( ) = k i L + k j. ( ) k j P X

Transcription

1 Appedix Problem 1 With two differet ad perfectly distiguishable demads for the fial product ad a fixed cost correspodig to a moopolically provided iput, fid the equilibrium prices ad the umber of firms i a moopolistic competitio framework. et K ( P = k i + k j ad K P j i = k i + k j be the differet demads for the moopolistic competitors product ad the fixed cost each of them faces. Each idividual moopolistic competitor will solve the problem (i max P K ( P k j + P K P k j j i j i If there are equal firms i the market, i equilibrium the followig coditio must apply j i (ii k i J = K J i J The first order coditios for the moopolistic competitor s problem ca therefore be writte as (iii (iv K + P K + P K P = 0 K P = 0 From (iii ad (iv it is possible to derive the equilibrium value for market demad elasticities (v = = 1 which is the reciprocal of the umber of firms. To solve the problem, it would be ecessary to fid the equilibrium umber of firms ad substitute it i the expressio for the equilibrium prices. To avoid excessive abstractio, we cotiue our aalysis workig with a liear-demad case. i

2 The liear case et K ( P = A P ad K P coditios for the problem are = A B P be the demad fuctios. The first order (vi (vii A P A B P P = 0 B P = 0 from where it is possible to obtai expressios for P ad P as fuctios of (viii P = (ix P = 1 A ( +1 1 A ( + 1 B ad a price ratio which depeds oly o the parameters of the demad fuctios (x P P = A A B It is straightforward to see that a ecessary ad sufficiet coditio for a sigle price to hold is ( A = A B Substitutig the results for P ad P ito the expressios for K ad K respectively gives (xi K = A A ( +1 = + 1 A (xii K = A A ( + 1 = + 1 A Now we ca replace equatios (viii, (ix, (xi ad (xii ito the zero-profits coditio ( A 2 (xiii Π = P K + P K = +1 where solvig for gives the equilibrium umber of firms B ( + 1 X A ii

3 (xiv = + ( A 2 + ( A 2 B 1 ad makes it possible to fid the values for the equilibrium prices. Cosumer Surplus I the liear demad case, cosumer surplus is give by (xv V 1 = 1 2 K A B P + K A B P Replacig (viii, (ix, (xi ad (xii ito the above expressio yields (xvi V 1 = A A A + which, after factorizig ad rearragig, ca be reduced to +1 A A B 1 +1 A B (xvii V 1 = Problem A 2 + A B With two differet ad perfectly distiguishable demads for the fial product ad a fixed cost correspodig to a moopolically provided iput, fid the equilibrium price ad the umber of firms i a moopolistic competitio framework, give the costrait that each competitor must charge the same price to all cosumers, regardless of their type. Usig the same otatio as i the previous case, with the differece that ow there will oly be oe price, P, the problem becomes (xviii max P K ( P k j + K ( P k j j i j i its first order coditio beig (xix K + K + P K P + K = 0 P iii

4 The liear case With the same fuctioal forms as i problem 1, the first order coditio becomes (xx ( A + A P( + B P( + B = 0 ad the equilibrium price as a fuctio of takes the form of (xxi P = 1 +1 ( A + A + B At this poit, it is coveiet to add horizotally both demads. Settig A = A + A, B = + B ad K = K + K, the added-up demad fuctio turs ito K = A BP. Isertig equatio (xxi ito it gives (xxii K = A A ( + 1 = + 1 A Substitutig equatios (xxi ad (xxii ito the zero-profits coditio (xxiii Π = allows to solve for 1 A 2 ( B P = 0 X (xxiv = + ad cosequetly to fid the value of P. A B 1 = + A + A 1 ( + B Cosumer Surplus I this problem, the liear-demad cosumer surplus is give by (xxv V 2 = 1 2 K A B P + K A B P Replacig expressio (xxi ito the demad fuctios allows to solve for K ad K iv

5 (xxvi (xxvii K = A K = A 1 ( ( +1 ( A + A + B ( A + A ( + B B Substitutig expressios (xxi, (xxvi ad (xxvii ito (xxv ad rememberig that A = A + A ad B = + B gives (xxviii V 2 = 1 A A B B A B 1 +1 which after some tedious algebra ca be reduced to A + A B 1 A ( +1 B 1 +1 B B A A B (xxix V 2 = 1 2 ( A A A 2 2 B B +1 Compariso betwee problems 1 ad 2: equilibrium umber of firms, equilibrium prices, cosumer surplus ad welfare Deotig with 1 ad 2 the equilibrium umber of firms i problems 1 ad 2 respectively ad usig the same demad fuctios, a ecessary ad sufficiet coditio for 1 to be greater tha 2 is (xxx ( A 2 ( + A 2 ( > A 2 + ( A 2 + 2A A B + B Addig up the right had side ad multiplyig both sides by ( B ( + B (xxxi ( A 2 B ( + B + ( A 2 ( + B > ( A 2 ( B + ( A 2 ( B + 2A A B Rearragig (xxxii ( A B 2 + ( A 2 > 2( A B ( A ad factorizig v

6 (xxxiii ( A B A 2 > 0 which is true if ad oly if (xxxiv A A B Equatio (xxxiv is false if ad oly if the price at which the quatity demaded is equal to zero is the same for both demad fuctios. I ay other case, the equilibrium umber of firms i the two price problem will be greater tha that holdig whe oly oe price is allowed. Note that if i equilibrium 1 were equal to 2, the P would be equal to P, ad both would be equal to P, as the coditio A ( = A B would hold. owever, should 1 be greater tha 2, P would take values betwee P ad P, fidig its equilibrium at a higher poit tha i the equal-umber-of-firms solutio. Figure i shows the solutio for either problem whe the coditio ( A = A B holds. A B = P A Figure i Equilibrium quatities ad prices whe both demads have the same itercept P = P K K K * A K * A K 2 et s ow tur our attetio to cosumer surplus. Settig Z = A expressio (xvii ca be rewritte as ( + A 2 ad usig (xiv, B vi

7 (xxxv 2V 1 = ad, after rearragig Z 1 Z 2 Z = 1 1 Z 2 Z (xxxvi 2V 1 = Z 2 Z + I a similar way, usig (xxiv, it is possible to write (xxix as 1 2 (xxxvii 2V 2 = Z 2A B A B ad rearragig A = Z 2A BP B ( X B B (xxxviii 2V 2 = Z 2A B + Substractig (xxxviii from (xxxvi = 2 (xxxix 2 V 1 V 2 from where A B Z P X (xl V 1 V 2 2 = A + A A + B ( + A 2 = ( B 2 +1 ( 1 +1 ad fially (xli V 1 V 2 = Expressio (xli shows that, with the same demad fuctios, the two price solutio always reduces cosumer surplus respect to the oe price solutio, except for the case where the equilibrium umber of firms is idetical for both problems. To aalyze the chage i welfare betwee the two price ad the oe price solutios we must also look at profits; as the moopolistic competitors always have zero profits, all beefits i the idustry are those of the moopolist sellig the iput X. Assumig the moopolist s costs are fixed, the differece i his profits are equal to the differece i the equilibrium umber of firms times the vii

8 uit price of the iput X. Therefore, the chage i welfare betwee the solutios of problems 1 ad 2 ca be writte as (xlii W = V 1 V 2 + ( 1 2 Dividig by (xliii W = V 1 V ad usig (xli (xliv W = 0 Therefore, society does ot gai or lose from passig from the oe price scheme to the two price scheme, as the loss i cosumer surplus is exactly offset by the moopolist s icreased profits. For further clarificatio, a simple umerical example follows. A umerical example et the demad fuctios be K = 20 2P ad K = 10 2P, ad = 5. The table gives the equilibrium values for prices, quatities ad umber of firms. Variable Two price case value Oe price case value P P P K K V W viii

9 Problem 3 With two differet ad perfectly distiguishable demads for the fial product, differet variable costs accordig to the type of customer to which the product is sold ad a fixed cost, fid the equilibrium prices ad the umber of firms i a moopolistic competitio framework. Keepig the usual covetios, the competitor s problem ow ca be expressed as (xlv K P max P k j + P j i K P with C allowig for a residual fixed cost. The first order coditios become k j C j i (xlvi K + P K P P X K P = 0 (xlvii K + P K P P X K P = 0 As i the precedig problems, we will ow go o with the liear case aalysis The liear case Usig the same demad fuctios as i problems 1 ad 2, the first order coditios take the form of (xlviii (xlix A P A B P P + = 0 B P + B = 0 from where prices as a fuctio of the equilibrium umber of firms may be obtaied. (l P = 1 A +1 B + P X (li P = A B + P X ix

10 The price ratio may therefore be writte as (lii P P = A + A B + Deotig with P 1 ad P 1 the equilibrium prices resultig from problem 1 ad with P 3 ad P 3 those emergig from this problem, from equatios (x ad (lii it is possible to write (liii P 1 P = P 3 1 P P 1 3 P 1 = P X which says that the equilibrium price ratio i problems 1 ad 3 will be the same if ad oly if the iput price ratio i problem 3 is equal to the equilibrium price ratio i problem 1. Usig the expressios for prices i (l ad (li, quatities may be writte as fuctios of (liv (lv K = A A P X = + 1 A [ ] K = A A B P X = +1 A B [ ] Substitutig (l, (li, (liv ad (lv ito the zero-profits coditio gives (lvi Π = 1 A + 1 B + P X +1 A [ ] +1 A [ ] A B + P X +1 A B [ ] +1 A B [ ] C = 0 ad after rearragig (lvii Π = + 1 [ ] A 2 A B P X + A B [ ] A B P X C = 0 from where it is possible to solve for the equilibrium umber of firms i terms of the parameters, ad, substitutig i equatios (l ad (li, fid the equilibrium prices. Further developmet of this aalysis will ot be coducted here, as o compariso betwee this problem ad the precedig oes is possible with exogeously set iput prices. x