ECON 841 T. J. HOLMES AND T. J. KEHOE INTERNATIONAL TRADE AND PAYMENTS THEORY FALL 211 EXAMINATION This exam has wo pars. Each par has wo quesions. Please answer one of he wo quesions in each par for a oal of wo answers. Please keep your answers for Par I and Par II separae. You can consul class noes, working papers, and aricles while you are working on he exam, bu you are asked no o discuss he exam wih anyone unil he exam period is finished for everyone. 1
Par I 1. Consider a wo-secor growh model in which he represenaive consumer has he uiliy funcion a1 a2 log( c 1c2 ). The invesmen good is produced according o a1 a2 k 1 dx1 x. 2 Feasible consumpion/invesmen plans saisfy he feasibiliy consrains where c x ( k, ) k 1 1 1 1 1 1 c x ( k, ), 2 2 2 2 2 2 k k k 1 2 1 2 1. The iniial value of k is k. All of he variables specified above are in per capia erms. There is a measure L of consumer/workers. a) Define an equilibrium for his economy. b) Wrie ou a social planner s problem for his economy. Explain how soluion o his social planner s problem is relaed o ha of he one-secor social planner s problem log c a1 1 s.. c k dk c, k k k. [You can wrie done a proposiion or proposiions wihou providing a proof or proofs, bu be sure o carefully relae he variables in he wo-secor model o he variables in he one-secor model.] c) Solve he one-secor social planner s problem in par b. [Recall ha he policy 1 funcion for invesmen has he form ( ) a k 1 k Adk where A is a consan ha you remember or can deermine wih a bi of algebra and calculus.] d) Suppose now ha here is a world made up of n differen counries, all wih he same i echnologies and preferences, bu wih differen consan populaions, L, and wih i differen iniial capial-labor raios k. Suppose ha goods 1 and 2 can be freely raded across counries, bu ha he invesmen good canno be raded. Suppose oo ha here is no inernaional borrowing. Define an equilibrium for he world economy. 2
e) Sae and prove versions of he facor price equalizaion heorem, he Solper- Samuelson heorem, he Rybczynski heorem, and he Heckscher-Ohlin heorem for his paricular world economy. a1 f) Le s c / y where y pk 1 p2 dk is world GDP per capia. Transform he firs-order condiions for he one-secor social planner s problem in par b ino wo difference equaions in k and s. Use he firs-order condiions for he consumer s problem of he equilibrium in par d o show ha i i i y y s y 1 y 1 s y y. y s 1 y 1 s y g) Use he soluion o he one-secor social planner s problem in par c o solve for s. Discuss he economic significance of he resul ha you obain. 3
2. Consider an economy where he consumers have Dixi-Sigliz uiliy funcions and solve he problem m max (1 ) log c log c( z) dz m s.. pc p( zczdz ) ( ) w cz ( ). Here 1 and 1. Furhermore, m is he measure of firms, which is deermined in equilibrium. Suppose ha good is produced wih he consan-reurns producion funcion y. a) Suppose ha he producer of good z akes he prices p( z ), for z z, as given. Suppose oo ha his producer has he producion funcion yz ( ) max xz ( ) ( z) f,. where xz ( ) is he firm s produciviy level and f. Solve he firm s profi maximizaion problem o derive an opimal pricing rule. b) Suppose ha good is produced wih he consan-reurns producion funcion y Suppose ha firm produciviies are disribued on he inerval x 1 according o he Pareo disribuion wih disribuion funcion F( x) 1 x, where 2 and /(1 ). Also suppose ha he measure of poenial firms is fixed a. Define an equilibrium for his economy. c) Suppose ha, in equilibrium no all poenial firms acually produce. Find an expression for he produciviy of he leas producive firm ha produces. Tha is, find a produciviy x 1 such ha no firm wih x( z) x produces and all firms wih x( z) x produce. Relae he measure of firms ha produce m o he measure of poenial firms and he cuoff x. d) Suppose now ha here are wo counries ha engage in rade. Each counry i, i 1, 2, has a populaion of i and a measure of poenial firms of i. Firms produciviies are again disribued according o he Pareo disribuion, F( x) 1 firm in counry i faces a fixed cos of exporing o counry j, j i, of f e where f f f. Each counry also imposes an ad valorem ariff on impors of e d differeniaed goods from he oher counry. The revenue from hese ariffs is redisribued in lump-sum form o he consumer in ha counry. Define an equilibrium for his world economy. x. A 4
e) Suppose ha he wo counries in par d are symmeric in he sense ha 1 2 and 1 2. Explain how o characerize he equilibrium producion paerns wih a cuoff value, or values, as in par c. [You should explain carefully how o calculae any cuoff values, bu you o no acually need o calculae i.] Compare his value, or hese values, wih ha in par c. Draw a graph depicing wha happens when a closed economy opens o rade. f) Discuss he srenghs and limiaions of his sor of model for accouning for firmlevel daa on expors. 5
Econ 841 Fall 212 Inernaional Trade and Paymens Theory T.J. Holmes and T.J. Kehoe Final Exam: Par II Quesion 1 Consider a model wih wo counries. Suppose labor is he only facor of producion and assume ha he number of workers ishesameineachcounry. Eachworker is endowed wih one uni of ime. There are wo secors, manufacuring and services. In he service secor, one uni of labor produces one uni of services. (Noe service produciviy is he same in boh counries.) The manufacuring secor follows Eaon and Korum (22). There are a coninuum of differeniaed manufacuring goods, [ 1]. Le be he produciviy of counry (we will make his endogenous below). This governs he disribuion of produciviy draws, so ha he c.d.f. of produciviy in counry is () =. Suppose here is a CES aggregaor of he differeniaed manufacuring goods, Z 1 = () 1 1, where 1+, and is a level of he manufacuring composie, while () is a quaniy of differeniaed good. Finally, uiliy of he manufacured good composie and services is Cobb-Douglas, = 1, where is he quaniy of services. 1
Le 1 be he iceberg cos of shipping beween he wo locaions. Le and be he quaniy of labor working in each secor a counry. Assume ha here is an exogenous exernal effec in produciviy, so ha = ( ), where ( ) an nondecreasing funcion, and (). Le behewagealocaion, le be he price index for he manufacured good composie a locaion. To simplify calculaions, we review some of he resuls of EK ha you can ake as given. Define Φ 1 and Φ 2 by Φ 1 = 1 1 + 2 2 Φ 2 = 1 1 + 2 2 Then equals = Φ 1, for a consan. Also, le 6=. Then he probabiliy ha counry is he lowes cos provider o counry equals = +, if 6=, and he probabiliy ha ishelowescosprovideroiselfis = + (1) Suppose =, so ha each counry is in auarky. Solve for he equilibrium, and deermine he price level of composie manufacured goods. (2) Suppose =1, so here is free rade. Show here exiss a symmeric equilibrium where 1 = 2. Wha is he price level of composie manufacured goods? (3) Suppose =1. Suppose ha 2 1. (This ensures ha boh locaions produce service goods.). Define an asymmeric equilibrium in which counry 1 has a higher manufacuring share han counry 2, 1 2, and derive a condiion 2
ha depends upon 1 ha characerizes an asymmeric equilibrium. Show ha 2 =is no possible in equilibrium. Show ha if () (), here exiss an asymmeric equilibrium. look wha happens locally around he symmeric equilibrium.) (Hin, ake your equilibrium condiion and Quesion 2 Suppose here are a uni measure of goods indexed by, and uiliy is Cobb-Douglas wih equals weigh on each good, where () is consumpion of. = Z 1 log (), There are wo counries. Le be he iceberg ransporaion cos, so ha o deliver one uni from one counry o he oher, unis mus be shipped. There are wo kinds of labor, high skill and low skill. A low skill worker is endowed wih 1 efficiency uni of labor. A high skill worker is endowed wih 1 efficiency unis of labor. Le and be he measure of high and low skill workers in each counry. A each locaion here exiss a freely available backyard echnology in which i is possible o produce one uni of each kind of good wih one efficiency uni of labor. In addiion, in counry 1, for good 1, here exiss an enrepreneur who has a 2 monopoly on a special producion echnology ha is more efficien han he backyard echnology. In paricular, he enrepreneur wih a special echnology for producing can produce 1 unis of, foreveryoneefficiency uni of labor. In counry 1, for 1, he only echnology for producing is he backyard echnology. 2 Counry 2 is he mirror image of counry 1 in ha for 1 2 here is an an 3
enrepreneur who can produce wih he special producion echnology, while for 1 2 he only echnology for producing in counry 2 is he backyard echnology. Thereisonefinal consrain. Le () and () be he employmen of an enrepreneur wih a special echnology for producing. There is a managemen consrain ha ()+ (). Assume firms engage in Berrand pricing. Focus on symmeric equilibria in which hewageasafuncionofskill, and, is he same in boh counries. (a) Consider firs he auarky case where =. Define an equilibrium. In equilibrium, he skill premium will eiher equal (he raio of he efficiency unis) or will be sricly greaer han. Derive a condiion on he underlying model parameers under which he skill premium mus necessarily be greaer han. (Hin: To undersand ou how his model works, i is useful o sar by assuming ha he consrain parameer is so large ha is is irrelevan and deermine he equilibrium in his case. Then reduce o where i is binding. I is also disinguish beween hecasewhere is small relaive o and where is large relaive o ) (b) Now consider wha happens if he economy is opened up o free rade, =1. In paricular: (i) Show ha if he skill premium equals wih free rade hen i also equals under auarky. (ii) Derive a condiion under which he skill premium equals under auarky bu mus be sricly greaer han under free rade. 4