtaned by the onopolst fro h/her. Margnal proft fro consuer : a change n the quantty suppled to consuer ples a change n the surplus the onopolst ust leave consuer to avod personal arbtrage. 4

45 * Π( ) * * * = u( ) c [ u( ) u( )] < 0 = 0 > 0 Startng fro * a reducton n the quantty suppled to consuer ncreases the proft because the surplus that the onopolst ust leave consuer to avod arbtrage s reduced. An output such that < < satsfes the followng: * Π( ) = u( ) c [ u( ) u( )] < 0 > 0 > 0 It s worthwhle for the onopolst to contnue reducng because the ncrease n profts fro the hgh-deand consuer (obtaned by leavng h/her wth a lower surplus) offsets the loss of profts fro the low-deand consuer obtaned by supplyng h/her a lower quantty. Π( ) = u( ) c [ u( ) u( )] = 0 > 0 In output the argnal gan, fro an nfntesal reducton n, fro the hgh-deand consuer by leavng h/her wth lower surplus s just equal to the argnal loss fro the low-deand consuer as a result of offerng a lower quantty. Moreover, the onopolst charges the low-deand consuer a prce (tarff) equal to the au wllngness to pay, thus leavng h/her wth no surplus: r = u ( ). (v) Under what condtons does the onopolst offer the good to both consuers? The onopolst wll decde to offer the good to both consuers f t obtans ore profts than by sellng the good only to the hgh-deand consuer. That s, the onopolst supples the good to both consuers f the followng s satsfed: 43

46 Π(0, ) Π(, ) * u ( ) c u ( ) c + u ( ) [ u ( ) u ( )] c * * * * r r [ u ( ) u ( )] u ( ) c If ths condton s not satsfed, the onopolst offers the good only to the hgh-deand consuer. Another equvalent way of lookng at the proble conssts of consderng the argnal proft of. If t were negatve for any level of Π( ) = u ( ) c [ u ( ) u ( )] < 0 > 0 > 0 then the onopolst would decde not to sell the good to the low-deand consuer gven that for any level of t would ncrease profts by reducng the quantty suppled to the low-deand consuer. (v) Graphc representaton (zero argnal cost) p u ( ) u ( ) = c = 0 * u ( ) = c = 0 * A B C u ( ) * * 44

47 Perfect prce dscrnaton * * ( r, ) (0,0) =, * * u( ) = u( ) = 0 c r = u ( ) A * * r = u ( ) A+ B+ C * * * * * Π = u( ) + u( ) A+ A+ B+ C * r * r No dentfcaton Assue that the onopolst does not know the dentty of the consuer and that t states a unque prce enu where t antans the prce-quantty bundles whch were optal under perfect prce dscrnaton. Consuer would have ncentves to engage n personal arbtrage. * * ( r, ) A * * ( r, ) A+ B+ C (0,0) Consuer Consuer 0 = A+ B+ C ( A+ B+ C) < A + B A= B * * * u( ) * u( ) r r Second-degree prce dscrnaton The followng condtons are satsfed wth equalty: r = u ( ) A( ) the onopolst charges consuer the area below hs/her nverse deand functon. u( ) r = u( ) r B( ) the onopolst leaves consuer wth a surplus B( ) (the nu) n order to avod arbtrage. 45

48 Frstly, we antan quanttes and only adjust the tarffs. * ( r, ) A * ( r, ) A+ C (0,0) Π (, ) = A+ C * * Π (, ) = A + A+ B+ C B * Π(, ) Π(, ) ( A A) + ( B B) > 0 * * * p u ( ) u ( ) B u ( ) u ( ) c A A B C u ( ) * * ( r, ) ( r, ) (0,0) Π( ) = u( ) c [ u( ) u( )] = 0 > 0 As we are assung that the argnal cost s zero: Π( ) = u( ) [ u( ) u( )] = 0 u( ) = u( ) u( ) > 0 46

49 p u ( ) u ( ) u ( ) B u ( ) u ( ) c = 0 u ( ) A C u ( ) * * * * Π (, ) = u( ) c + u( ) [ u( ) u( )] c A + A+ B+ C B * 0 0 The decson to supply the good only to the hgh-deand consuer. p Π( ) = u ( ) c [ u ( ) u ( )] < 0 > 0 > 0 B u ( ) ( r, ) * * A+ B+ C (0,0) A C u ( ) * * 47

50 .8. Thrd-degree prce dscrnaton () Defnton and contet. () Proft azaton. The rule of the nverse of elastcty. () A coparson of profts wth the case of unfor prcng (sngle onopoly prcng). (v) Effects on socal welfare. ()Defnton and contet There s thrd-degree prce dscrnaton when consuers belongng to dfferent groups or subarkets are charged dfferent prces, although each consuer pays the sae prce for each unt bought. Ths s probably the ost coon type of prce dscrnaton. Eaples: dscounts to students, senor ctzens etc. The onopolst receves an eogenous sgn whch allows t to dstngush perfectly separated arkets or subarkets: = 0. p j Ths s a type of drect dscrnaton: the onopolst states dfferent prce enus for consuers belongng to dfferent groups or arkets. Identfcaton: the onopolst classfes each consuer n a group. () Proft azaton. The rule of the nverse of elastcty We consder the sple case of = : the onopolst classfes consuers n two groups wth nverse deand functons p ( ) and p p < = The onopolst ( ), wth ( ) 0,,. can establsh dfferent prces n the two arkets but wthn a arket t s not possble to dscrnate prces. The azaton proble s: 48

51 , Π(, ) a p ( ) + p ( ) c.( + ) Π = p( ) + p( ) c= 0 () () MR = MR = c Π = p( ) + p( ) c = 0 () p + p = c ( ) ( ) ( ) p ( ) p( )[ + ] = c p ( ) p( )[ + ] c ε ( ) = p( )[ ] = c ε ( ) c p( ) = =,. ε ( ) Therefore, p( ) > p( ) ff ε( ) < ε( ). As a consequence, the onopolst charges the hghest prce n the arket wth the lower prce elastcty (n absolute value). () A coparson of profts wth the case of unfor prcng (sngle onopoly prcng) The onopolst s proft under thrd-degree prce dscrnaton s at least as hgh as the proft under unfor prcng. The reason s sple: under thrd-degree prce dscrnaton the fr can always choose equal prces f that s the ost proftable opton. 49

52 (v) Effects on socal welfare ) What s the proble? ) Bounds of the change n socal welfare. 3) Applcatons: a) Lnear deand. b) Openng of arkets. ) What s the proble? Ths secton copares thrd-degree prce dscrnaton and unfor prcng fro a socal welfare pont of vew. In general, a oveent fro unfor prcng to thrd-degree prce dscrnaton benefts soe agents and hars others. Benefted by T-DPD: the onopolst and the consuers n the hgher-elastcty arket (gven that the prce s reduced by dscrnaton). Hared by T-DPD: the consuers n the lower-elastcty arket (gven that the prce s ncreased by dscrnaton). Therefore, the effect on socal welfare s ndeternate. ) Bounds of the change n socal welfare Assue for the sake of splcty that there are only two arkets and we start fro an aggregate utlty functon u ( ) + u ( ) + y + y, where and are the consuptons of good by the two groups and y s the oney to be spent on other goods ( y = y+ y). 50

53 u and u are strctly concave. The nverse deand functons are gven by p ( ) = u ( ) and p ( ) = u ( ). If C (, ) s the cost of supplyng and we can easure the socal welfare as: W(, ) = u ( ) + u ( ) C(, ) Consder two confguratons of output (, ) and 0 0 (, ) whose prces are ( p, p ) and 0 0 ( p, p ), respectvely. Assue that the ntal set of prces corresponds to unfor prcng (the onopoly sngle prce) p 0 = p 0 = p 0 and that p and p are the prces under thrd-degree prce dscrnaton. Consder the oveent fro we have (see Append): 0 to. Due to the strctly concavty of u 0 0 p( ) = p Δ u( ) < u( ) + u( ) ( ) () Δ u < pδ 0 p Δ >Δ u > p Δ 0 0 u( ) < u( ) + u ( ) ( ) () Δ u > pδ p( ) = p Δ 0 0 p ( ) = p Δ u( ) < u( ) + u( ) ( ) () Δ u < pδ 0 p Δ >Δ u > p Δ 0 0 u( ) < u( ) + u( )( ) () Δ u > pδ p( ) = p Δ (3) (4) By addng (3) and (4) we get p Δ + p Δ >Δ u +Δ u > p Δ + p Δ 0 0 where 5

54 Δ u =Δ u +Δu ; Δ = ; Δ = 0 0 p = p ( ) = u ( ); p = p ( ) = u ( ); p = p ( ) = u ( ); p = p ( ) = u ( ). The change n socal welfare s gven by: Δ W = W(, ) W(, ) = u( ) u( ) + u( ) u( ) [ C(, ) C(, )] Δu Δu ΔC =Δ u +Δu ΔC Therefore p Δ + p Δ Δ C >Δ W > p Δ + p Δ Δ C 0 0 If argnal cost s constant: Δ C = c( + ) c( + ) = cδ + cδ 0 0 Therefore the bounds of the change n socal welfare becoe: 0 0 ( p c) Δ + ( p c) Δ >Δ W > ( p c) Δ + ( p c) Δ (5) Upper bound Lower bound Gven that p = p = p the bounds of the change n socal welfare are: Δ 0 ( p c)( Δ +Δ ) >Δ W > ( p c) Δ + ( p c) Δ (6) Upper bound Lower bound - Upper bound: ths ples that a necessary condton for thrd-degree prce dscrnaton to ncrease socal welfare, Δ W > 0, s that t should ncrease total output. Assue on the contrary that Δ =Δ +Δ 0. Gven that 0 ( p c) 0 > then (4) Δ W < 0. - Lower bound: ths ndcates that a suffcent condton for thrd-degree prce dscrnaton to ncrease socal welfare s that the su of the changes n output weghted by the dfference between the prce under dscrnaton and the argnal cost ust be postve. Graphcally, for the case of a sngle arket, the bounds would be: 5

55 p 0 0 p p ( p c) Δ >Δ W > ( p c) Δ c 0 3) Applcatons a) Lnear deands Assue that the deands are gven by a ( p) = p,,, b b = and that the argnal cost s zero, c = 0. The proft azaton proble under thrd-degree prce dscrnaton s: a p( p) + p( p) p, p Π a a a = ( p ) + p ( p ) = 0 p p = 0 p = ; = p b b b b Π a a a = ( p ) + p ( p ) = 0 p p = 0 p = ; = p b b b b The total output s: a a ab + a b = + = + = b b bb Under unfor prcng: a p ( p) + p ( p) Π a a = ( p) + ( p) + p ( p) + p ( p) p + p p p = 0 p b b b b b b 0 ab + ab p = ; ( b + b ) p 53

56 a ab + a b ab + ab ab a b ab + ab a b = = = 0 b b ( b+ b) b( b+ b) b( b+ b) a a b + ab a b + a b a b ab a b + a b ab = = = 0 b b ( b+ b) b( b+ b) b( b+ b) The total output s: ab+ ab ab ab + ab ab = + = b( b+ b) b( b+ b) abb + a( b ) abb + abb + a ( b) abb = bb ( b + b ) abb + a( b) + abb + a( b) ( ab + ab )( b+ b) ab + ab = = = bb ( b + b ) bb ( b + b ) bb Therefore, total output s the sae under both prcng polces. That s, Δ =Δ +Δ = 0, or, equvalently, Δ. = Δ The bounds would be Δ 0 ( p c)( Δ +Δ ) >Δ W > ( p c) Δ + ( p c) Δ (6) = 0 < 0 Socal welfare therefore decreases: Δ W < 0. As we show below, the above result depends crucally on the assupton that all arkets are served under unfor prcng. 54

57 b) Openng of arkets Iagne that the two arkets deands are lke those n the graphc. p 0 p = p p ( p ) ( p ) If the onopolst had to sell at a unfor prce, t would have to reduce the prce n arket by such an aount that the decrease n profts n that arket would not be offset. Therefore, ( p c)( Δ +Δ ) >Δ W > ( p c) Δ + ( p c) Δ (6) 0 p = 0 > 0 = 0 > 0 > 0 = 0 > 0 > 0 Gven that the lower bound n (4) s postve then Δ W > 0. But not only does socal welfare ncrease, n fact thrd-degree prce dscrnaton Pareto donates unfor prcng. A ove fro unfor prcng to thrd-degree prce dscrnaton ples an ncrease n the onopolst s profts, an proveent for consuers n arket and no change for consuers n arket. 55

58 Append If u s a strctly concave functon for any and y the followng s satsfed: u uy u y y ( ) < ( ) + ( )( ). The tangents always rean above the functon when t s strctly concave. u u ( ) u( y ) u ( ) uy ( ) u lnear u ( y) = y u u u y y u y lnear ( ) = ( ) + ( ) ( ) u u u y y u y strctly concave ( ) < ( ) + ( ) ( ) y 56

59 Chapter. Non-Cooperatve Gae Theory Introducton The Theory of Non-Cooperatve Gaes studes and odels conflct stuatons aong econoc agents; that s, t studes stuatons where the profts (gans, utlty or payoffs) of each econoc agent depend not only on hs/her own acts but also on the acts of the other agents. We assue ratonal players so each player wll try to aze hs/her proft functon (utlty or payoff) gven hs/her conjectures or belefs on how the other players are gong to play. The outcoe of the gae wll depend on the acts of all the players. A fundaental characterstc of non-cooperatve gaes s that t s not possble to sgn contracts between players. That s, there s no eternal nsttuton (for eaple, courts of justce) capable of enforcng the agreeents. In ths contet, co-operaton aong players only arses as an equlbru or soluton proposal f the players fnd t n ther best nterest. For each gae we try to propose a soluton, whch should be a reasonable predcton of ratonal behavour by players (OBJECTIVE). We are nterested n Non-Cooperatve Gae Theory because t s very useful n odellng and understandng ult-personal econoc probles characterzed by strategc nterdependency. Consder, for nstance, copetton between frs n a arket. Perfect copetton and pure onopoly (not threatened by entry) are specal non-realstc cases. It s ore frequent n real lfe to fnd ndustres wth not any frs (or wth a lot of frs but wth just a few of the producng a large part of the total producton). Wth few frs, copetence between the s characterzed by strategc consderatons: each fr takes ts decsons (prce, output, advertsng, etc.) takng nto account or conjecturng the behavour of the others. Therefore, copetton n an olgopoly can be seen as a non-cooperatve gae where the frs are the players. Many predctons or soluton proposals arsng fro Gae 57

60 Theory prove very useful n understandng copetton between econoc agents under strategc nteracton. Secton defnes the an notons of Gae Theory. We shall see that there are two ways of representng a gae: the etensve for and the strategc for. In Secton 3 we analyze the an soluton concepts and ther probles; n partcular, we study the Nash equlbru and ts refneents. Secton 4 analyzes repeated gaes and, fnally, Secton 5 offers concludng rearks... Basc notons There are two ways of representng a gae: the etensve for and the strategc for. We start by analyzng the an eleents of an etensve for gae.... Gaes n etensve for (dynac or sequental gaes) An etensve fro gae specfes: ) The players. ) The order of the gae. 3) The choces avalable to each player at each turn of play (at each decson node). 4) The nforaton held by each player at each turn of play (at each decson node). 5) The payoffs of each player as a functon of the oveents selected. 6) Probablty dstrbutons for oveents ade by nature. An etensve for gae s represented by a decson tree. A decson tree coprses nodes and branches. There are two types of node: decson nodes and ternal nodes. We have to 58

61 assgn each decson node to one player. When the decson node of a player s reached the player chooses a ove. When a ternal node s reached the players obtan payoffs: an assgnent of payoffs for each player. EXAMPLE : Entry gae Consder a arket where there are two frs: an ncubent fr, A, and a potental entrant, E. At the frst stage, the potental entrant decdes whether or not to enter the arket. If t decdes not to enter the gae concludes and the players obtan payoffs (fr A obtans the onopoly profts) and f t decdes to enter then the ncubent fr, A, has to decde whether to accoodate entry (that s, to share the arket wth the entrant) or to start a utually njurous war prce. The etensve for gae can be represented as follows: β E α NE E (0, 0) Ac. α A G.P. (4, 4) β (-, -) β Players: E and A. Actons: E (to enter), NE (not to enter), Ac. (to accoodate), G.P. (prce war). Decson nodes: α. Ternal nodes: β. 59

62 (, y): vector of payoffs. : payoof of player E; y: payoff of player A. At each ternal node we have to specfy the payoffs of each player (even though soe of the have not actually anaged to play). Assuptons: () All players have the sae percepton of how the gae s. () Coplete nforaton: each player knows the characterstcs of the other players: preferences and strategy spaces. () Perfect recall: each player reebers hs/her prevous behavour n the gae. Defnton : Inforaton set The nforaton avalable to each player at each one of hs/her decson nodes. I D L A (.,.) M (.,.) B R (.,.) S (.,.) I D L M L M (.,.) (.,.) (.,.) (.,.) Gae Gae In gae, player has dfferent nforaton at each one of hs/her decson nodes. At node A, f he/she s called upon to play he/she knows that the player has played I and at B he/she 60