SUPPLY. Start with the short run in the very short run there is no supply response:

Transcription

1 UPPLY tart wth the short run n the very short run there s no supply response: Fg. Perfect competton:. Large # of frms,. sellng homogeneous product 3. p max. 4. Prce takers 5. Perfect nformaton 6. No transacton cost. 7. Free Entry and Ext hort Run upply Curve: hort Run upply Curve: When suppler are able to react to prce changes the demand curve upward slope. For an ndustry we can sum up the supply curves of each frm n the ndustry to get the ndustry supply cuvres. Assumng prefect competton the MC curve wll be the frm supply curve. Fg.

2 Combnaton of all ndvdual frms. hort Run upply Functon: Q ( P, v, w) n  q ( P, v, w) Where P prce v functon prce of captal w functon prce of labor hort Run elastcty of supply: e s, p %DQs % DP dqs dp P Q s Unlke demand, e s,p >0 wq Example from book: TC 4v Under perfect competton, frms are prce takers so demand perfectly elastc (horzontal demand curve) so P MR. max P MR MC Fg. 3 MC PMR wq 00 p fp MC f q 400 w If w 4 then q 50P If there are 00 dentcal frms, market supply: Q s wll be: 00  q 00 50P 5000P

3 00P If w ncreases to 5 then the new frm supply curve s: q 40 p 5 The market supply curve s: Q s 00 Â Elastcty of market supply: e sp q 4000P d s P P 4000P 4000 P Q Q Q From upply functon f Q s 4000P f e sp Relatonshp between market frm and ndvdual demand: s s s Fg. 4! D` d d ` D D D Fg. 4 Market Frm Indvdual ` ` Mathematcal model of upply and Demand: Q D 0 P(_) where _ shft parameter (ncome, etc) Q (P,_) where _ parameter (nput prces, no. of producers n the market) D P D P < 0, P >0 P f D D a a (?)0 (?)0 b any sgn Equlbrum: Q D Q Total dfferentaton to both sdes: dq D D P * dp + D a * da

4 dq P * dp + b * db Mantenance of equlbrum requres: dq D dq f D P * dp + D a * da P * dp + b * db Assume b 0 only demand parameter changes: D P *dp + D a *da P * dp f -D P *dp + P *dp D a * da f (-D P + P )*dp D a * da dp da D a - D P + P The sgn of the partal s not determned. P Da a Elastcty: e P,a f Multpled by f ep,a a P - D P P P a a P P D a - D P a P a Q Qa Multply by (Q/Q) f e P,a ( P - DP ) P ep-e QP D a Q Prce shft functon elastcty depends on prce elastcty of supply and demand. e LONG RUN LONG RUN COMPETITION EQUILIBRIUM A perfectly comp. Industry s n long run equlbrum f there are no ncentves to enter or leave the ndustry. The no. of frms wll be such that: P MC AC f no economc proft. Constant cost: entry does not affect LRAC horzontal Increasng cost: entry ncreases cost of nputs (LRAC upward slopng) Decreasng cost: entry reduces cost of nputs (LRAC downward slopng) FIRM FIRM MC AC Q

5 Consumer and Producer surplus Fg. 8 C P C +P Q [ U ( Q) - P Q] + PQ - Ú P( Q) dq U ( Q) - Ú P( Q) dq U(Q): utlty functon P(Q): long run supply.. 0 f U (Q) P(Q) so max. occurs when margnal value of Q for consumer s equal to the market prce. We could also just take the dfference of the area under the demand curve and supply curve for: Q Q Q 0 Q Ú 0 P ( Q) d( Q) - d Q Ú 0 P ( Q) d( Q) s Welfare analyss Restrctons or prce controls Quantty restrcton

6 Fg. 9 P Prce controls In fgure 9 above the prce floor causes excess supply. Ths also cause a shft from C to P of the rectangle above and a DWL f the trangle above. In fgure 0 below we can see as the prce floor get closer to the equlbrum prce there s less DWL and less C s transfer to producers. Fg. 0 Other example wth long run effects n supply: Now consder a prce celng at the current equlbrum prce of P. If demand ncreases the equlbrum prce would be P 3. There would be a transfer from P to C and DWL. Wth tme the supply curve wll become more elastc or just shft to the left, causng less supply at each prce. Fg. P3 P P L Tax Incdence P D P t Ths s the dfference pad by consumers and receved by producers.

7 Total dff.: dp D dp dt f dp dp D dt Market equlbrum f dq D dq f D p dp D p dp Pluggng n from above f D p dp D p (dp D - dt) f P t D p p - D p e e - e D where D p D and P p P Dong the same for supply P t ed e - e D where: e D 0 and e 0 P P f D 0 and 0 t t P If e d 0 then d (all tax pad by consumer) t P - If e d - then t (all tax pad by producer) Dvdng: P t - e t consumer(tax payer) wll be. D - so the more elastc the functons are the better off the PD e + t Fg. t

8 Trade restrctons: (Note: Unlke your text, t s assumed the domestc market s large enough to affect the world prce.) Fg. 3 L P * A L W P w B D C Q Pm Q * Q w Area ABC: net gan by consumer Area P * BDP w : transfer from domestc prod. to consumers Wthout restrcton domestc supplers produce Q dom Fg. 4 ame Graph P`` P` III II IV I Area I: net P.., gan to foregn producers Area II: transfer from domestc to foregn Area III: Transfer from P to C Area IV: DWL to C

9 Fg. 5 Tarff transfer from Domestc cons.to E LDOM LW -t L DWL for Domestc consumers DWL for Foregn producers Now assume the domestc demand does not the world prce (perfectly elastc world supply). Government Revenue DWL for Domestc consumers

10 Now assume the domestc demand does not affect the world prce (perfectly elastc world supply). Government Revenue LDOM P w + t P w DWL for Domestc consumers

11 Chapter 6 (Ncholson) General equlbrum Under the assumpton of perfect competton the unt one prce..to hold.no transacton costs. Edge worth box Frst assume fx amount of nputs (captal and labor) Fg. 6 Total captal Isoquants Total labor Fg. 7 Labor for X Labor for Y K for Y Total K K for X Total Labor Effcency n producton Fg. 8

12 Producton possblty fronters Fg. 9 Y PPF show those alternatve combnatons of outputs that can be produced wth fxed quanttes of nput f those nputs are employed effcently. X Rate of product Transformaton (RPT) slope of PPF (trade off between X and Y ) RPT Y - X hape of PPF uppose we bare a cost functon for ths economy that depend on the amount of X and Y at ths level of nputs: C C (X, Y) Total dff. C C X X C + Y Y I. Know RPT: RPT Y X If dc 0 than C Y RPT - X X C Y MC MC X Y Dmnshng returns Opportunty costs

13 Guns and butter No captal Producton functon: / X L X Y LY Labor force 00 f L Y + L X 00 f X + 4Y 00 Total dff. f X dx + 8Y dy 0 f Equlbrum Prces Y - dx RPT 4 dy X Fg. 0 N N Fg. Excess demand for X A B P x /P y Producer wll produce at A but consumer would buy at B. Excess demand for X so prces adjust to: Fg. P x /P y

14 MR P - P X Y RPT Example: X + 4Y 00 () U XY RPT 4Y X MR - X Y Y X RPT 4 MR X 4Y () X Y From () and () f X +X 00 f X 00 f X Y 00 f Y Comparatve statstcs If utlty functon changes to: U X 3/4 Y /4 RPT MR Techncal Progress Fg. 3 E E

15 Tarffs and trade Fg. 4. Y tot Y E - P /P Y dom P x /P y X dom X E X T After nteracton trade we mport X (X T X dom ) and export Y (Y tot - Y dom. ). General equlbrum prces Assume the followng nverse demand functon: D (P,.., P n ) The whole set of prces: P (P,,P n ) The demand for good s: D (P) D (P * ) (").. In other words, are there a set of prces n whch all markets are n equlbrum? To see ths let defne excess demand n market as: E D (P) D (P) E D (P * ) D (P * ) 0 Ths mples homogenety of degree zero f all markets are nclude.(ncludng factor markets) f all prces are double quantty n equl. stays the same. W.`s Law: n  P ED ( P) 0 Cannot have excess demand or excess supply of all goods.

16 Chapter 7 Effcency of P.C. Pareto effcent allocaton: An allocaton of resources s pareto effcent f t s not possble to make one person better off wthout makng someone else worse off. Productve effcency (techncally effcent): An allocaton of resources s effcent n producton f no further reallocaton would permt more of one good to be product wthout necessarly reducng the output of some other goods. ngle Frm nput choce X f(k x, L x ) K K x L full employment of resources L x amout devoted to X producton. Y g( K y, Ly ) g( K - K x, L - LK ) [ Y - g( K - K, L L )] f ( K x, Lx ) + l x - K where Y g( K, L) K x f K - lg K 0 f L x L - lg L 0 l Y - g(k - K X,L - L Y ) 0 f f f K L g g K L Usng results from chapter, RT X RT Y One Market,Two frms: same good X f (K, L ) + f (K, L ) K K + K and L L + L Wrte as: X f K, L ) + f ( K - K, L - ) ( L X K f K f + K 0 nce added captal for frm must come from frm f f K - f K

17 X f f - K K K and X f f - L L L 0 f f K f K f L f L Comparatve advantage (beneft from trade) Let total resources be fxed at 00. The followng PPf for countres: w wne, c cloth. 8w + 4c 00 w + c 00 England Portugal c RPT E - w Totally dfferentatef 8 dw + 4dc 0 - dc f 8dw -4dc f RPT E dw dc RPT P - dw Portugal has absolute cost advantage but England has a comparatve advantage n cloth. Pror to trade assume resources are dvded equally amount cloth and wne accordng to some preference functon: England: w 6.5 C.5 Portugal: w 5 C 5 England producton of only cloth c 5, Portugal w 70/ 35 and c 30/ 5 Total wne producton 35 > Total cloth producton > Exchange economy Fg. 6 Y

18 Effcency n product mx T(X, Y) f PPF (mplct) U X, Y + l(t(x, Y)) X U X T + l X 0 Y U Y + l T Y 0 l T (X,Y) 0 U X U Y T X T Y MR -dy/dx RPT fg. 5 Market mperfectons departng from P.C. Imperfect competton Market power Externaltes: thngs not accounted for n prces Publc goods Monopoly- Characterstcs A sngle seller. A sngle frm produces all ndustry output. The monopoly s the ndustry. Unque Product- Monopoly output s perceved by customers to be dstnctve and preferable to ts mperfect substtutes Blockaded entry and ext- Frms are heavly restrcted form enterng or leavng the ndustry Barrers to entry sources of monopoly Techncal barrers to entry declnng cost on relatve porton of cost curve (natural monopoly)

19 pecal knowledge ( protected ) Legal barrers Patents, Copyrghts Exclusve franchse (publc utlty) Creaton of barrers: buyng ownershp of unque lobbyng for legal protecton Imperfect dssemnaton of nformaton- Cost, prce, and product qualty nformaton are wthheld from unnformed buyers. Proft Max. lke all companes: MC MR Here monopolst face downward slopng demand curve. Fg. 7 MC P AC MR D Q Inverse elastcty rule at proft max. pont: (P MC)/P -/e P over MC Ths s called markup Proft: Fg. 8 MC P C A E MR AC D

20 Monopoly Market demand: Q 000 0P Inverse: P 00 Q/0 TC 0.05Q TR P*Q 00Q Q /0 MR 00 Q/0 MC 0.Q f Q * 500 and P * 75 TC 0.5(500) AC TC/Q 500/ P (P * - AC)*Q TR TC (75 45)* Markup prce over MC P 75 MC 0.Q 50 Markup Elastcty rule: e Q,p (dq/dp)*(p/q) -0*75/500-3 (P MC)/P /3 P 3/*MC How could output be affected f fxed cost ncreased? How could output be affected f varable cost ncreased? Allocaton of resources Compare wth Perfect Compatton Fg. 9 MC Transfer from C to P DWL AC

21 Fg. 30 A Area ABC: DWL B C MC AC MONOPOLY AND QUALITY Competton or monopoly power can be exercsed along other dmensons. Denote X as qualty, where: P P < 0 and > 0 Q X P P(Q, X)Q C(Q, X) Demand s a functon of Q and X Cost s a functon of Q and X P P( Q, X ) + Q - C 0 Q Q Q P Q - C X 0 X X The second st order condton states that the decrease revenue from a qualty decrease must equal the lower MC of the qualty ncrease. Compared to Perfect Competton they may actually have the same Q but dfferent X. PRICE DICRIMINATION Perfect P D.:( st degree) sell at max. prce to all, steel all consumer surplus: Fg. 3 Example: Consder the prevous case. How we wll sell to each at the max. prce?

22 What ths means s the LAT buyer wll buy exactly where MC P and that s how much Q we sell. P 00 Q/0 0.Q MC Q * 666 >500 TR Ú Q 0 * Q P( Q) dq 00Q TR 00(666) (666) /40 00(0) 0 / TC 0.05(666) P TR TC 3333 At the margn (last consumer) we sell at P MC 66.6 Market separaton (3 rd degree P.D) eparate nto markets. Max. Proft: P ( + ) Pj ( + ) MC f e e Fg. 3 j P P j Ê ˆ Á + Ë e j Ê ˆ Á + Ë e P j P AC MC Market : Q 4 P Market : Q 4 P ngle Prce under perfect competton pmc Total Q Q + Q where P P so Q 4 P + 4 P 48 3P fp6-(/3)q and MR 6 /3Q f MCAC 6 Then Q 5, P and P TR TC99 If 3rd degree prce dscrmnaton between the market s possble. MR 4 Q and MR Q MR MR MC f Q 9, Q 6, P 5, P 9 P TR +TR TC

23 P P Q +P Q AC*(Q) 9*5 + 9*6 6*5 99 where Q +Q 5 ngle Prce under perfect competton pmc Q 48 3P MR 6 /3Q Q 5, P Dead weght loss n market P w 0.5(P MC)*(Q PC Q ) 0.5(5 6)(8 9) 40.5 nce n market Q PC 8 f P MC 6 Dead weght loss n market DW 0.5(P MC)*(Q PC Q ) 0.5(9 6)( 6) 9 where Q PC n market DW + DW 49.5 DW loss sngle prce: 0.5(P MC)(Q PC Q) 0.5( 6)(30-5) 37.5 Q PC 30 Dscrmnaton through prce selecton nd degree prce dscrmnaton Quantty dscounts: - mnmum purchase requrements - cover charge - te-n sale Two part tarff (prce): Fxed fee added to per unt prce. An example s a cover charge then a per unt cost for drnks. so TR can be wrtten as T(Q) A + PQ Total prce per unt s T A P + P Q Q uppose we look at the demand curve agan n the 3 rd degree prce dscrmnaton case: Example: Q 4 P Q 4 P Under perfect competton P6, Q 8, Q Consumer surplus n the second market s C 0.5(-6)36. Because ths s the market segment wth the smallest demand ths s the most the prce dscrmnatng monopolst could charge 36 as cover charge (admsson)and prce of 6. P T( Q ) + T( Q ) AC*( Q +Q ) 7

24 P 36+6 Q Q 6*( Q +Q ) 7 Ths however s not the best prcng scheme. Note P A+A-(P-MC) *( Q +Q ) (A) should equal C for consumer at the hgher prce C.5(Q )(-P).5(4 P)(-P) o P *0.5(4 P)(-P)-(P-6) *( Q +Q ) Notng that demand s Q +Q 48-3P when they everyone s charged the same prce P *0.5(Q )(-P)-(P-6) *( 48-3P) P 8P-P results n P9, Q 5, Q 6 and A0.5(4 P)(-P)9 so T(Q)9+9Q P 8 whch s less 3 rd degree but has the aspect of farness snce all are charge the same.

25 Regulaton of Monopoly (fx graph) Fg. 33 P A C AC MR MC If AC above demand curve proft s not possble. Fg. 34 AC MR MC D Above Green Area Loss Below we have two set of consumer payng two prces one the monopoly prce the other where P AC Fg. 35 P P AC P 3 MR MC Here consumers payng P subsdze consumer payng P or P 3. RATE OF RETURN REGULATION Producton functon: Q f(k, L)

26 Pf (K,L) -WL K Pf ( K, L) -WL -VK Pf(K, L) WL VK + l (WL + K - Pf (K,L) If l 0 monopoly wthout regulaton If l ( - V)K t means the company should hre an nfnte amount of captal. If the rate of return s above the cost of captal (V). Hence 0 < l < * L Pf L -W + l (W - Pf L )0 * K Pf K - V + l ( - Pf K ) 0 * l WL - K - Pf (K,L) 0 For labor Pf L w, as wll hold for any frm whch proft maxmzes. The last formula s obtaned by addng and subtractng V/(-l) from the proceedng expresson. For captal: ( - l) Pf K V - l f Pf K V - l - l V l( -V ) - - l nce l < and > V Pf K <V whch means a regulated monopoly hres more captal than under an unregulated condton.(lower productvty) IMPERFECT COMPETITION Olgopoly A market structure characterzed by few sellers and nterdependent prce/output decsons Few sellers. A handful of frms produce the bulk of ndustry output. Homogeneous or unque product. Olgopoly output can be perceved as homogeneous or dstnctve. Blockaded entry and ext. Frm are heavly restrcted from enterng or leavng the ndustry. Imperfect dssemnaton of Informaton. Cost, prce and product qualty nformaton are wthheld from unnformed buyers. P f(q) f(q,+q +..+ q n ) Pq - TC q) f ( Q) q - TC ( q ) f ( q + q q ) q - TC ( q ) ( n

27 CARTEL MODEL Max. proft for market (lke monopoly) then splt legal. P PQ - [TC (q ) + TC (q ) +..+ TC n (q n )] q P + P q + q qn ) - MC ( q ) 0 MR( Q) - MC ( q ) 0 q ( Vablty:. legal. nformaton on demand and cost 3. Chslng (cheatng) DUOPOLY MODEL Cournot soluton Frm recognzes f q affects prce but does not realze prce change affects the competton s quantty. He beleves P q j 0 but 0 q q P he beleve n maxmzng the followng P + q - MC ( q ) 0 q q P o output wll be less then P.C. but snce q < 0 output s greater than under a q monopoly or cartel soluton. PRICE LEADERHIP One frms sets prce the other follows. Example: Natural prng Demand : Q q + q 0 P P 0 Q TC 0 P TR TC 0Q Q Cartel soluton: 0 - Q 0 Q Q 60, P 60, P 3600 Cournot

28 q q Assume each frm thnks. 0 q q P 0 q q P Pq (0 q q )q 0q q q q P Pq (0 q q )q 0q q q q q 0 - q - q 0 q 0 - q - q 0 These are called reacton functon; they show how each wll react to the other output level. q q 0 f q q f 0 q q 0 f 0 3q f q 40 q P 40 P 600 LEADERHIP One frm as leader set prces and the other follows: P P MC D Q L Q T tackelberg Leadershp Model q q (0 q )/ - q

29 q P Pq 0q q q q 0 - q - q - q 0 q q 0 3/q q 0 ubstute q (0 q )/ nto ths dervatve. 0 3/q 60 + q / 0 60 q 0 f q 60, q 30 P 30 P 800, P 900 tackelberg model does not say who should be the leader, f both act as a leader both wll produce Q C Q C + Q C > Q T Prce below P.C. If both are followers, cournot soluton. PRODUCT DIFFERENTIATION How the product s perceved Heterogeneous Product group: Output of sets of frms consttute a product group f the substtutablty n demand among the products (cross prce elastcty) s very hgh relatve to other goods. Frm s choces: can choose how much to spend on dfferentates. TC TC I (q, z ) P g(q, P j, z, z j ) P P q TC (q, z ) q P + q P TC - q q 0 z P z q TC - z 0 b/c P dependes on q j & z j, the demand functon become complcated. patal dfferentaton Locaton choces

30 Monopolstc competton: bascally same assumptons as P.C. but wth product dfferentaton: Monopolstc Competton A market structure characterzed by a large number of sellers of dfferentated products. Characterstcs Large # of buyers and sellers. Each frm n the ndustry produces a small porton of ndustry output, and each customer buys only a small part of the total. Product heterogenety. The output of each frm s perceved by customers to be essentally dfferent from, though comparable wth, the output of other frms n the ndustry Free entry and ext. frms are not restrcted from enterng or leavng the ndustry Perfect dssemnaton of nformaton. Cost, prce, and product qualty nformaton s known by all buyers and all sellers n the market. MC AC D MR Ths proft brngs now frms a lower demand for your product untl zero proft: D Assume all frms have same TC C 9 + 4q ame demand

31 n- q -0.0(n -)P + 0.0 P j n f P P j then quantty has to equal q 303/n snce n- P j  P j (n -) If all P j are equal If so, then P. P (303/n) [9 + 4 *( 303/n)] For the moment lets gnore ths Pluggng n q and C n the proft functon P. P q - C We get n- È P q - c P -0.0(n -)P + 0.0 P j Í Î Í n q Pluggng the demand functon n for q and passng P thru we get È n- P Í -0.0(n -)P P Î Í Â P j P n È Ê n (n -)P + 0.0 P j ˆ Í Í Á Ë n Î [ ] Takng the dervatve w.r.t. P P P È Í- 0.0( n -)() P Î - 4 (- 0.0( n -)) 0 n  - P j n now set P P j P ( n -)() P + 0.0( n -) P + - 4( - 0.0( n -)) 0 P olvng for P ( n -) P - 0.0( n -) P - 4( - 0.0( n -)) n n

32 +0.0(n -)P 303 n P 303-4( -0.0(n -)) n 0.0(n -) - 4( -0.0(n -)) n( n -) f 4 P nce P q - C 0 for all Monopolstcally Compettve frms whch means we can plug our results for P and q we just got when P P J Ê P q - c Á Ë n(n -) + 4 ˆ Ê 303 ˆ Ê Á ˆ Á Ë n Ë n n 0, P P j 7, q 3, P I 0 To get ths result we need some ugly algebra afterward we we stll need to plug ths nto a programmable calculator. n (n ) *303 9n 3 9n n 3 n 30300*303/9 n(n n) Plug nto programmable calculator. n0. Game Theory. PERFECTLY CONTETABLE MARKET If entry and ext are freq. and/or outsde potental entrant can enter by cuttng prces and stll make proft. If such proft opportuntes exsted potental entrants would take advantage of them. Natural Monopoly TC 00Q Q P 000 5P P 00 0.Q MR Q MC 00

33 Q M 50 and P M 50 Contestable soluton: Here prce must equal average cost Q 000 5P 000 5(AC) 000 5( /Q) /Q f Q /Q 0 Q 500Q f Q C 400 P C 0 Nash Equlbrum Notaton U A (a *, b * ) U A (a, b * ) for all a à A And U B (a *, b * ) U B (a *, b ) for all b à B Ths frst equaton reads; the Utlty A receve from choosng a strategy of a* f B chooses b* s at least a great than when choosng a strategy of a when B chooses b*. Nash Equlbrum strateges A par of strateges (a *, b * ) represent an equlbrum soluton to a player game f a * s an optmal strategy gven the equaton above for A aganst b * and b * s an optmal strategy for B aganst a *. Consder the followng game wth a par of possble strateges. If A chooses L then he gets 7 unts of utlty f B also chooses L. If A chooses L and B chooses then A gets 6 utls. If A chooses then he gets 6 unts of utlty assumng B chooses L. If A chooses and B also chooses then A gets 6 utls. Now consder B. If B chooses L then he gets 5 unts of utlty f A also chooses L. If B chooses L and A chooses then A gets 4 utls. If B chooses then he gets 4 unts of utlty assumng A chooses L. If B chooses and A also chooses then B gets 3 utls. Dormtory Game (game tree) L 7.5 B L A 5.4 L B

34 The form of the decson tree mples B must choose after A. In ths case we assume B does not know what A has chosen. Below can be a smultaneous verson of the game above n normal form or one can move frst. Frst move A: L A L B Notce that B s always better off choosng L no matter what A chooses. Ths s B s domnant trategy. Therefore f A knows ths, he wll also chose L snce 7>6. The L and L strategy s a Nash equlbra. The other strateges do not offer a Nash equlbra snce n each one a player can mprove hs stuaton by agreeng to one strategy then swtchng. No Nash equlbrum: A B R P R 0,0,- -, -, 0,0,- P,- -, 0,0 The above game has no Nash equlbrum snce any combnaton of strateges poses a stuaton n whch one player can mprove hs stuaton by swtchng. Here there are two equlbra M B M. 00 A 00

35 If one chooses M then the other s better off also choosng M. If one chooses then the other s better of also choosng. There are Nash equlbra here. Mxed strateges: games wth only fnte strateges n whch the pure strateges are played wth certan probabltes. Lets assume the A wll chose M wth probablty (r) and wth probablty (-r). B wll chose M wth probablty (s) and wth probablty (-s). A M M B, 0,0 0,0, r -r s -s The expected utlty for A wll be E(U A )r*s* + r*(-s)*0 + (-r)*s*0 +(-r)*(-s)* E(U A )-r-s+3rs-s-r(3s-). If s>(/3) then A wll chose M (r). If s<(/3) then A wll chose (r0). If s(/3) A s ndfferent between and M snce E(U A )/3 n all stuatons. To see ths set r n the equaton above and compare t the stuaton where r0. -s-(3s-) -s-0(3s-). The same can be easly done for B. ee text. ub-game perfect equlbrum: a Nash equlbrum n whch the strategy choces of each player do not nvolve non-credble threats. That s, no strategy n such equlbrum requres a player to carry out an acton that would not be n ts nterest at the tme the choce s made. Repeated games: Bertrand Equlbrum Assumng equal MC, homogeneous goods.

36 The game conssts of each frm undercuttng ts rval untl they can now longer do so wthout makng profts negatve. The outcome wll be P A P B MC. Because f: P A P B > MC one can ncrease proft by lowerng prce. Notce the dfference under Cournot P > MC P MC Cournot Bertrand Infnte Bertrand Game If A chooses P A then B can choose P A > P B > MC and get all the market. If t s nfnte a twn trgger becomes possble P A P B P M where P M monopoly prce Now the queston s whether lowerng your prce ths perod s worth lower profts n all other perods. If not you receve than / of the monopoly proft from now on, f you cut proft you receve all monopoly profts ths perod and zero n the future. > ( m m d m d m d m The frm wll not cut ts prce as long as ) m < m Ê ˆ Á Ë- d Where d s the dscount factor. The /(-d) comes from the nfnte sum rule assumng monopoly proft stays the same each subsequent perod. If d>(/) then the frm wll not cut prces. If d(/) then the frm s ndfferent. Consder a stuaton wth more frms and the followng demand, MC, FC and proft. Q P MC 0, P m 40000, FC0 Wth more frms: m > ( m +d m +d m d m ) n

37 40000 > Ê ˆ Á n Ë -d f n - d < n ENTRY, EXIT AND TRATEGY Now consder the Cournot natural sprng example wth a fxed cost of $784. Assume further that A has frst mover advantage. The reacton functon of B s unchanged: (Only varable cost effects short run q.) q B (0 q A )/ fp 0 q A - q B P B Pq B 784 P(0 q A )/ 784 (0 q A q B )(0 q A )*/ 784 (0 q A )[0 q A (0 q A )(/)] Ê0 - q A ˆ f B Á Ë ettng proft of B to zero f (0 q A )*/ 8 o as long as q A 64 B wll not enter the market snce profts wll be zero or less. At the prce of 64 the profts for A are P A 56* Entry and ncomplete nformaton In ths case whether B enters depends on the subjectve prob. You assgn a probablty the exstng frm s hgh or low cost. P prob. of low cost and P prob. of hgh cost P B 3 f A s hgh cost P B - f A s low cost E(P B ) P*3 + ( P)*(-) f P>/4 E(P B )>0 so B enters It s therefore n A s nterest to make B beleve he s low cost weather he s or not. The problem occurs when B knows ths. He knows t s n your nterest to at low cost and you wll act ths way no matter what.

38 Predatory prces: Drvng others out of busness wth low prces. Then charge monopoly prces. Bayesan Nash Equlbrum: a strategy par a * b * wll be a Bayesan Nash equl. Provdng that a * maxmzes A expected utlty when B play b * and vse versa. [ ] " a` Ã A E[ U A (a *, b *, t A, t B )] Â B A (t B )U(a *,b *,t A,t B ) > E U A ( a,b *,t A,t B ) t B Where t B represents the type of player B may be. B A (t B ) s the subjectve probablty that A puts on B beng of a certan type. BAYEIAN NAH EQUIL. EXAMPLE: Inverse demand: P 00 q A - q B Assume ntally that MC A MC B 0 f q A q B 30 P A P B 900 Now assume Imperfect nformaton: MC A 0, MC B ether 6 or 4, unknown to A ubjectve prob. 50% for each so E(MC).5(6)+.5(4)0 nce MC s constant and equal to AC P B (P MC B )q B (00 q A q B - MC B )q B Proft max. P B q B 0. fq B * (00 - MC B q A )/ If B s hgh cost MC6 q BH * (84 q A )/ If B s hgh cost MC4 q BL * (96 q A )/ We can now calculate a expected proft. E(P A ) q A [0.5(00 MC A q A q BH ) + 0.5(00 MC A q A q BL )] E(P A ) (90 q A 0.5q BH 0.5q BL )q A Frst order condton P A q A 0 : f q A * (90 0.5q BH -0.5q BL )/ f q A 30, q BH 7, q BL 37 MECHANIM DEIGN AND AUCTION

39 Game theory can help obtan a desred results under varous economc stuatons. Consder an aucton wth economcs agents bddng. Let V A denote the value agent A places on the product and let V denote the same for agent. The goal of the seller of ths product s to get the best prce. The beneft(proft) for agent A s denote below. A beleve all other agents place a value on the product that s unformally dstrbuted between 0 and. If agent A bds hgher than anyone else then hs beneft s. P A (V A b A )Prob. (b A > b ) "..n-, Where If agent A does not have the hghest bd P A 0 f b > b A for any Prob. (b A > b,..n) Prob(b A > k V,,.,n) Where k s the fracton of the valuaton agent bds for the product. Now consder the followng property of the unform dstrbuton. Prob(b A > k V ) Prob(b A /k > V ) Therefore Ú b A k 0 dv A b A /k Prob. (b A > b,..n) ( b A ) b A b A... b A b n- A k k k k n- By symmetry we can conclude that k k " o the Expected proft of A s n- b A P A (V A b A ) k n- Maxmzng, the optmal bd for A should be. n- P A 0 n - f ba VA b A n k n-

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