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# International Trade EconS 571

WSU

GPA 3.58

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This 21 page Class Notes was uploaded by Maurine Kuhic on Thursday September 17, 2015. The Class Notes belongs to EconS 571 at Washington State University taught by Mark Gibson in Fall. Since its upload, it has received 54 views. For similar materials see /class/205982/econs-571-washington-state-university in Economic Sciences at Washington State University.

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Date Created: 09/17/15

MONOPOLISTIC COMPETITION MODEL Key ingredients quot c taste for variety of differentiated goods 1 J 0 Consumer utility log 00 lplog 0 Production of differentiated products y j l b maXZj f 0 increasing retumsfixed costs Assume that good 0 the agricultural good is produced with perfect competition and the constant returns to scale production function y 0 g 0 a but that there are n manufactured goods that are produced with monopolistic competition and the increasing returns to scale production function specified above The representative consumer solves maX log 00 lplogZlcf s t p000 Zlpjcj S w c 20 0 0 u 6 MRS equals price ratio u co p0 pel 9 cl Z1CJ P1 1 60 1 Producer of good 139 solves the consumer s problem to find the indirect demand function 61p PIC n p c 2116 po 0 n of n p C Zia p 211 summ1ng over 1 n c 2116 po 0 multiplying by c CO 2quot p161 l s1mp11fy1ng p0 CO 2 poo0 w from previous equation and budget constraint w c 0 Zpo Indirect demand function Pro ts of rm 139 ply wbx wf revenuevariable costs xed costs We suppose that the rm chooses its output y to maximize its pro ts assuming that the outputs of all other rms are constant and that prices will adjust to clear the markets of each good This is the Cournot competition assumption To maximize profits the rm sets MR M C We set C y in the indirect demand function this is the assumption that the price of good 139 adjusts to clear the market for good 139 and plug this function into expression for profits M w 6 y yI wbyI wf p 2 Z n To maximize profits the rms sets the rst derivative of this expression equal to 0 that is MR M C 0 w 21 yj 1ny yf pyf 1 2 ZWY wb0 Set w l as numeraire Since rms are symmetric we know that there is an equilibrium in which y 7 if y gt 0 2 quot7quot2 2m 2 2 n2 pn 1 2n2b 7 Y bn P P 2ny 2ny pn l Znr pr l W J b 7 The pro ts of typical firm are 2 pot 1 b py y f 2quot M f We assume that there is free entryexit until pro ts equal zero 2fn2 1 pn p 0 The pro ts of typical rm are Y pot 1 b py y f 2quot M f We assume that there is free entryexit until pro ts equal zero 2fn2 1 p7n p70 n 1 pJ1 p22 4p2f 4f Equilibrium An equilibrium of the monopolistic competition model is the number of manufacturing rm ft a price e for the agricultural good a price 7 for each manufacturing rm that operates at a positive level a wage rate vii a consumption plan 80 51 828 production plans J30 20 for the agricultural good and f2 2 j for each manufacturing firm that operates at a positive level such that 0 Given 0j71f72f7 and ft the consumer chooses 5081525 to solve maX log 00 1plogZlcf s t ocoz f7jcj SW C 20 pO wso 0if ogt0 0 Given the indirect demand function p 01 c c that comes from solving the representative consumer s utility maximization problem firm j chooses f2 to solve maX pjA1yjj yj fubyj vif o jjj fvbjj vTfS 0 0 if f2 gt 0 where f7 f7j1fjjn yo 20 o y lbmaX2 f 0j12ft o 5 jj0l2ft o Z15 e Numerical example b1f2 p12 249 n 2 245 42452 42454 8 n 7 7 15 E 23333 p0 w1 y0 245 Utility log 245210g715m 74960 Homogenous of degree one representation of utility a real income index eXp12log 245210g715 2 eXp1274960 4244 An integral number of firms There is a problem with our concept of equilibrium if the number of firms ft does not turn out to be an integer Suppose for example that b1f2p12 2490 Then when we solve n 245 quot2452 42454 8 we obtain ft 622342 How do we interpret this solution There are two approaches that we could take 1 We could restrict ft to be an integer and let it be the largest number of rms for which pro ts are nonnegative In this case however there can be positive pro ts in equilibrium These pro ts need to be earned by someone If we give them to the representative consumer then the consumer s budget constraint becomes oco Zlf7jcj S WZH where fl are pro ts Everything becomes a more complicated even in this simple model with only one market with monopolistic competition Things become much more complicated in applied models with many such markets 2 We could think of ft as being an integer up until we compute the number of rms at which we point we simply calculate a real number This is the approach that economists typically use in applying this sort of model Reinterpreting the model as a model of international trade We can reinterpret this model as a model of international trade among countries that are identical except for their sizes as measured by their labor forces Z Consider the numerical example in which b 1 f 2 p 1 2 and there are two countries one in which 21 441 and the other in which 22 49 We can think of these countries as being the United States and Canada respectively In the integrated equilibrium of the world economy p0w1 2 n 245 W 622342 pn 1z 612342490 219367 y 2an 226223422 pel Wi bquot M20327 2nyquot 2ny pn 1 612342 yo 6 245 2p0 21 To calculate consumption of each variety in each country we just divide the world production of the variety 7 proportionally In country 1 for example 1 51 J f 19367 17430 Z Z 490 We also divide the production and the consumption of the agricultural good proportionally 1441490ft 441490622342 560108 772 49 490 49 490622342 62234 and 3 4414900 441 490245 2205 49490f0 49490245 245 Strictly speaking there is nothing in this model that pins down the location of production of the agricultural good We are calculating a symmetric equilibrium Trade Equilibrium quotI A A1 1 fl p0 p W CO C 23 o 7 country 1 560108 10 20327 10 2205 17430 2205 2205 19367 39367 country 2 62234 10 20327 10 245 01937 245 245 19367 39367 Utility 1 log 2205 2log 6223421743012 142133 L22 log 245 2log6223420193712 98190 Real income index em 122005 e11 2 13557 Notice that not surprisingly the real income in country 1 is 9 times greater than that in country Gains from Trade To calculate the gains from trade we can compute the autarky equilibria for both countries We have already calculated this equilibrium for country 2 Autarky Equilibrium r 1133 5 W Go 5 93 5 0 J7 7 country 1 561075 10 20363 10 2205 19300 2205 2205 19300 39300 country 2 70000 10 23333 10 245 15000 245 245 15000 35000 Utility 1 log 2205 2log5610751930012 141080 L22 log 2452log 705 2 74960 Real income index e11 2 115748 em 4244 The smaller country country 2 has the most to gain from trade In country 1 real income goes up by 54 percent 122005115748 10541 In country 2 real income goes up by 2194 percent 135574244 31944 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 1 2010ch 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 wVE4 n c0 1 22 uw 7239 uw 7239 62 1 7p 1 7p pzlquot joquotpz391quot dz39 pzlquot Plquot where 1ep n i p P I0 pz 1quot 612 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 239 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 62 max pzczx fPZ 242 242 z 1 39P 1 i i i i f pz1quot P1quot xzpzlquot Plquot taking P as given The solution is 1 272 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 I713 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 A11 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 i i i a 17p quot2 lip n3 1 9 P J 612 1 J 612 dz 0 P99 0 sz 0 sz lep 1 L L L P P lx11quot nzxfquot n3x3139 To determine l we solve Zni 1 1 p x1 p161 x 1 L1 L L L f0 1 F991 7719 quot2xquot quot3981 quot J 1 7r1pxf f 2 0 71199 7596 7596 21 1 pyn nz 3n3 j I f XI x 1 To determine 7239 we solve CZ CS znz p262 f 75 p363f x2 x3 A c c c n1p101 1fn2p202 2fn3p303 3f x1 x2 x L L L 72391 phlequot Mix W r21n2r13fu 1 P quot2 5f 7139 L L L 7119 quot296 3963 nyn1 p anzraf Plugging in the expression for 1 we obtain L 17 17 Ff n2 21 21 x1 x1 and rt 1 a p n2x 1 pyn2n3 L L 1 1 120 p 11p n2 quot 1 p quot 1 pm f xl 11pl xl 11pl 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 1 Ex f0 xdFx f xyx39y39ldx y l 7 and the variance is yxrwz y 2 Ex2 Ex2 fe dFxJ39lwx yxiyildxr 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 Z 1 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 L PH 71quot px1quot dFx nquot px1quot yx39de pewep H L 2 mp 1p7x 71pp pewep 17p L Pi 2 rm 1 p7x 71pp 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 71 p p 7r p7l p p 70x 1 p p 71 p pewep f 1 quotlpm 1 COO u 17rp p003 panpr m We calculate the cutoff productivity 7 m p7lpp f Ll 0 pewep pf f f P7Cf n1 pr 1 f x 7lpp7 If f 0 n7 T 71pp 7r Notice that this expression depends on pro ts 7239 which we can calculate as 7239 nJ pxcx fdFx quotJ W f xereldx rigf 1 1 no A no n 7 p pipyziff 7139 J xlipyxi7ildx an39f yxi7ildx rigf 1 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 71 p pyn 11p 11p n7f 11p 711p p 7py which implies that 1 i p 7 1 m p p 7p 71pp n7f 7pnf x7 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 f 7pnf 7lt1 71pp nflt71pp 7py that is if the xed costs of having all potential rms produce is suf ciently low In this case L p m 71pp y1 p py n z py1 p pynx MM 1 quot n1p7 L px1 P1 px1quot rm 1 p7 The calculation of total pro ts becomes Cx L lip II anpxcx ch fdFx quotI W f xryrldx p 7239 n qux39 dx anwo yx39y39ldx 1 y1 p pynr m 1 n1 pyn nf Z 1 m nf 1 1 pm Notice that the pro ts of a rm with productivity x 1 are 1 pu nf 1 Z m p mi 1 1 pm J f p1c1 C1 f Wi 1 f quot1 p7 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 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 17 x i 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 le 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 1 i 1 1 P H n I p xdFx n I pxdFx 2 NH 1 171 171 P raj dpx yx dxn2J px yx dx L pewep i L pewep P1 quot11 1 p7x 1 20 pl 1 p7x 1 71pp 71pp 1C1d pewep L pewep i L P1 rap 1 pWEd 139 7WD p 1 M7 139 71pp 71pp L 977mm 1 pylrp pl 1p7 Ed 1 n2ri1quot fu 1 P 17p 2 71 p p The demand in country 1 for goods produced by a rm in country 1 with productivity x 2 x1 01 is M71 75 2 p7lpJZ7096a 17p 1 610C i 9 i pewep 39 pix1quot P11quot 1p7nlid 1 nzri1quot fu 139 J We calculate an expression for the cutoff productivity Ed cl39 p ya m p QME 1 1 p11xmcllx 13x f pemrp 1 epl 9017mm fd 0 a H 1 E H pxld xld 1 p7 quot1x101 n272 x22 L cl 71pp4 l quot p11x cllxm IE d f 9407 7 prmrp de0 raid 1 nzri1quot fu 1 Similarly we calculate an expression for the cutoff productivity in i L 2 2 2 lip 17p 2 2 r 6 x2 71pp 2 2rl x 10109901092 1 71 1 12 MW 1 HM 12 0 7 quotin 1 rarf1quot ig 1 The expression for 72391 is n1 n1 jpixcix 61x dFxraI pfmcfm 6100 Foe x x w 71pp1 ax 74 III quot1 2m pewep i pewep f yx dx 7 quotlid 17 7120 H fu 17 i L Ive 71pp 2 2rf1quot x1quot f xwdx 71 pewep i pewep 2 7 quotin 17 711le17 17 warp i pinup 17p4 maid 1399 17mm zrarf1quot E2 1quot 7 7 1 per 1 9 1 per 1 9 977079 i m quot1xldyfd xleyfe quot1 17quot quot2Ti 1quot E2 17quot quot2 17quot 711712179Ze 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 x1 01 x12 III f2 1 calculations let us consider the symmetric case where 2 Z n1 r12 n and T If r In this case y1 p pynf fd 0 pewep i pewep my 2 1 sz i L 17p y1 p pynr 0 pewep i pewep f2 my 2 Hi n17p minim er Notice that kp 9 T xd fa and that L pewep i pewep pl39 1 p7n fa 1 THE 1 Pa 71 p p The equation for 7239 can be rewritten as 7 n lipm wrn 5 2 fd 977079 H179 fd 9 E 977079 r lfe quot f P 1 7r 17p en Notice the similarity between this expression and the analogous expression for the closed economy model The equation for fa can be rewritten as L 71pp fj 0 1 pewep f pewep 1 77p 17 lip lip L m xd 1 139 J xd d 1 77 W 7 1quot 7 p p d fd p 0 pewep pewep n7fd quot 7 quot J 71P 7lpp 7012 quot 7 Cd Plugging this expression into the expression for 7239 we obtain pm 7r 7 7r p uz 7 P which implies that 71P 7 71pp fd quot x 1 pewep pewep quot7pfd quot T397f2 quot J A model with costly entry Consider the closed economy case The entry condition is jpxcx 3 f Foe 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 iLT 71pp We now calculate an expression for n f MFA f x HdX pewep I x x na pw p pewep 1pff 1 pewep 7 71 W 1p quot W quotf nyf M W 71 Notice that this implies that

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