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# Note for EconS 571 with Professor Gibson at WSU

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Date Created: 02/06/15

MONOPOLISTIC COMPETITION WITH HETEROGENEOUS FIRMS There is a continuum of rms that produce differentiated products Consumers have utility functions that exhibit love for variety and solve the maximization problem max l Lzlogc0 longczp dz p 0 st poo0 J pzczdz w2 7239 62 2 0 Here 7239 are pro ts of the rms which are owned by the consumers The solution to this problem is w7r 0 M 0 w7r w7r 62 f4 7p 2 1 1272a fuzsz 1272 where 1ep p P Ion pz39 dz39 Good 0 is produced with the constant returns production function y0 0 and sold in a competitive market We set p0 w l as numeraire Firm 139 has the production function y2 max xz z f 0 Notice that rms have potentially different productivity levels xz The rm solves the pro t maximization problem c z 7239 Y 7239 max ma a f P2 1 f pz1quot PH xzpzlquot P1 taking P as given The solution is 102 pm A model with a nite number of productivity levels Suppose that there are 3 different productivity levels x3 gt x2 gt x1 gt 0 and that there is a measure 71 of potential rms of each productivity level x j Suppose too that rms exit until remaining rms all earn nonnegative pro ts Depending on parameters there are 6 different possibilities 1 A subset of firms with productivity x3 produces and earns 0 pro ts This subset has measure 773 S n3 2 All rms with productivity x3 produce and earn nonnegative pro ts No rm with productivity x2 can earn nonnegative pro t 3 All rms with productivity x3 produce and earn positive pro ts A subset of rms with productivity x2 produces and earns 0 pro ts This subset has measure I92 S n2 4 All rms with productivity x3 produce and earn positive pro ts All rms with productivity x2 produce and earn nonnegative pro ts No rm with productivity cl can earn nonnegative pro t 5 All rms with productivities x2 and x3 produce and earn positive pro ts A subset of rms with productivity x1 produces and eams 0 pro ts This subset has measure l S n1 6 All rms with productivities x2 and x3 produce and earn positive pro ts A11 rms with productivity cl produce and earn nonnegative pro ts Notice that possibility I is just the DiXitStiglitz model with homogenous rms To illustrate how to compute equilibria we suppose that we are in case 5 lep p 79 79 P a 1 n2 1 n3 1 P J L pdz L pdzJ L pdz 0 P99 0 sz 0 sz To determine l we solve u 6 c p Hq g p pa p f0 x 1 1 F99 Mp quot296 Maxi L mzma m r f0 L L L quot1991 P n2xr L L 17 1 20 pmwnxm g P f a a To determine 7239 we solve 7r1pfaxl n2xi 7 Hex rlln2713f 1 P 1quot2quot3f 7139 LLL n wm wm w nyn1 p anzraf Plugging in the expression for 1 we obtain L L 17 17 Ff n2 21 21 x1 x1 and sTllpm 1 pyn2n3 A 1mm xi 1 P x3 1pl 1 1 quot1 f my quot2 1 1 pm quot3 1 1 pm Notice that if n1 r12 r13 713 and x1 x2 x3 this collapses to the usual formula for homogenous rms A model with a continuum of productivity levels Suppose that there is a measure 71 of potential rms Firm productivities are distributed on the interval x Z 1 according to the Pareto distribution with distribution function F x 1 x which has the density function dFx yx39H Notice that the mean of x is no W474 y 1 Ex f0 xdFx f x7x397 1dx y 1 l 7 and the variance is 2 2 L2 LZ 2196 Ex Ex jlx y1 dFxjlx y Jyx dx y2 oo L 2 1 71 2 2 7 Ex 7Ex 7m For the variance to be nite we require that y gt 2 As we will see we also require that 7 gt p 1 p We can think of restricting productivities to satisfy x 21 as a normalization of units relating labor to consumption of differentiated goods by xing the minimum productivity If we want to normalize units in some other way we could replace the distribution function with Fx1 t97x397 for x Z 9 which has the density function dFx 719796 There are now only two possibilities 1 There is a level of productivity f gt 1 for which rms earn 0 profits The set of rms with productivities x 2 f produce This set has measure r077 2 All rms produce and earn nonnegative pro ts Case 1 We start by supposing that there is a cutoff productivity f where rms earn 0 pro ts and calculate i no i no A PH 71quot px1quot dFx nquot px1quot yx39de p pewep Pi npq p7x 1 71pp L pewep quotp1quot 1p7f 1 71 p p x Notice that we require yl p gt p for P to be nite The demand for goods produced by a rm with productivity x is 1 h n yaw pm n py1 p py6nxl 60C 1 1 L m m px1 P1quot one nJH G MW 1 n1 pm 1 We calculate the cutoff productivity 7 1 cf p 71pp ght 1 1 pxcx f pu f0 x px x n1 p7x 1 7lpp 7r f 0 quot7 f n7f 7 71pp 7r Notice that this expression depends on pro ts 7239 which we can calculate as L w cx w 71pp x1 n n pxcx fwx n f x 7 1dx rigf 1 71pp7r pewep rigf 1 7rn A no J xl pyx39 dx an yx39y39ldx n 1p7rnf397f 2 1 m quotFf 11p Notice how similar this expression is to the analogous expression for the model with a nite number of productivity levels 1mm nf 71 p pyn 1mm y1 p pyn 11p 11p n7f 11p 711p p 7py which implies that 1 Y W m p p 7p 7lpp n7f 7 WW W Case 2 Notice that we are wrong to guess that there is a cutoff productivity f where rms earn 0 pro ts if the value that we calculate for f is less than 1 1 7 f 7 WW lt1 71pp 1 Y quotf lt 7 p p a 7 P1 that is if the xed costs of having all potential rms produce is suf ciently low In this case 7 L pa 71 p p cx MY 71 p Punpy1pp2 x quot1p7 L i 1 L px1 P1 px1quot rm 1 p7 The calculation of total pro ts becomes L lip II anpxcx ch fdFx quotI W f xryrldx 7rn y1 p pyn wa yxy1 dxnfjwyx1 dx m 1 1 n1 pyn nf Z 1 m nf 1 1 pm Notice that the pro ts of a rm with productivity x 1 are 71pp 1p nf f 71pp i1f 11p p1c1c1f plt quotPM p MlmpW y pm 1 1 1 pogo CO f ny1 1 py Aha pm quot7 1 7 p1c1 c1 f M nf gt 0 n711p yW A two country model with a continuum of productivity levels Suppose now that there are two countries 139 l 2 Let each country has a population of Z and a measure of potential rms of n Firms productivities are distributed according to the Pareto distribution F x l x Afirm in country 139 faces a xed cost ofexporting to country j j at 139 of f2 where f gt fd f and an iceberg transportation cost of 1391 1 Z 0 The solution to the rm s pro t maximization problem is to set J 139 PK 96 px In each country there are three possibilities I There are two cutoff levels of productivity le gt Em gt1 Firms with le earn 0 pro ts exporting Firms with Ed earn 0 pro ts producing for the domestic market The set of rms with x 2 f1 produce for the domestic market and for export The set of rms with 3 Z x 2 EM produce for the domestic market only The set of rms with x lt fm cannot earn nonnegative profits and do not produce 2 There is one cutoff level of productivity 2 gt1 Firms with f earn 0 pro ts exporting The set of rms with x Z Z produce for the domestic market and for export The set of rms with f 2 x Z I produce for the domestic market only and earn nonnegative pro ts 1 3 All rms produce for the domestic market and for export They earn nonnegative profits doing both Suppose that we are in case 1 We calculate the price index in country 1 P1 i quotJ Pi xdFx quot2i 1 wa 0 22 p 1 p 1 no no lip P11 P rajm px1quot yxi dx n2 J yxi dx L WNW i L 7079 p1 rap 1 pm 750 p 1 pm 71pp 71pp 71d L p7717p i L pewep P1 rap 1 pWEd 139 7WD p 1 M7 139 71pp 71pp L m i m p pl 1p7 raid 1 n2ri1quot fu 1 mm The demand in country 1 for goods produced by a rm in country 1 with productivity x 2 x1 01 is 71pp cx MIMI p7lP P 1 lxl f L i pewep i pewep 39 pix1quot P11quot 1p7nlid 1 nzri1quot fu 139 J We calculate an expression for the cutoff productivity Ed 01F p ya m p QME 1 1 PiOGUCiOCm If 12 W70 1 pl paw fa0 xld 1 T1 pxld xld 1 p7 quot1x101 1 p n272 pxu p L cl 71pp MANN p11x cllxm 1 fd P71p 1 1977114 f 2039 7 quotEd 1 nzri1quot fu 1 Similarly we calculate an expression for the cutoff productivity in i L 7202072 71 P P52 7T27121 pf992cf992 1 L 1 MW 1 HM 0 2 7 quotin 1 rarf1quot ig 1 The expression for 72391 is no 6 x no c2 x 7a n1 m pxcx 2 JdFxm I pfxcfx J Foe lO Ive 71pp1 ax 7 1 III 2m pewep i pewep f yx dx 7 quotlid 1 quot2T l pfzg 1 i L w 71pp 2 7r2rf1quot x1quot H nllrm pewep i pewep j x dx 7 quotin 1 quot17121 R 1 977079 i 977079 7 17 pll maid 1399 17 pfz zrarf1quot E2 1quot 7 7 1 m 1 m m 1 m quotllxldmmley l quot1 17quot MUDH E2 17quot quot2 17quot 7117121 p 17quot There are analogous expressions for 7 E2 and 72392 This provides us with a system of 6 d fig and 72392 To simplify the equations is 6 unknowns to be solved for 97101 x12 III f2 1 calculations let us consider the symmetric case where 2 Z n1 r12 n and T If r In this case p y1 p pynf7 fd 0 pewep i pewep my 2 1 sz i L 17p y1 p pynr pewep i pewep f2 my 2 Hi n17p minim er kp p 2 p pewep p pewep 17p 17p lrp 17p p 1 p7n xd 1 x2 71pp Notice that 5 lg and that PM ll The equation for 7239 can be rewritten as 1 7 f 7quot n17p2n7n xm r e xi J ft 977079 977079 H179 n17p2n7nfd P w quot jfd quot E Notice the similarity between this expression and the analogous expression for the closed economy model The equation for fa can be rewritten as p 71pp 70 1 pewep fd 0 pewep i f 77p 17 17p lip 2 m xd 1 139 J xd d 71pp fi f 1 0 pewep pewep 1 m fd quot 14 quot 71P f771pp fd quot d pewep pewep 39 m fd quot r397f2 quot Plugging this expression into the expression for 7239 we obtain pm 7r 7 7r 0 77px which implies that 71P W 71pp fd quot d pewep pewep 39 quotVmt fa quot T397f2 quot 12 A model with costly entry Consider the closed economy case The entry condition is 6x IpxcxTf Fx where is the entry cost In this formulation 7239 0 that is since expected pro ts are equal to the cost of entry there are no pro ts net of entry costs in equilibrium As before we can obtain an expression for the cutoff productivity 7 l 7 f L 71 p p We now calculate an expression for n 1 w p 71pp xl 1 1 n l 3 f xyldx rt1p7f 1 pewep 1pff 1 pewep 7 71f 1 W 1p quot W quotf nyf M W 71 Notice that this implies that i 71pp 13

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