EconS 501 Micro Theory I 1 Recitation #12 Imperfect Competition

Transcription

1 EconS 50 Mcro Theory I Rectaton # Iperfect Copetton Exercse 5.3 (NS). [ON YOUR OWN] Ths exercse analyzes Cournot copetton when frs have dfferent argnal costs. Ths departure fro dentcal frs allows the student to shft around fr s best-responses ndependently on a dagra. Let c be the constant argnal and average cost for fr (so that frs ay have dfferent argnal costs). Suppose deand s gven by P Q. a. Calculate the Nash equlbru quanttes assung there are two frs n a Cournot arket. Also copute arket output, arket prce, fr profts, ndustry profts, consuer surplus, and total welfare. ANSWER: Let us fnd the output level that fr selects when axzng ts profts n ths Cournot duopoly ax q ( q q ) cq q :qq c q q c q best response for consuer Lkewse qc q best response for consuer Solvng sultaneously for q and q, whch yelds an output for fr of q c c q 4q q c c c c c q 3 Felx Munoz-Garca, School of Econocs Scences, Washngton State Unversty, Pullan, WA, Eal: funoz@wsu.edu.

2 c c c And slarly, an output for fr of q. 3 Further, aggregate output s c c c c c c c ( c c ) Q q q 3 3 3, whch ples that equlbru prce s and that fr s profts are and those of fr are As a consequence, aggregate profts are and consuer surplus s ( c ) P c c c 3 Q, ( c c ) c 9 qp cq, ( c c ) c 9 qp cq, ( c c ) ( c ) c c c c 9 ( c ) CS c c Q p 8 ( ), and aggregate welfare s thus the su of aggregate profts and consuer surplus, c c c W CS. b. Represent the Nash equlbru on a best-response functon dagra. Show how a reducton n fr s cost would change the equlbru. Draw a representatve soproft for fr. ANSWER: Pont E n followng fgure represents the Nash equlbru. The curved lne represents fr s soproft.

3 q BR (q ) E E BR (q ) q The reducton n fr s argnal cost shfts ts best response outwards ( q c q ) and shfts the equlbru fro E to E. Fr wll produce ore for any gven q. Exercse 5.7 (NS). Ths exercse analyzes the Stackelberg gae both wth and wthout the possblty of entry-deterrng nvestent. Assue as n Proble 5. of NS that two frs wth no producton costs, facng a drect deand functon Q 50 P, choose quanttes q and q. [In Proble 5., frs were assued to copete a la Cournot, sultaneously selectng quanttes. In that context, you can easly show that every fr s best response functon s q 75 q j.] a. Copute the subgae-perfect equlbru of the Stackelberg verson of the gae n whch fr chooses q frst and then fr chooses q. ANSWER: We solve the gae usng backward nducton startng wth fr s acton. Fr oves second and best responds to fr s choce. We saw q fro Proble 5. (b) that fr s best-response functon s q 75. We substtute ths back nto fr s proft functon so that fr s akng ts optal choce gven what t expects fr to do 3

4 q q[50 ( qq)] q50 q75 :50q75 q. q q 75 Takng the frst-order condton wth respect to q and solvng yelds q 75. Substtutng ths back nto fr s best-response functon yelds q q b. Now add an entry stage after fr chooses q. In ths stage, fr decdes whether or not to enter. If t enters t ust snk cost K, after whch t s allowed to choose q. Copute the threshold value of K above whch fr prefers to deter fr s entry. ANSWER: If fr accoodates s entry, the outcoe n part (a) arses. P=50 Q. P = There are no producton costs, thus proft for fr = PQ=37.575=,8.5. When fr produces q f fr best responds to q, and enters t wll generate proft p qk (50 q) / 4 K. In order to deter entry ths proft ust be less than or equal to 0. Settng fr s proft functon equal to zero (50 q ) K 4 q 50 K 0 The threshold value of q 50 K. Fr s proft fro operatng alone n the arket and producng ths output s QP= ( 50 K )( K ), whch exceeds,8.5, the proft n part (a), f K 0.6. (as can be shown by graphng both sdes of the nequalty) Exercse 5.9 (NS). Ths exercse exanes the Herfndahl ndex of arket concentraton. Many econosts subscrbe to the conventonal wsdo that ncreases n 4

5 concentraton are bad for socal welfare. Ths proble leads students through a seres of calculatons showng that that the relatonshp between welfare and concentraton s not ths straghtforward. One way of easurng arket concentraton s through the use of the Herfndahl ndex, whch s defned as: n H s where s q Q Where s s fr s arket share. The hgher s H, the ore concentrated the ndustry s sad to be. Intutvely, ore concentrated arkets are thought to be less copettve because donant frs n concentrated arkets face lttle copettve pressure. We wll assess the valdty of ths ntuton usng several odels. a. Recall fro class the equlbru output levels n the n-fr Cournot gae. If you don t reeber t, don t worry, here t s agan for you to practce. Fnd equlbru output level for every fr, the resultng aggregate output, and the arket prce. In addton, evaluate consuer surplus, ndustry proft, and total welfare n ths equlbru. Once you are done, copute the Herfndahl ndex for ths equlbru. ANSWER: Fr s proft s q ( a bq bq c) wth assocated frst-order condton a b bq c 0. Ths s the sae for every n fr so we ay pose syetry [ Q ( n ) q ]. Pluggng n the syetry condton n the above frst-order condton, yelds a bb( n) q c0 ( a c) q ( n) b Further, aggregate output s na ( c) Q, ( n) b Thus plyng that equlbru prce s ( a nc) P, ( n ) And that aggregate (ndustry) profts are n ( a c) n b ( n), where ndvdual profts are just /n share of aggregate profts snce frs are syetrc. As a consequence, consuer surplus s 5

6 CS b ( n) n ( a c) whle overall socal welfare (.e., the su of aggregate profts plus consuer surplus) s n ( a c) W ( n) b. Because frs are syetrc, s / n, thus we can solve for the Herfndahl ndex H n( ). n n b. Suppose two of the n frs erge, leavng the arket wth n- frs. Recalculate the Nash equlbru and the rest of the tes requested n part (a). How does the erger affect prce, output, proft, and total welfare. Copute the Herfndahl ndex for ths equlbru. ANSWER: We can obtan a rough dea of the effect of erger by seeng how the varables n part (a) change wth a reducton n n. Per-fr output, prce, ndustry proft, and the Herfndahl ndex ncrease wth a reducton n n, caused by the erger. Total output, consuer surplus, and welfare decrease wth a reducton n n, caused by the erger. c. Put the odel used n parts (a) and (b) asde and turn to a dfferent setup: that of Proble 5.3, where Cournot duopolsts face dfferent argnal costs. Use your answer to Proble 5.3(a) to copute equlbru fr outputs, arket output, prce, consuer surplus, ndustry proft, and total welfare, substtutng the partcular cost paraeters c c. Also copute the Herfndahl ndex. 4 ANSWER: Substtutng c c / 4 nto the answers for 5.3, we have q / 4, Q /, P /, / 8, CS / 8, and W / 4. Also, H /. d. Repeat your calculatons n part (c) whle assung that fr s argnal cost c falls to 0 but c stays at 4. How does the erger affect prce, output, proft, consuer surplus, total welfare, and the Herfndahl ndex. ANSWER: Substtutng c 0 and c / 4 nto the answers for 5.3, we have q 5 /, q /, Q 7 /, P 5/, 9 / 44, CS 49 / 88, and W 07 / 88. Also, H 9/ 49. 6

7 e. Gven your results fro parts (a)-(d), can we draw any general conclusons about the relatonshp between arket concentraton on the one hand and prce, proft, or total welfare on the other? ANSWER: Coparng part (a) wth (b) suggests that ncreases n the Herfndahl ndex are assocated wth lower welfare. The opposte s evdenced n the coparson of part (c) to (d): welfare and the Herfndahl ncrease together. General conclusons are thus hard to reach. Exercse 5.0 (NS). [ON YOUR OWN] Ths exercse extends the Inverse Elastcty Prcng Rule (IEPR) fro a arket structure wth only one fr (onopoly) to arket structures wth ore than one fr. It derves an alternatves for of the IEPR we know under onopoly that we can apply nto a Cournot odel of quantty copetton. a. Use the frst-order condton (Equaton 5.) for a Cournot fr to show that the usual nverse elastcty rule for Chapter holds under Cournot copetton (where the elastcty s assocated wth an ndvdual fr s resdual deand, the deand left after all rvals sell ther output on the arket). Manpulate Equaton 5. n a dfferent way to obtan an equvalent verson of the nverse elastcty rule: P MC s q where s P e Q QP, where s s fr s arket share and e QP, s the elastcty of arket deand. Copare ths verson of the nverse elastcty rule to that for a onopolst fro the prevous chapter. ANSWER: Equaton 5. can be rearranged as follows: Equaton 5. PQ ( ) P( Qq ) C ( q) 0 q P C Pq, P P P C P ' q dp / dqq P P P where q, P s the elastcty of deand wth respect to fr s output. The second equalty uses the fact that P dp / dq dp / dq. Multplyng nuerator and denonator by Q, we can also rearrange Equaton 5. as q, P 7

8 P C P' q dp / dqq dp / dq Q q s P P P P Q QP, MWG.C.0. Consder a J-fr Cournot odel n whch frs costs dffer. Let cj( qj) jc ( qj) denote fr j s cost functon, and assue that c () s strctly ncreasng and convex. Assue that... j. (a) Show that f ore than one fr s akng postve sales n a Nash equlbru of ths odel, then we cannot have productve effcency; that s, the equlbru aggregate output Q s produced neffcently. ANSWER: Each fr chooses ts output q 0 to axze ts profts pq ( q) q cq ( ) Takng frst-order condtons wth respect to q, and assung an nteror soluton, yelds p(q)+p (Q)q =α cq ( ) The left-hand sde represents the argnal revenue that, for a gven Q - output produced by all other frs, fr obtans when t ncreases ts ndvdual producton q. Snce p (Q)<0, the argnal revenue les below the deand curve p(q); as llustrated n the fgure below. The rght-hand sde, n contrast, represents the argnal cost of fr, whch graphcally s ncreasng n q gven convexty n costs. You can vsually notce that a gven ncrease n α, for nstance f fr j has α j >α then the argnal cost curve shfts upwards, thus crossng the argnal revenue of fr j to the left-hand sde,.e., q j <q. 8

9 Iagne the case where fr dffers fro fr j we ay calculate the dfference n argnal cost between the two frs as jc ( qj) c ( q) Fro our frst-order condtons we know that after cancelng out the p( Q) ters ths s equvalent to p( Q)( q qj) Iagne the partcular case where j so that also q qj. Because p ( Q) < 0 the ter p( Q)( q qj) s postve. Ths ples that jc ( qj) c ( q). Thus the argnal cost for fr j s greater n ths case, plyng that argnal costs across frs are not necessarly equalzed. Lkewse aggregate output s not necessarly produced effcently. Indeed, for aggregate output Q to be effcent, we would need that the su of argnal costs concdes wth the argnal revenue curve descrbed above. In the settng n whch frs ndependently choose ther output levels, every fr s output s ncreased untl the pont n whch the argnal revenue s equal to fr s argnal costs. If frs argnal costs were constant and syetrc, then ther aggregate output would be effcent. However, gven that frs argnal costs are ncreasng (convexty), aggregate output s not necessarly effcent. (b) If so, what s the correct easure of welfare loss relatve to a fully effcent (copettve) outcoe? [Hnt: Reconsder the dscusson n Secton 0.E] ANSWER: The correct easure of welfare loss relatve to a fully effcent outcoe n ths case s equal to the loss of consuer surplus due to noncopettve prcng plus the hgher producton cost due to productve neffcency. (c) Provde an exaple n whch welfare decreases when a fr becoes ore productve (.e., when j falls for soe j). [Hnt: Consder an proveent n cost for fr n the odel of Exercse.C.9.] Why can ths happen? ANSWER: Let s use our results fro Exercse.C.9 n order to provde an exaple n whch welfare decreases when a fr becoes ore productve. Fro.C.9 the Cournot equlbru output and prce levels are a c c q 3b a c c q 3b acc p 3 Total profts of the two frs can now be coputed as ( p c ) q ( pc ) q (a 5c 5c a( c c ) 8 cc )/9b Consuer surplus can be coputed as 9

10 q q q q 0 p( qdq ) ( aqbq / ) ( acc)(5 acc) /8b 0 Addng up total profts and consuer surplus, and dfferentatng wth respect to c, we obtan Surplus 9c 9c 4a c 9b 4 Ths dervatve s postve when c c ( 9) a. Ths wll occur when fr s costs are uch greater than fr s costs. In ths case a decrease n c reduces socal welfare. The reason s that when c slghtly falls, fr steals ore busness fro fr, whch rases producton neffcency. When c s substantally larger than c, ths effect actually donates the ncrease n consuer surplus due to a lower prce. MWG.C.. Consder two strctly concave and dfferentable proft functons ( q, q ),, j {,}, defned on q [0, q]. j j j (a) Gve suffcent condtons for the best-response functons bj( q j) to be ncreasng or decreasng. ANSWER: Assue that ( q, qj) 0 for,. Where the subscrpt eans dfferentate twce wth respect to the frst eleent. Every fr wll axze proft by fndng ts best response functon b (q j ). We can evaluate the sgn of fr s best response functon (to check f t ncreases or decreases wth ts rval s output q j ) by applyng the plct functon theore, as follows b( qj) ( b( qj), qj) : qj qj ( b( qj), qj) Therefore, snce the nuerator satsfes ( q, qj) 0 for, by defnton, b( qj) the sgn of s the sae as the sgn of q ( b ( ), ) qj qj. In other words, fr j s best-response functon s ncreasng (decreasng) when the cross-dervatve s postve (negatve, respectvely). (b) Specalze to the Cournot odel. Argue that a decreasng (downward-slopng) best-response functon s the noral case. ANSWER: In the Cournot odel 0

11 ( q, q ) p( q q ) q c( q ) j j Then dfferentatng further wth respect to q j (, ) ( ) ( ) q q p q q q p q q j j j whch s negatve f p() s downward slopng and not to convex. Ths sees lkely for the nverse deand functon, thus the noral slope of the best response functons n the Cournot odel s negatve. MWG.D.. Consder an nfntely repeated Bertrand duopoly wth dscount factor. Deterne the condtons under whch strateges of the for n (.D.) sustan the onopoly prce n each of the followng cases: Fro page 40 (.D. n MWG): p f all eleents of Ht equal ( p, p ) or t Pjt ( Ht ) c otherwse (a) Market deand n perod t s x ( ) t t p x( p) where 0 s the rate of growth of deand across perods. ANSWER: Monopoly proft n perod t s ax x( p)( pc) ax x( p)( pc) p t t t If a fr devates at perod t, t can obtan n that perod, and t wll get zero forever after. If t does not devate, ts payoff s t ( ) t0 ( ) Monopoly prce can be sustaned when devaton fro the strategy s not proftable. Devaton s not proftable f and only f or ( ) p Hence, the nal dscount factor supportng cooperaton decreases n the rate of growth of deand,.e., cooperaton can be sustaned under a larger set of dscount factors as deand grows faster across perods. [See fgure below].

12 (b) At the end of each perod, the arket contnues to exst wth probablty [0,]. ANSWER: If a fr devates, t can obtan n that perod, and t wll get zero forever after. If t does not devate, ts payoff s t0 t ( ) ( ) Therefore, devaton s not proftable f and only f or ( ) Thus, cooperaton cannot be sustaned under any dscount factor (between zero and one) when the probablty that deand contnues exstng s relatvely low, but can be sustaned when the probablty that deand contnues exstng s suffcently hgh (and decreases as ths probablty gets closer to 00%). [See fgure, where and are both saller than.]

13 (c) It takes K perods to detect and respond to a devaton fro the collusve agreeent. ANSWER: If a fr devates, t can obtan K K t ( ) t0 ( ) In the next K perods, and t wll get zero forever after. If t does not devate, ts payoff s t t0 ( ) Therefore, devaton s not proftable f and only f K ( ) or ( ) ( ) Hence, the ore perods of te K that a cheatng fr reans undetected by ts colludng partners, the ore attractve cheatng becoes. Cooperaton therefore can only be sustaned under ore restrctve sets of paraeter values. [See fgure]. K 3

14 K 4