Market Power and Strategy

Transcription

1 Notes on Market Power and Strategy Aprl 03 Iñak Agurre Departamento de Fundamentos del Análss Económco I Unversdad del País Vasco

2 Inde Chapter. Monopoly Introducton.. Proft mamzaton by a monopolstc frm... Lnear demand and constant elastcty demand..3. Comparatve statcs..4. Welfare and output..5. Prce dscrmnaton..6. Frst-degree prce dscrmnaton..7. Second-degree prce dscrmnaton..8. Thrd-degree prce dscrmnaton. Chapter. Non-Cooperatve Game Theory Introducton... Basc notons... Etensve form games.... Strategc form games... Soluton concepts for non-cooperatve game theory.... Domnance crteron.... Backward nducton crteron...3. Nash equlbrum...4. Problems and refnements of Nash equlbrum..3. Repeated games.

3 .3.. Fnte temporal horzon..3.. Infnte temporal horzon..4. Conclusons. Chapter 3. Olgopoly Introducton 3.. The Cournot model Duopoly Olgopoly (n frms) Welfare analyss. 3.. The Bertrand model Homogeneous product Heterogeneous product Leadershp n the choce of output. The Stackelberg model Colluson and the stablty of agreements Short-term colluson The stablty of agreements under a fnte temporal horzon and under an nfnte temporal horzon.

4 Chapter. Monopoly Introducton We say that a frm s a monopoly f t s the only seller of a good (or goods) n a market. Problem: t s not easy to defne good and market. A frm may become a monopoly by varous reasons: - Control over raw materals. - Acquston of the eclusve sellng rghts (by a patent, by a publc aucton etc.). - Better access to the captal market. - Increasng returns of scale etc. In contrast wth a perfectly compettve frm whch faces a perfectly elastc demand (takng prce as gven), a monopolst faces the market demand. Therefore, a frm wth monopolstc power n a market t s aware of the amount of output that t s be able to sell t s a contnuous functon of the prce charged. Put dfferently, the monopolstc frm takes nto account that a reducton n output wll ncrease the prce that can be charged. In consequence, a monopolst has the power to set the market prce. Whle we can consder a compettve frm as a prce taker, a monopolst s prce decson-maker or prce setter... Proft mamzaton () The problem of proft mamzaton n prces and n quanttes. Frst order condtons. Second order condtons. A graphcal nterpretaton of the proft mamzaton problem. () Interpretaton of margnal revenue. 3

5 () Margnal revenue equals margnal cost condton. (v) Output and demand elastcty. (v) Lerner Inde of monopolstc power. (v) Graphcal analyss. (v) Second order condtons. () The problem of proft mamzaton n prces and n quanttes There are two types of constrant that restrct the behavour of a monopolst: a) Technologcal constrants summarzed n the cost functon C(). b) Demand constrants: (p). We can wrte the proft functon of the monopolst n two alternatve ways: - Π ( p) = p( p) C( ( p)) by usng the demand functon. - Π ( ) = p( ) C( ) by usng the nverse demand functon. The demand, (p), and the nverse demand, p(), represent the same relatonshp between prce and demanded quantty from dfferent ponts of vew. The demand functon s a complete descrpton of demanded quantty at each prce whereas the nverse demand gves us the mamum prce at whch a gven output may be sold n the market. ma Π( p) ma Π( ) p 0 m m p m m m m = ( p ) p = p( ) 4

6 The problem of proft mamzaton as a functon of prce ma Π( p) ma p( p) C( ( p)) p p Π p = p + p p C p p = ( ) ( ) ( ) ( ( )) ( ) 0 Π ( p) = ( p) + p ( p) C ( ( p)) ( p) C ( ( p)) ( p) < 0 The problem of proft mamzaton as a functon of the output ma Π( ) ma p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p ( ) C ( ) = 0 Π ( m ) = 0 Frst order condton. Π ( ) = p ( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). Π m Π ( ) = 0 Π (0) > 0 m Π( ) 5

7 () Interpretaton of margnal revenue Margnal revenue, r ( ), s: r ( ) = p( ) + p ( ) () Addtonal revenue from sellng an addtonal unt. Loss of revenue from sellng unts already produced at a lower prce. () Margnal revenue equals margnal cost condton The proft-mamzng output level (nteror soluton) satsfes: Π ( m ) = r ( m ) C ( m ) = p( m ) + p ( m ) C ( m ) = 0 () At the monopolstc optmal output the margnal proft s zero, Π ( m ) = 0; that s, an nfntesmal change n the level of output mantans proft unchanged. An output level such that Π (.) > 0 does not mamze profts: an (nfntesmal) ncrease n output would ncrease profts. In a smlar way, a level of output such that profts: a (nfntesmal) decrease n output would ncrease profts. Π (.) < 0 does not mamze At the proft-mamzng level of output margnal revenue equals margnal cost, m m r ( ) C ( ); = that s, an nfntesmal change n the level of output changes revenue and cost equally. (In other words, an nfntesmal ncrease n the level of output ncreases revenue and cost by the same amount and an nfntesmal decrease n the level of output reduces revenue and cost by the same amount). An output level such that r (.) C (.) > does not mamze profts: an (nfntesmal) ncrease n output would ncrease revenue more than 6

8 cost (therefore ncreasng profts). Lkewse, a level of output such that r (.) C (.) < does not mamze profts: a (nfntesmal) decrease n output would reduce cost more than revenue (therefore ncreasng profts). (v) Output and elastcty: ε ( ) We seek to show that at the monopoly output the prce-elastcty of demand s or more. Frst, we defne the prce-elastcty of demand n absolute value: - as a functon of prce: p p ( p ) ε ( p) = ( ), (3) - as a functon of output : ε p( ) ( ) =. p ( ) (4) We net represent margnal revenue as a functon of the prce-elastcty of demand: r p p ( ) = ( ) + ( ) (5) p ( ) r ( ) p( ) = + p ( ) (6) r ( ) = p( ) ε ( ) (7) In the monopoly output margnal revenue and margnal cost are equal: ( ) ( ) ( ). ε ( ) r p C = = (8) Gven that the margnal cost s by defnton non-negatve (zero or more) then the margnal revenue must be non-negatve. Ths occurs when the prce-elastcty of demand n absolute value s or more. That s: 7

9 ( ) 0 ( ) 0 ( ). ε ( ) p C p ε ( ) 0 (v) Lerner nde of market power Now we obtan the Lerner nde of monopoly power (or market power) also called the relatve prce-margnal cost margn. From condton (8) we obtan: p p( ) ε ( ) ( ) = C ( ). By rearrangng we get: p( ) C ( ) =. (9) p( ) ε ( ) Therefore, the Lerner nde (and therefore monopoly power) s a decreasng functon of the prce-elastcty of demand n absolute value and when ε ( ) = (as t would occur f the frm behaved as a perfectly compettve frm) market power would be zero, p p( ) ( ) C ( ) = 0. (v) Graphcal representaton p C ( ) m p m Π r ( ) p( ) m 8

10 Margnal revenue, r p p ( ) = ( ) + ( ), s located below nverse demand gven that the nverse demand functon s downward slopng, p ( ) < 0. That s, r ( ) < p( ) for > 0, but both functons have the same ntercept, r (0) = p(0). The proft of the monopolst (when there s no fed cost) s gven by: m m m m m m m m m m C( ) m ( ) p C( ) p C ( z) dz p m 0 Π = Π = = = (v) Second order condtons Interpretaton We assume for the sake of smplcty 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 margnal revenue has to be lower than the slope of the margnal cost: d r d ( ( )) d( C ( )) < d In other words, the margnal revenue curve must cross margnal cost from above. 9

11 r, C d r d ( ( )) d( C ( )) > d d r d ( ( )) d( C ( )) < d C ( ) r ( ) m Cases. Strctly conve cost or lnear cost: C ( ) 0 (ncreasng or constant margnal cost) a) Strctly concave demand or lner demand: p ( ) 0 Π ( ) = p ( ) + p ( ) C ( ) < 0 < b) Strctly conve demand: p ( ) > 0 r ( ) = p ( ) + p ( ). We need to check < 0 > 0 r < C ( ) ( ).. Strctly concave cost: C ( ) < 0 (decreasng margnal cost) We always have to check whether r < C ( ) ( ). 0

12 .. Lnear demand, constant elastcty demand and constant margnal cost () Lnear demand and constant margnal cost Inverse demand: p( ) = a b ( a > 0, b > 0 ). Producton cost: C( ) Margnal revenue: = c ( c 0 ). ( a > c ) r ( ) = a b. Slope of nverse demand: p ( ) = b Slope of margnal revenue: d r d ( ( )) = b Strctly concave proft functon: Π ( ) = r ( ) = b < 0. Margnal proft at zero: Π = p C = a c > (0) (0) (0) 0. Proft mamzaton: m m m m a c ( ) ( ) = b r = C a b = c m m m m m a + c Monopoly prce: p = p( ) p = a b p = Monopoly profts: ( ) ( ) [ ( ) ] [ ] a c a c a Π m = Π c m = p m c m = p m c m = = b 4b () Constant elastcty demand and constant margnal cost Demand: ( p) = Ap b ( A > 0, b > ). Producton cost: C( ) = c ( c > 0 ). Prce-elastcty of demand: ε p ( p) p Ap ( b+ ) ( p) = ( p) = bap = b. b Inverse demand: b b p( ) = A.

13 Margnal revenue: ( b ) b b r ( ) = A. b Slope of margnal 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 mamzaton: Π (0) = > 0. m m ( b ) ( ) ( ) ( ) b m ( ) b m ( ) b b r = C r = A = c = A b c b ( b ) b b b m m b b b b b ( ) = A c = A c ( b ) ( b ) Monopoly prce: b b m m m m b m b b b b b p = p( ) p = A ( ) = A A c p = c ( b ) ( b ) Monopoly profts: b b m m m m m m c b b b ( b ) ( ) [ p( ) c] [ p c] A c A c ( b ) Π = Π = = = = b ( b ) ( b ) By solvng the problem of proft mamzaton as a functon of prce, we obtan the Lerner nde: m p c m = p ε ( p)

14 Under constant elastcty demand the Lerner nde becomes m p c m = and t s p b straghtforward to obtan the monopoly prce. Then t s easy to obtan the monopoly output and the monopoly profts..3. Comparatve statcs We now study how the monopoly prce and output respond to a change n producton costs. Economc ntuton tells us that an ncrease n margnal cost should ental a reducton n output and an ncrease n prce. We assume that the margnal cost s constant (and there s no fed cost). The cost functon s gven by C( ) = c. ma Π( ) ma p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p ( ) c = 0 () m ( c) the monopoly output s an mplct functon of the margnal cost. Π ( ) = p ( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). We can analyze the change n monopoly output due to a change n margnal cost n two equvalent ways: () By completely dfferentatng condton () wth respect to and c. p ( ) + p ( ) d dc = 0 We get: 3

15 d = <0 dc p ( ) + p ( ) < 0 CºO () Therefore, an nfntesmal ncrease n margnal cost reduces output and an nfntesmal reducton n margnal cost ncreases output. m () By usng the fact that the optmal output for the monopolst ( c ) s an mplct functon m of margnal cost. Therefore, by defnton, ( c ) satsfes the frst order condton; that s, p c c p c c ( m ( )) + m ( ) ( m ( )) = 0 By dfferentatng wth respect to margnal cost: p ( m ( c)) m ( c) + m ( c) p ( m ( c)) m ( c) = m m m m p ( ( c)) + ( c) p ( ( c)) ( c) = Rearrangng: m ( c) = 0 m m m p ( ( c)) + ( c) p ( ( c)) < Fnally the change n prce due to the change n margnal cost s: < 0 dp ( ) = dp d = p >0 dc d dc p ( ) + p ( ) < 0 CºO (3) Eamples () Lnear demand m m a + c dp p = = dc 4

16 dp p ( ) = = dc p ( ) + p ( ) Under lnear demand the change n prce s half the change n margnal cost: dp = dc = 0 () Constant elastcty demand m m 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 demand the ncrease n the monopoly prce s greater than the ncrease n margnal cost: dp > dc..4. Welfare and output () The representatve consumer approach. Quas-lnear utlty. () Mamum wllngness to pay. Margnal wllngness to pay. () The demand functon s ndependent of ncome. (v) Socal welfare functon and socal welfare mamzng output. (v) Total surplus, consumer surplus and producer surplus. (v) Effcency condtons n the presence of several consumers or markets. 5

17 (v) A comparson between monopoly output and effcent output by usng the proft mamzaton problem. (v) A comparson between monopoly output and effcent output by usng the socal welfare mamzaton problem. () Irrecoverable effcency loss. () The representatve consumer approach. Quas-lnear utlty We wll follow the representatve consumer approach to analyze welfare and evaluate monopoly from a socal welfare pont of vew. Under ths approach, t s assumed that market demand (p) s generated by mamzng the (quas-lnear) utlty of a representatve consumer. Consder an economy wth two goods, and y. Good s produced n the monopolstc market whle we can nterpret the good as the amount of money to be spent on the other good by the consumer once he/she has spent the optmal amount of money on good. We assume that the representatve consumer has a Quas-lnear Utlty Functon: U y u y u u u (, ) = ( ) + ( (0) = 0; (.) > 0; (.) < 0) () Mamum wllngness to pay and margnal wllngness to pay Mamum wllngness to pay, R( ) : the mamum amount of money that the consumer s wllng to pay for unts of the good. He/she pays the mamum f he/she s ndfferent between consumng unts by payng R( ) and not consumng the good, thus usng all hs/her ncome endowment to consume the other goods.that s: U (, m R( )) = U (0, m) 6

18 Note that the consumer must be ndfferent and, therefore, the above condton must be satsfed wth equalty. If for eample U (, m R ( )) > U (0, m) then the consumer would wsh to pay a greater amount to R ( ) and f hgher than hs/her mamum wllngness to pay. Gven that the utlty functon s quas-lnear then: Therefore, under quas-lnear utlty: U (, m R ( )) < U (0, m) then R ( ) would be U (, m R( )) = U (0, m) u( ) + m R( ) = u(0) + m R( ) = u( ) u( ) Mamum wllngness to pay Margnal wllngness to pay: ths s the change n mamum wllngness to pay due to an nfntesmal change n the quantty consumed. u ( ) Margnal wllngness to pay () The demand functon s ndependent of ncome L(, y, λ ) ma u( ) + y, y ma u( ) + y + λ m y p s. a y + p = m, y, λ [ ] L u p = ( ) λ = 0 = λ = 0 y L = m y p = 0 λ L p = u ( ) Inverse demand funton 7

19 The demand functon (p) s the nverse of ths functon and therefore satsfes the frst order condton: p = u ( ( p)) Demand functon Property of the quas-lnear utlty functon: the demand functon s ndependent of ncome. By dfferentatng wth respect to p we get: = u ( ( p)) ( p) ( ) = < 0 negatve slopng demand u ( ( p)) p < 0 (v) Socal welfare functon and socal welfare mamzng output In ths subsecton we ustfy the use of W ( ) = u( ) C( ) as the socal welfare functon. We consder the problem of obtanng the allocaton that mamzes the utlty of the representatve consumer wth a resources constrant: we nterpret the producton cost of good as the amount of good y to whch must be gven up n order to have the good. ma u( ) + y, y s. a y = m C( ) By replacng y n the obectve functon we get: ma u( ) + m C( ) ma u( ) C( ) cons tan te 8

20 Therefore the socal welfare mamzng problem becomes: ma W ( ) ma u( ) C( ) 0 0 W (0) = u (0) C (0) > 0 ( ) = ( ) ( ) = 0 ( ) = 0 (3) Frst order condton. e W u C W ( ) = ( ) ( ) < 0 Strctly concave welfare functon (regular case). W u C Therefore, the welfare mamzng output or effcent output satsfes e e e W ( ) = 0 u ( ) = C ( ). Under constant margnal cost the effcency condton becomes: e u ( ) = c, That s, at the effcent output margnal wllngness to pay equals margnal cost. (v) Total surplus, consumer surplus and producer surplus The functon W ( ) = u( ) C( ) can be nterpreted as the total surplus; that s, the dfference between mamum wllngness to pay and producton cost. By defnton the followng s satsfed: u( ) u(0) u ( z) dz = 0 = 0 C( ) C(0) C ( z) dz = 0 = F = 0 Therefore mamzng u( ) C( ) s equvalent to choosng the level of output that mamzes the area below the nverse demand and above the margnal cost. 9

21 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: [ ] [ ] W ( ) = u( ) C( ) = u( ) p + p c EC ( ) EP( ) The consumer surplus, CS(), measures the dfference between mamum wllngness to pay and the amount of money actually pad. The producer surplus, PS(), measures the profts of 0

22 the frm (when there are no fed costs). Therefore, effcent producton also mamzes the addton of the consumer surplus and the producer surplus. p e CS( ) C ( ) e PS( ) p( ) e (v) Effcency condtons n the presence of several consumers or markets We net analyze the problem of obtanng a Pareto effcent allocaton when we consder an economy wth two consumers under quas-lnear utlty, u ( ) + y, and an endowment of m, =,.. We mamze the utlty of one agent (for eample consumer ) whle mantanng constant the utlty of the other (consumer ), gven a resource constrant (margnal cost, c, s assumed to be constant). ma u ( ) + y, y,, y s. a u ( ) + y = u y + y = m + m c.( + ) By substtutng y and y n the obectve functon the problem becomes:

23 ma u ( ) + u ( ) c.( + ) + m + m u, From the frst order condtons we get: e u ( ) c = 0 = = e e e u ( ) u ( ) c Effcency condton (4) = u ( ) c 0 (v) A comparson between monopoly output and effcent output by usng the proft mamzaton problem. ma Π( ) ma p( ) C( ) 0 0 Π (0) = p(0) C (0) > 0 p(0) > C (0) Π ( ) = p( ) + p ( ) C ( ) = 0 Π ( m ) = 0 Frst order condton. Π ( ) = p ( ) + p ( ) C ( ) < 0 Strctly concave proft functon (regular case). ( ) 0 e Π ( )? ( ) 0 m 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. m Π ( ) = 0 Π ( ) < 0 Π ( ) < Π ( ) > Π ( ) < 0 e e m e m ( ) ( ) 0 d Π 0 ( ) d Π < < Π

24 Π m Π ( ) = 0 e Π ( ) < 0 m e (v) A comparson between monopoly output and effcent output by usng the socal welfare mamzaton problem. ma W ( ) ma 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 m W ( )? W ( ) < 0 m u ( ) m m m m m W ( ) = u ( ) C ( ) = p ( ) > 0 m p( ) < 0 By defnton of monopoly output. 3

25 e W ( ) = 0 W ( ) > 0 W ( ) < W ( ) > W ( ) < 0 m e m e m dw ( ) ( ) < 0 < 0 ( ) d W W W m W ( ) > 0 e W ( ) = 0 m e (v) Irrecoverable effcency loss (IEL). e m e m e ( ) ( ) [ ( ) ( )] [ ( ) ( )] [ ( ) ( )] m 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 m CS( ) C ( ) m p e PS( ) m PS( ) p( ) p( ) e m 4

26 p C ( ) m p IEL p( ) m e.5. Prce dscrmnaton () Defnton. () The ncentve to dscrmnate prces. () Condtons. (v) Types of prce dscrmnaton (Pgou, 90). (v) Eamples. (v) The model. () Defnton There ests prce dscrmnaton when dfferent unts of the same good are sold at dfferent prces ether to the same consumer or to dfferent consumers. Dscusson - Dfferences n qualty: passenger transport, cultural or sportng events etc. 5

27 - A sngle prce may be dscrmnatory and dfferent prces not. We say that there s no prce dscrmnaton when the dfference between the prces pad by two consumers for a unt of the good eactly responds to the dfference n the cost of provdng them wth the good. () The ncentve to dscrmnate prces At the proft-mamzng level of output margnal revenue equals margnal cost, m m r ( ) C ( ); =.e., an nfntesmal change n the level of output changes revenue and cost equally. That s: p( ) + p ( ) = C ( ) m m m m () Addtonal revenue from sellng an addtonal unt. Loss of revenue from sellng unts already produced at a lower prce. The monopolst would be reluctant to sell more unts f t does not have to reduce the prce. Therefore, there are ncentves to try to capture a hgher proporton of the consumer surplus ncentves to dscrmnate prces. p C ( ) m p The ncentve to dscrmnate prces: to capture a hgher proporton of socal surplus. m Π r ( ) p( ) m 6

28 () Condtons Two condtons are needed for a frm to be able to dscrmnate prces: a) The frm must be able to classfy consumers (whch depends on nformaton). b) The frm must be capable of preventng the resell of the good (whch depends on the possbltes of arbtrage and on transacton costs). The smplest case occurs when a frm receves an eogenous sgn (age, locaton, occupaton, etc.) whch allows t to classfy consumers nto dfferent groups. It s more dffcult to classfy accordng to an endogenous category (e.g., quantty purchased or the tme of purchase). In that case the monopolst must establsh prces n such a way that consumers classfy themselves n the correct categores. (v) Types of prce dscrmnaton (Pgou, 90) ) Frst-degree prce dscrmnaton or perfect dscrmnaton. The seller charges a dfferent prce for each unt equal to the mamum wllngness to pay for that unt. Ths requres full nformaton concernng consumer preferences and no arbtrage. The monopolst succeeds n etractng the complete consumer surplus. ) Second-degree prce dscrmnaton (or nonlnear prcng). Prces dffer dependng on the number of unts of the good but not across consumers. Each consumer faces the same prce catalogue but prces depend on the quantty purchased (or on another varable, e.g., product qualty). Eamples: volume dscounts. Self selecton. 7

29 3) Thrd-degree prce dscrmnaton. Dfferent prces are charged to dfferent consumers but each consumer pays a constant amount (the same prce) for each unt. The frm receves an eogenous sgn whch allows t to classfy consumers nto dfferent groups. Ths s the most frequent type of prce dscrmnaton. Eamples: dscounts for students, senor ctzens, etc. Identfcaton. Another way of classfyng prce dscrmnaton s to dstngush between drect prce dscrmnaton and ndrect prce dscrmnaton. Second-degree prce dscrmnaton s a case of ndrect dscrmnaton (consumers face a unque prce schedule and they classfy themselves by ther choces) whle frst-degree prce dscrmnaton and thrd-degree prce dscrmnaton would be drect dscrmnaton. In the case of thrd-degree prce dscrmnaton the frm gves dfferent prce menus for consumers belongng to dfferent groups or markets. (v) Eamples It s more dffcult to fnd real markets where there s no prce dscrmnaton than markets where such dscrmnaton ests. Although t s often not possble to dstngush clearly what type of prce dscrmnaton ests t s an nterestng eercse to thnk about what type of prce dscrmnaton s been practced n the followng cases. - Two-part tarffs: telephone, Internet, electrcty, cable televson, etc. - Dfferent electrcty rates for ndustral use and domestc use. - Dscounts n museums, magazne subscrptons, cultural and sportng events, for chldren, young people or senor ctzens. - Volume dscount n publc transport. 8

30 - 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 supermarkets, etc. - Home-servce food, tele-shoppng etc. (v) The model We study the three types of prce dscrmnaton by usng a very smple model. Assume that there are two potental consumers wth quas-lnear utlty functons: u ( ) + y, =,. u (0) = 0, =,. u ( ) : mamum wllngness to pay of consumer =,. u ( ) : margnal wllngness to pay of consumer =,. We say that the consumer s a hgh-demand consumer and that the consumer s a lowdemand consumer f the followng s satsfed: u ( ) > u ( ) u ( ) > u ( ) Thus, consumer s a hgh-demand consumer and consumer s a low-demand consumer f both the mamum wllngness to pay and the margnal wllngness to pay are hgher for consumer than for consumer for any quantty of the good. 9

31 The comparson between consumers of mamum wllngness to pay and margnal wllngness to pay only makes sense for the same level of output. Moreover, the comparson has to be made for any level of output. p u ( ) u ( ) u ( ) u ( ) u ( ) u ( ) u ( ) > u ( ) u ( ) > u ( ) The margnal cost of the monopolst s assumed 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 dscrmnaton or perfect prce dscrmnaton () Defnton and contet. () The case of a sngle consumer. () Observatons. Is the quantty suppled by the monopolst effcent? (v) The case of two consumers. (v) Does the monopolst supply effcent outputs to consumers? The monopolst supples a hgher quantty to the hgh-demand consumer (proof). (v) What would happen f the monopolst were not able to dentfy consumers? 30

32 () Defnton and contet The seller charges a dfferent prce for each unt of product and equals the mamum wllngness to pay for that unt. Ths requres full nformaton on consumer preferences and no arbtrage of any knd. In partcular, the monopolst needs to be able to dentfy consumers when they buy the good. (Classc eample: a vllage doctor). () The case of a sngle consumer The monopolst supples a prce-quantty bundle monopolst proposes a take t or leave t choce: * * ( r, ) whch mamzes profts. The * * ( r, ) (0,0). The consumer ether pays * r for * unts or does not receve the good. The mamzaton problem of the monopolst s: ma r c r, s. a u( ) r () Constrant () can be equvalently wrtten as u( ) r 0 : the consumer has to obtan a nonnegatve surplus from good. Ths type of constrant s known as partcpaton restrcton or ndvdual ratonalty restrcton. Gven that the monopolst wshes to mamze profts t wll choose the hghest possble tarff r and, therefore, condton () wll be satsfed as equalty: r = u( ). The problem thus conssts of: 3

33 Π( ) ma 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 monopolst effcent? The monopolst produces a Pareto-effcent output, * e =, gven that t supples a quantty such that the margnal wllngness to pay equals the margnal cost. (Revew the problem of mamzng socal welfare and compare wth the problem we have ust solved). However, the monopolst obtans the entre socal surplus. p Π = Π = = = m * * * e e e ( ) u( ) c u( ) c W ( ) m Π c * e u ( ) = C ( ) b) The monopolst produces the same quantty that t would produce f t behaved as a perfectly compettve frm. If t took prce as a parameter then ts output decson would be 3

34 p( ) = c but gven that utlty s quas-lnear then p = and consequently ( ) u ( ) u ( ) = c. However, the dstrbuton of trade gans would be ust the opposte. c) We mght obtan the same results by usng a two-part tarff. p * * T ( ) = A + p = u( ) c + c A * * * m Π = T ( ) c = u( ) c c m Π u ( ) = * e C ( ) d) We would obtan the same result f the monopolst sold each unt to the consumer at a dfferent prce equal to hs/her mamum wllngness to pay for that unt. Assume that we break producton down nto n equal portons of sze so as = n. The mamum wllngness to pay for the frst unt (of sze ) s gven by: u(0) + m = u( ) + m p u(0) = u( ) p The mamum wllngness to pay for the second unt s: u( ) + m p = u( ) + m p p u( ) = u( ) p And so on. We would obtan the followng sequence of equatons: u(0) = u( ) p u( ) = u( ) p u( ) = u(3 ) p... u(( n ) ) = u( n ) pn 3 33

35 n Addng and takng nto account that u (0) = 0 we get u( n ) = p. When the sze of the = unts becomes nfntesmal, we obtan that proposng a take t or leave t choce to the consumer s equvalent to sellng hm/her each (nfntesmal) unt at a prce equal to the margnal wllngness to pay for t. p = * u( ) u 0 ( z) dz * p( z) c * u ( ) C ( ) (v) The case of two consumers The monopolst supples consumer, =,, wth a prce-output bundle * * (, ) r n order to mamze profts. The monopolst gves consumer, =,, a take t or leave t choce: * * ( r, ) (0,0). Consumer, =,, ether pays * r for * unts or does not receve the good. The mamzaton problem of the monopolst s: ma r + r c.( + ) r,, r, s. a u ( ) - r 0 u ( ) - r 0 r = u ( ) r = u ( ) proft mamzaton Therefore, the problem becomes: 34

36 ma 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 monopolst supply effcent outputs to consumers? The monopolst supples a hgher quantty to the hgh-demand consumer (proof) The monopolst offers effcent outputs: = and * e * e. = (Revew the problem of obtanng a Pareto-effcent allocaton and compare wth the problem we have ust solved) We net demonstrate that the monopolst offers a hgher quantty to the hgh-demand consumer: * > *. * u( ) = c * * * u( ) = u( ) < u( ) * u( ) = c Consumer s the hghdemand consumer: u ( ) > u ( ) 35

37 Therefore, * * u( ) < u( ) but gven that functon u s strctly concave then d( u( )) < 0 d and n consequence > * *. (v) What would happen f the monopolst were not able to dentfy consumers? (Ths subsecton serves to ntroduce the analyss of second-degree prce dscrmnaton). Assume now that the monopolst s not able to dentfy consumers when they go to buy the good. That s, the monopolst cannot propose personalzed supples and s therefore restrcted to statng a sngle prce menu. Assume that t states a prce menu by usng the tarffs and quanttes whch are optmal under perfect prce dscrmnaton: ( r, ) * * * * ( r, ) (0,0) where r = u ( ) and * * r = u ( ). We can see that the hgh-demand consumer has * * ncentves to buy the bundle desgned for the low-demand consumer. * * * * * * 0 = u( ) r < u( ) r = u( ) u( ) > 0 Incentve to engage n personal arbtrage. The surplus obtaned by consumer f he/she buys the bundle desgned for hm. The surplus that consumer would obtan f he/she buys the bundle desgned for consumer. 36

38 .7. Second-degree prce dscrmnaton (or non-lnear prcng) (Keywords: no dentfcaton, unque prce menu and self selecton). () Defnton and contet. () Partcpaton restrctons and self selecton restrctons. Interpretaton. () Demonstraton of what constrants are satsfed wth equalty. Interpretaton. (v) The proft mamzaton problem. (v) Observatons. Does the monopolst supply effcent quanttes? The monopolst offers a lower-than- effcent quantty to the low-demand consumer (Proof). (v) Under what condtons does the monopolst offer the good to both consumers? (v) Graphc representaton. () Defnton and contet The prces dffer dependng on the number of unts bought but not from one consumer to another. We consder a contet where the monopolst knows the preferences of the consumers (t knows the preference dstrbuton functon) but s unable to dentfy consumers when they go to buy the good. So the frm s oblged to establsh a unque prce menu and to allow consumers to self classfy or self select. In ths sense we can say that there s ndrect prce dscrmnaton. The consumers face the same prce schedule but prces depend on quantty (or some other varable, e.g. the qualty of the good) bought. 37

39 () Partcpaton restrctons and self selecton restrctons. Interpretaton The obectve of the monopolst s to optmally desgn the prce menu n such a way that each consumer chooses the prce-quantty bundle desgned for hm/her. ( r, ) ( r, ) (0,0) Consumer Consumer Restrctons for the monopolst - Partcpaton restrctons (or ndvdual ratonalty constrants) u ( ) r 0 () u ( ) r 0 () These restrctons guarantee that each consumer wshes to buy the good. Each consumer obtans at least as much utlty by consumng the good as by not consumng. Put dfferently, each consumer obtans a non-negatve surplus by purchasng the good. - Self selecton restrctons (or ncentve compatblty constrants) u ( ) r u ( ) r (3) u ( ) r u ( ) r (4) These restrctons guarantee that each consumer prefers the prce-quantty bundle desgned for hm/her to the prce-quantty bundle desgned for the other consumer. Put dfferently, these constrants avod personal arbtrage: each consumer gets as least as great a surplus by choosng the bundle desgned for hm/her as he/she does by choosng the bundle desgned for the other consumer. 38

40 () Demonstraton of what constrants are satsfed wth equalty. Interpretaton We now arrange constrants accordng to each consumer. () y (3) r u ( ) () ( ) ( ) () r u u + r () y (4) r u ( ) (3) ( ) ( ) (4) r u u + r The monopolst wshes to mamze 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 assumpton that consumer s the hgh-demand consumer and consumer the low-demand consumer ( u ( ) > u ( ) and u ( ) > u ( ) ) s suffcent to determne what constrants are bndng. ) Demonstraton that (4) s satsfed wth equalty and (3) wth strct nequalty. Assume that (3) s satsfed wth equalty and, therefore, that r = u( ). Then (4) r r u ( ) + r r u ( ). Gven that consumer s the hgh-demand consumer u( ) > u( ) then r u( ) > u( ). That s, r > u ( ) whch means that restrcton () would not be satsfed whch s a contradcton. (The fact that the partcpaton constrant of the hgh-demand consumer s satsfed wth equalty s not compatble wth the fact that the low-demand consumer buys the good). As a concluson, (3) s not bndng and (4) s satsfed wth equalty: r = u ( ) u ( ) + r (5) 39

41 ) Demonstraton that () s satsfed wth equalty and () wth strct nequalty Assume that condton () s satsfed wth equalty and, therefore, that r = u( ) u( ) + r. By substtutng r from condton (5) we get: r = u( ) u( ) + u( ) u( ) + r = r whch mples u ( ) u ( ) = u ( ) u ( ) u t dt = ( ) u ( t) dt [ u ( t) u ( t)] dt = 0 But ths contradcts the assumpton that consumer s the hgh-demand consumer, u ( ) > u ( ). Therefore, () s not bndng and () s satsfed wth equalty: r = u ( ) (6) Interpretaton The monopolst charges consumer a tarff equal to hs mamum wllngness to pay gven that the low-demand consumer has no ncentve to engage n personal arbtrage. Gven that the hgh demand consumer has ncentve to engage n personal arbtrage (and to mmc the low-demand consumer) the monopolst charges hm/her the mamum prce that nduces hm/her to choose the bundle desgned for hm/her (the amount of money that ust leaves hm/her ndfferent between hs/her bundle and that desgned for the low-demand consumer). 40

42 We now show (n a dfferent more ntutve way) why the monopolst must provde a postve surplus to the hgh-demand consumer. Consder the self selecton constrant for the hgh-demand consumer: u ( ) r u ( ) r (4) Note that the rght sde of ths constrant s postve condtonal on the low-demand consumer s wshng to buy the good. That s, f we choose the mamum value for r condton (4) would be: u ( ) r u ( ) u ( ) > 0 gven that consumer s the hgh-demand consumer (whch mples that the partcpaton restrcton of consumer cannot be satsfed wth equalty). But gven that the monopolst must allow the hgh-demand consumer to obtan a postve surplus, t decdes to leave the consumer wth the mnmum possble surplus, ust that amount such that the hgh-demand consumer s ndfferent between hs/her bundle and the bundle desgned for consumer. That s, rearrangng restrcton (5): u ( ) r = u ( ) u ( ) > 0 Gven that the low-demand consumer has no ncentve to engage n personal arbtrage the monopolst charges hm/her the mamum that he/she s wllng to pay r = u ( ). (v) The proft mamzaton problem ma r + r c.( + ) ma r + r c.( + ) r,, r, r,, r, u( ) - r 0 () r = u( ) (6) s. a s. a u( ) - r 0 () r = u( ) [ u( ) r] (5) u ( ) - r u ( ) - r (3) u ( ) - r u ( ) - r (4) 4

43 By substtutng we get:, Π(, ) ma u ( ) + u ( ) [ u ( ) u ( )] c.( + ) Π = u( ɶ ) c [ u( ɶ ) u( ɶ )] = 0 (7) Π = u( ɶ ) c = 0 (8) The tarffs are gven by: rɶ = u ( ɶ ) rɶ = u ( ɶ ) [ u ( ɶ ) u ( ɶ )] (v) Observatons ) The monopolst provdes the hgh-demand consumer wth the effcent quantty and leaves hm/her wth a postve surplus. Condton (8) mples u ( ɶ ) = c and, therefore, the monopolst offers the effcent quantty e to the hgh-demand consumer ɶ = (revew Pareto-effcency condtons). Moreover, the monopolst charges hm/her a prce (a tarff) lower than hs/her mamum wllngness to pay leavng hm/her wth a postve surplus equals to that whch he/she would obtan f he/she chose the bundle desgned for consumer. rɶ = u( ɶ ) [ u( ɶ ) u( ɶ )] and hs/her surplus would thus be: u( ɶ ) rɶ = [ u( ɶ ) u( ɶ )]. ) The monopolst offers the low-demand consumer a quantty lower than the effcent quantty and leaves hm/her wth no surplus. Π = u( ɶ ) c [ u( ɶ ) u( ɶ )] = 0 (7) > 0 4

44 Gven that consumer s the hgh-demand consumer [ u ( ɶ ) u ( ɶ )] > 0 and then from condton (7) we get u ( ɶ ) > c. By defnton, the effcent output satsfes u ( ) = c, and as e a consequence u ( ɶ ) > u ( ). The mamum wllngness to pay s a strctly concave e functon: e u( ɶ ) > u( ) ɶ < d( u( )) < 0 d e We net look at the ntuton of ths result. We nterpret the margnal proft of and evaluate t at dfferent producton levels. Π = u ( ) c [ u ( ) u ( )] * > 0( < ) > 0 Margnal proft from consumer : a change n the quantty suppled to ths consumer mples a change n the proft obtaned by the monopolst from hm/her. Margnal proft from consumer : a change n the quantty suppled to consumer mples a change n the surplus the monopolst must leave consumer to avod personal arbtrage. * Π( ) * * * = u( ) c [ u( ) u( )] < 0 = 0 > 0 43

45 Startng from * a reducton n the quantty suppled to consumer ncreases the proft because the surplus that the monopolst must leave consumer to avod arbtrage s reduced. An output such that ɶ < < satsfes the followng: * Π( ) = u( ) c [ u( ) u( )] < 0 > 0 > 0 It s worthwhle for the monopolst to contnue reducng because the ncrease n profts from the hgh-demand consumer (obtaned by leavng hm/her wth a lower surplus) offsets the loss of profts from the low-demand consumer obtaned by supplyng hm/her a lower quantty. Π( ɶ ) = u( ɶ ) c [ u( ɶ ) u( ɶ )] = 0 > 0 In output ɶ the margnal gan, from an nfntesmal reducton n, from the hgh-demand consumer by leavng hm/her wth lower surplus s ust equal to the margnal loss from the low-demand consumer as a result of offerng a lower quantty. Moreover, the monopolst charges the low-demand consumer a prce (tarff) equal to the mamum wllngness to pay, thus leavng hm/her wth no surplus: rɶ = u ( ɶ ). 44

46 (v) Under what condtons does the monopolst offer the good to both consumers? The monopolst wll decde to offer the good to both consumers f t obtans more profts than by sellng the good only to the hgh-demand consumer. That s, the monopolst supples the good to both consumers f the followng s satsfed: Π(0, ) Π( ɶ, ɶ ) * u ( ) c u ( ɶ ) cɶ + u ( ) [ u ( ɶ ) u ( ɶ )] c * * * * rɶ rɶ [ u ( ɶ ) u ( ɶ )] u ( ɶ ) cɶ If ths condton s not satsfed, the monopolst offers the good only to the hgh-demand consumer. Another equvalent way of lookng at the problem conssts of consderng the margnal proft of. If t were negatve for any level of Π( ) = u ( ) c [ u ( ) u ( )] < 0 > 0 > 0 then the monopolst would decde not to sell the good to the low-demand consumer gven that for any level of t would ncrease profts by reducng the quantty suppled to the low-demand consumer. 45

47 (v) Graphc representaton (zero margnal cost) p u ( ) u ( ) = c = 0 * u ( ) = c = 0 * A B C u ( ) * * Perfect prce dscrmnaton * * ( 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 Assume that the monopolst does not know the dentty of the consumer and that t states a unque prce menu where t mantans the prce-quantty bundles whch were optmal under perfect prce dscrmnaton. Consumer would have ncentves to engage n personal arbtrage. 46

48 * * ( r, ) A * * ( r, ) A+ B+ C (0,0) Consumer Consumer 0 = A + B + C ( A + B + C) < A + B A = B * * * u * ( ) u ( ) r r Second-degree prce dscrmnaton The followng condtons are satsfed wth equalty: r = u ( ) A( ) the monopolst charges consumer the area below hs/her nverse demand functon. u( ) r = u( ) r B( ) the monopolst leaves consumer wth a surplus B( ) (the mnmum) n order to avod arbtrage. Frstly, we mantan quanttes and only adust 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 ( ) * * 47

49 ( rɶ, ɶ ) ( rɶ, ɶ ) (0,0) Π( ɶ ) = u( ɶ ) c [ u( ɶ ) u( ɶ )] = 0 > 0 As we are assumng that the margnal cost s zero: Π( ɶ ) = u( ɶ ) [ u( ɶ ) u( ɶ )] = 0 u( ɶ ) = u( ɶ ) u( ɶ ) > 0 p u ( ɶ ) u ( ɶ ) u ( ɶ ) B ɶ u ( ) u ( ɶ ) c = 0 u ( ɶ ) A ɶ C u ( ) ɶ * * * * Π ( ɶ, ɶ ) = u( ɶ ) c ɶ + u( ) [ u( ɶ ) u( ɶ )] c Aɶ + A + B + C Bɶ *

50 The decson to supply the good only to the hgh-demand consumer. p Π( ) = u ( ) c [ u ( ) u ( )] < 0 > 0 > 0 u A B ( ) C u ( ) ( r, ) * * A+ B+ C (0,0) * *.8. Thrd-degree prce dscrmnaton () Defnton and contet. () Proft mamzaton. The rule of the nverse of elastcty. () A comparson of profts wth the case of unform prcng (sngle monopoly prcng). (v) Effects on socal welfare. ()Defnton and contet There s thrd-degree prce dscrmnaton when consumers belongng to dfferent groups or submarkets are charged dfferent prces, although each consumer pays the same prce for each unt bought. Ths s probably the most common type of prce dscrmnaton. Eamples: dscounts to students, senor ctzens etc. 49

51 The monopolst receves an eogenous sgn whch allows t to dstngush m perfectly separated markets or submarkets: = 0. p Ths s a type of drect dscrmnaton: the monopolst states dfferent prce menus for consumers belongng to dfferent groups or markets. Identfcaton: the monopolst classfes each consumer n a group. () Proft mamzaton. The rule of the nverse of elastcty We consder the smple case of m = : the monopolst classfes consumers n two groups wth nverse demand functons p ( ) and p p < = The monopolst ( ), wth ( ) 0,,. can establsh dfferent prces n the two markets but wthn a market t s not possble to dscrmnate prces. The mamzaton problem s:, Π(, ) ma 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 ( ) 50

52 c p ( ) = =,. ε ( ) Therefore, p ( ) > p( ) ff ε( ) < ε ( ). As a consequence, the monopolst charges the hghest prce n the market wth the lower prce elastcty (n absolute value). () A comparson of profts wth the case of unform prcng (sngle monopoly prcng) The monopolst s proft under thrd-degree prce dscrmnaton s at least as hgh as the proft under unform prcng. The reason s smple: under thrd-degree prce dscrmnaton the frm can always choose equal prces f that s the most proftable opton. (v) Effects on socal welfare ) What s the problem? ) Bounds of the change n socal welfare. 3) Applcatons: a) Lnear demand. b) Openng of markets. ) What s the problem? Ths secton compares thrd-degree prce dscrmnaton and unform prcng from a socal welfare pont of vew. In general, a movement from unform prcng to thrd-degree prce dscrmnaton benefts some agents and harms others. 5

53 Benefted by T-DPD: the monopolst and the consumers n the hgher-elastcty market (gven that the prce s reduced by dscrmnaton). Harmed by T-DPD: the consumers n the lower-elastcty market (gven that the prce s ncreased by dscrmnaton). Therefore, the effect on socal welfare s ndetermnate. ) Bounds of the change n socal welfare Assume for the sake of smplcty that there are only two markets and we start from an aggregate utlty functon u( ) + u( ) + y + y, where and are the consumptons of good by the two groups and y s the money to be spent on other goods ( y = y + y ). u and u are strctly concave. The nverse demand functons are gven by p ( ) = u ( ) and p ( ) = u ( ). If C(, ) s the cost of supplyng and we can measure 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. Assume that the ntal set of prces corresponds to unform prcng (the monopoly sngle prce) p = p = p and that p and p are the prces under thrd-degree prce dscrmnaton. Consder the movement from 0 to. Due to the strctly concavty of u we have (see Append): 5

54 0 0 p ( ) = p u( ) < u( ) + u( ) ( ) () u < p 0 p > u > p 0 0 u( ) < u( ) + u ( ) ( ) () u > p p ( ) = p 0 0 p ( ) = p u( ) < u( ) + u( ) ( ) () u < p 0 p > u > p 0 0 u( ) < u( ) + u( ) ( ) () u > p p ( ) = p (3) (4) By addng (3) and (4) we get p + p > u + u > p + p 0 0 where u = u + u ; = ; = 0 0 p = p ( ) = u ( ); p = p ( ) = u ( ); p = p ( ) = u ( ); p = p ( ) = u ( ). The change n socal welfare s gven by: W = W (, ) W (, ) = u( ) u( ) + u( ) u( ) [ C(, ) C(, )] u u C = u + u C Therefore p + p C > W > p + p C 0 0 If margnal cost s constant: C = c( + ) c( + ) = c + c 0 0 Therefore the bounds of the change n socal welfare become: 0 0 ( p c) + ( p c) > W > ( p c) + ( p c) (5) Upper bound Lower bound 53

55 Gven that p = p = p the bounds of the change n socal welfare are: ( p c)( + ) > W > ( p c) + ( p c) (6) Upper bound Lower bound - Upper bound: ths mples that a necessary condton for thrd-degree prce dscrmnaton to ncrease socal welfare, W > 0, s that t should ncrease total output. Assume on the contrary that = + 0. Gven that > then (4) W < 0. 0 ( p c) 0 - Lower bound: ths ndcates that a suffcent condton for thrd-degree prce dscrmnaton to ncrease socal welfare s that the sum of the changes n output weghted by the dfference between the prce under dscrmnaton and the margnal cost must be postve. Graphcally, for the case of a sngle market, the bounds would be: p 0 0 p p ( p c) > W > ( p c) c 0 54

56 3) Applcatons a) Lnear demands Assume that the demands are gven by a ( p ) = p,,, b b = and that the margnal cost s zero, c = 0. The proft mamzaton problem under thrd-degree prce dscrmnaton s: ma 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 a b + a b = + = + = b b b b Under unform prcng: ma 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 a a b + a b a b + a b a b a b a b + a b a b = = = 0 b b ( b + b ) b ( b + b ) b ( b + b ) a a b + a b a b + a b a b a b a b + a b a b = = = 0 b b ( b + b ) b ( b + b ) b ( b + b ) 55

57 The total output s: a b + a b a b a b + a b a b = + = b ( b + b ) b ( b + b ) a b b + a ( b ) a b b + a b b + a ( b ) a b b = b b ( b + b ) ab b + a( b ) + abb + a( b ) ( ab + ab )( b + b ) ab + a b = = = b b ( b + b ) b b ( b + b ) b b Therefore, total output s the same under both prcng polces. That s, = + = 0, or, equvalently,. = The bounds would be 0 ( p c)( + ) > W > ( p c) + ( p c) (6) = 0 < 0 Socal welfare therefore decreases: W < 0. As we show below, the above result depends crucally on the assumpton that all markets are served under unform prcng. b) Openng of markets Imagne that the two markets demands are lke those n the graphc. p 0 p = p p ( p ) ( p ) 56

58 If the monopolst had to sell at a unform prce, t would have to reduce the prce n market by such an amount that the decrease n profts n that market would not be offset. Therefore, ( p c)( + ) > W > ( p c) + ( p c) (6) 0 p = 0 > 0 = 0 > 0 > 0 = 0 > 0 > 0 Gven that the lower bound n (4) s postve then W > 0. But not only does socal welfare ncrease, n fact thrd-degree prce dscrmnaton Pareto domnates unform prcng. A move from unform prcng to thrd-degree prce dscrmnaton mples an ncrease n the monopolst s profts, an mprovement for consumers n market and no change for consumers n market. Append If u s a strctly concave functon for any and y the followng s satsfed: u < u y + u y y ( ) ( ) ( )( ). The tangents always reman above the functon when t s strctly concave. u u( ) u( y ) u( ) u( y) u lnear u ( y) = y u u = u y + y u y lnear ( ) ( ) ( ) ( ) u u < u y + y u y strctly concave ( ) ( ) ( ) ( ) y 57

59 Chapter. Non-Cooperatve Game Theory Introducton The Theory of Non-Cooperatve Games studes and models conflct stuatons among economc agents; that s, t studes stuatons where the profts (gans, utlty or payoffs) of each economc agent depend not only on hs/her own acts but also on the acts of the other agents. We assume ratonal players so each player wll try to mamze hs/her proft functon (utlty or payoff) gven hs/her conectures or belefs on how the other players are gong to play. The outcome of the game wll depend on the acts of all the players. A fundamental characterstc of non-cooperatve games s that t s not possble to sgn contracts between players. That s, there s no eternal nsttuton (for eample, courts of ustce) capable of enforcng the agreements. In ths contet, co-operaton among players only arses as an equlbrum or soluton proposal f the players fnd t n ther best nterest. For each game we try to propose a soluton, whch should be a reasonable predcton of ratonal behavour by players (OBJECTIVE). We are nterested n Non-Cooperatve Game Theory because t s very useful n modellng and understandng mult-personal economc problems characterzed by strategc nterdependency. Consder, for nstance, competton between frms n a market. Perfect competton and pure monopoly (not threatened by entry) are specal non-realstc cases. It s more frequent n real lfe to fnd ndustres wth not many frms (or wth a lot of frms but wth ust a few of them producng a large part of the total producton). Wth few frms, competence between them s characterzed by strategc consderatons: each frm takes ts decsons (prce, output, advertsng, etc.) takng nto account or conecturng the behavour of the others. Therefore, competton n an olgopoly can be seen as a non-cooperatve game where the frms are the players. Many predctons or soluton proposals arsng from Game 58

60 Theory prove very useful n understandng competton between economc agents under strategc nteracton. Secton defnes the man notons of Game Theory. We shall see that there are two ways of representng a game: the etensve form and the strategc form. In Secton 3 we analyze the man soluton concepts and ther problems; n partcular, we study the Nash equlbrum and ts refnements. Secton 4 analyzes repeated games and, fnally, Secton 5 offers concludng remarks... Basc notons There are two ways of representng a game: the etensve form and the strategc form. We start by analyzng the man elements of an etensve form game.... Games n etensve form (dynamc or sequental games) An etensve from game specfes: ) The players. ) The order of the game. 3) The choces avalable to each player at each turn of play (at each decson node). 4) The nformaton held by each player at each turn of play (at each decson node). 5) The payoffs of each player as a functon of the movements selected. 6) Probablty dstrbutons for movements made by nature. An etensve form game s represented by a decson tree. A decson tree comprses nodes and branches. There are two types of node: decson nodes and termnal nodes. We have to 59

61 assgn each decson node to one player. When the decson node of a player s reached the player chooses a move. When a termnal node s reached the players obtan payoffs: an assgnment of payoffs for each player. EXAMPLE : Entry game Consder a market where there are two frms: an ncumbent frm, A, and a potental entrant, E. At the frst stage, the potental entrant decdes whether or not to enter the market. If t decdes not to enter the game concludes and the players obtan payoffs (frm A obtans the monopoly profts) and f t decdes to enter then the ncumbent frm, A, has to decde whether to accommodate entry (that s, to share the market wth the entrant) or to start a mutually nurous war prce. The etensve form game 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 accommodate), G.P. (prce war). Decson nodes: α. Termnal nodes: β. 60