Advanced International Economics
Advanced International Economics ECON 871
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Econ 871 LECTURE NOTES Lukasz Drozd Fall 2008 Contents 1 International Trade 1 11 Introduction 1 12 Patterns of Trade in the Aggregate Data 2 13 Armington Model 3 14 Dornbusch Fisher and Samuelson Model 22 15 Hecksher Ohlin Model 24 16 Eaton and Kortum Model 26 17 New Trade Theory 55 18 Krugman Helpman Model 56 19 Melitz Model 62 110 Bernard Eaton Jensen and Kortum Model 74 2 International Business Cycle 80 21 Introduction 80 22 Business Cycle Measurement 82 23 Prototype International Business Cycle Model 84 24 Basic Supply Side Extensions 110 25 Calibration 116 26 Quantitative Comparison of Theory and Data 149 27 Drozd and Nosal Model 157 28 Other Extensions 158 CONTENTS 3 Papers for Student Presentations 160 List of Figures 12 13 14 15 16 21 22 23 24 25 26 27 28 29 210 An example of isolated small country and implications for aggregate price level Canadian Producers and Consumers Pay the lceberg Cost Frechet density function Price schedules and specialization pattern in the EK model Price schedules and specialization in the Armington model Decomposition of the distribution Time horizon of the analysis Comparison of real exchange rate with terms of trade linearly de trended data Share of working age population and population growth in the US Structure of NlPA Accounts GDl GDP statistica discrepancy breaken down by income type Calibrated values of 04 1959 2005 Calibrated values of 6 1959 2006 Calibrated values of 77 1959 2006 Calibrated values of 6 for 0 27 1959 2006 Comparison of KY ratio between model BGP path and US data initial value matched by construction iii 108 119 125 128 129 132 132 140 LIST OF FIGURES 211 Properties of International Business Cycles Volatility and Comove ment List of Tables 21 22 23 24 25 26 Comparing CA US and US CA gravity equations 16 Real export and real import pricess 109 Relative volatility of the terms of trade 110 Summary of the puzzles on prices 111 Parameter values for the quarterly model 133 Volatility Ratio in a Cross Section of Countries 136 Comparison of Models with Data 159 Chapter 1 International Trade 11 Introduction International trade theory addresses the question why in the long run countries engage in the exchange of goods and services Depending on the answer models of trade can be classi ed into three major categories Ricardian models ii Hecksher Ohlin models and iii Monopolistic competition models The rst two categories are referred to as the traditional trade theory In these models countries trade because there are intrinsically different and by the logic of Ricardian comparative advantage trade allows them to take advantage of these differences In particular in the Ricardian models the technologies to produce each good differ and in the Hecksher Ohlin models the factor endowments differ Both features result in a situation of comparative advantage and lead to a partial or a complete specialization The monopolistic competition models referred to as the new trade theory depart from this traditional approach in at least two important respects First in the new trade theory countries are no longer intrinsically different ex ante but still trade and specialize ex post The key idea is that trade and specialization allows them to access a larger variety of goods while at the same time exploit economies of scale CHAPTER 1 INTERNATIONAL TRADE 2 present in producing them The second important difference is the modeling methodology In contrast to the traditional trade theory the new trade theory is a strictly positive theoryl Namely it attempts to directly describe and mirror the exact market and institutional structure that we see in the data and is silent about the deep rooted frictions that could give rise to this structure endogenously The weakness of such approach is the need for a more extensive data based justification of its more complex structure but its strength is a more natural mapping between the theory and the data In trade theory it has paid off by allowing researchers to extend trade facts to producer level facts and as Samuel Kortum puts it by building on the rmlevel stylized facts the resulting aggregate theory is likely to be more credible both as a description of reality and as a tool for policy analysis We should also mention that initially the new trade theory was a theoretical re sponse to the empirical observation that most trade takes place between very similar countries industrial countries and most importantly the observation that these countries tend to trade very similar categories of manufactured goods called intra industry trade Popular at the time Hecksher Ohlin models could not sensibly deal with observation In this respect monopolistic competition models still have an edge over other theories Even though Ricardian models can sensibly deal with intra industry trade they are silent about the source of the underlying technology differ ences that lead to this phenomenon In the future the ongoing integration of the Ricardian theory with the theory of innovation and growth is likely to ll this gap 12 Patterns of Trade in the Aggregate Data to be completed 1Positive theory directly characterizes what is Normative theory focuses on what ought to be CHAPTER 1 INTERNATIONAL TRADE 3 13 Armington Model Armington model is a Ricardian model2 Each country in the Armington model is assumed to be ef cient in producing just one good and in nitely inef cient in producing all the other This assumption makes the comparative advantage structure somewhat trivial but the model becomes very tractable In this section we will study the predictions of a basic multicountry Armington model and apply them to understand trade flows between countries Model Economy There are N countries or regions and N goods in the world Each country has the technology to produce only one good from the set 1 N and can not produce all the other goods by assumption Production factors are assumed immobile across countries and all markets are competitive In terms of notation we assume that country n produces good n Geography is modeled here by an iceberg transportation cost that is intended to capture the notion of trade barriers between countries regions lceberg transporta tion cost dmv between country n and i implies that d units of good must be shipped from country n in order for one unit to arrive in country i In what follows the following properties of the iceberg cost will be assumed symmetry d dm 2 1 ii no cost within the country d 1 iii triangle inequality d lt d 51 all my 1N 2Since this model is the baseline framework adopted in the open economy macro it is particularly important to know how it ts into the broadly de ned trade theory CHAPTER 1 INTERNATIONAL TRADE 4 Households In each country n 1 N there is a stand in household that has preferences described by a CBS aggregator given by U lt Z 11 i1N where 039 is the elasticity of substitution between the goods 039 gt 1 and 04 is the weight of each good 04 1 Each household is assumed to inelastically supply its endowment of Ln units of labor Given market wage 111 and a schedule of prices pm for each good the problem of the household in country n is to maximize 11 subject to the budget constraint given by Z wnLn H 12 i1N where 11 are the pro ts paid out by the local rms in equilibrium 1 1 will be zero Firms In each country there is a stand in competitive rm that takes all prices as given The rm employs labor supplied by the home households produces goods and sells these goods both at home and abroad Production technology is assumed to be subject to constant returns to scale The pro t function of a stand in rm from country i is given by Hi Z pmym39 wiln 13 i1N where y is the amount of good i sold in each country n sold there at price pm and l is labor input The rm s objective is to maximize 13 subject to production CHAPTER 1 INTERNATIONAL TRADE 5 constraint 2 dniyni S li 14 i1N where the left hand side denotes the total quantity produced and the right hand side is the production function Market Clearing and Feasibility Market clearing requires that the supply of each good equals the demand for each good Cm it an 15 and the supply of labor equals to the demand for labor Ln lquot7 all n 16 Equilibrium The de nition of equilibrium is as follows De nition 1 Competitive equilibrium in this economy is 0 prices wmpm o and allocation cmym lquot7 such that 0 given prices cm solves the household s problem 0 given prices 3 th solve the rm s problem 0 and all markets clear Proposition 2 The competitive equilibrium allocation exists and is unique CHAPTER 1 INTERNATIONAL TRADE 6 Proof Competitive equilibrium allocation if exists must be Pareto ef cient by 1st Welfare Theorem and so the allocation must solve the planning problem given by maximization of 11 subject to 14 15 and 16 Since this planning problem involves a maximization of a continuous and concave objective function subject to a convex and compact constraint set the solution to the planning problem is unique Thus by 2nd Welfare Theorem the competitive equilibrium exists and is unique I Exercise 3 Show that in equilibrium the following version of the law of one price must hold pni all 77 17 quot397 Predictions for Trade In its general formulation the Armington model can not be solved analytically and so we have to resort to a partial characterization of the equilibrium The proposition below derives the model s key predictions the patterns of trade and how they depend on geography We will refer to this prediction as the gravity equation3 In general the gravity equation is an equation characterizing how trade shares expenditure shares of one country on some other country s goods are related to income levels and various measures capturing trade costs Proposition 4 In the Armington model the share of expenditures of country n on goods imported from countryi in total expenditures of country n is given by the fol lowing equation Xm 7 Xi dm X 3X HPquot 1 17 3In applied and atheoretical contexts a similar equation has been extensively used link trade to income distance and other characteristics of countries It proved to be successful in capturing the actual patterns of trade Here we will look at these results in light of the predictions of the model The simplest empirical gravity equation regresses the volume of trade between bilateral pairs of countries regions on their bilateral distance income and various dummy variable common border language etc It works really well in terms of tting the data However since such simple gravity equation is different from the one derived from our model 7 and in principle we would like to use it to perform counterfactual experiments the rst order task is to understand the theory behind it rst CHAPTER 1 INTERNATIONAL TRADE 7 where X pmcm dre expenditures of country n on goods from country i Xquot ZXM are total expenditures of countryi on all goods equal to country n s income wnLn and P Zpf is the ideal CPI price index price level weighted by the actual consumption share of each good Proof Note that the household s problem can be summarized by the following Lagrangian En V al0 7 1 Pmcm39 wquot 18 11N 11N where by de nition of the Lagrange multiplier Pin is the shadow value of one unit of income in terms of the composite consumption U and H is the shadow price of a unit of composite consumption U Using the order conditions to this problem 85 A A 1 p v ac UT 0470 7 Pi 0 all i 19 lt is easy to link this multiplier to prices 1a A Ll P 411473quot Un 15 0quot 1 1 1U A U1 of E U i E i i 1 U71 1 U71 L771 170 7 71 P E mph 7 E at an E arequot 10 L PnE mom 1 110 i We refer to it as an ideal price index Next from the rst order conditions Pm 7 Ugagc P 7 quot i ni7 n PM a t 71 F Un 04 an CHAPTER 1 INTERNATIONAL TRADE 8 and the de nition of expenditures Xm E pmcm we derive Xm g pm H X711 04139 Summing up the above expression wrt j we obtain4 Calif 111 Multiplying both sides of equation 111 by Xquot7 and summing up wrt n we use the law of one price and the balanced trade condition implied by 12 X1Y1ZXM 113 n to obtain avplia i l Zn 1 Xn 77 114 The above equation links prices 111 to aggregate variables We use it to sub stitute for prices in 111 after using 112 and derive 170 Xm39 a1p11 17 Xn 115 n n r TU n n 4The demand for each good 239 in country n is given by C c a M mi 2 R B CHAPTER 1 INTERNATIONAL TRADE 9 In addition in the special case when the iceberg transportation cost dm dm is symmetric we can show that Z1 Xn PH 2 X 116 n Pquot n i 7 and instead of 115 obtain an even simpler expression cumbersome to derive 170 Xamp h Zan lDiPn I Exercise 5 Consider the following expenditure minimization problem N EU 6131111320 plot subject to Swami U c 2 0 alli1N where pgs denote prices EU are total expenditures given U Digs are the preference weights 039 is the elasticity of substitution and U is the composite good consumption level or simply utility Assume that pgs 0450 and U are all strictly positive a Show that EU is homogenous of degree 1 EuU uEU all u gt 0 and thus takes the form P gtlt U where P b Prove the Envelope Theorem in the context of the problem stated in a i e show that E U A where A is the Lagrange multiplier on the constraint in a Then use the conclusion from point a to say E U P and thus by Envelope Theorem CHAPTER 1 INTERNATIONAL TRADE 10 to say A P Using it solve for E1 which together with a shows N 1 EW Zml lf i1 This is an alternative way of deriving the price index to the one we did in the proof of the proposition above c Show that the expenditures minimization problem with E U Income is equivalent the underlying utility maximization problem given by 2 LI L U 13354 2 WW 1 i1N subject to Z pici Income i1N Exercise 6 Suppose that the preferences of the household are instead described by U WW 2 at i1N where ONT is the consumption of the local nontradable good services Assume that production technology of the nontradable good is linear and assume that labor is perfectly mobile across the two sectors In the extended model derive the gravity equation by modifying each step in the above proof accordingly HINT Use the fact that this is a CobbDouglas aggregator and so it implies that nontradable goods have a constant share in the overall consumer expenditures You should get exactly the same gravity equation with total expenditures replaced by total expenditures on all tradable goods The existence of the rst few terms in equation 17 should be intuitive In fact we should expect the share of expenditures on good i in total expenditures of country n are positively related to the size of country i measured by income or labor CHAPTER 1 INTERNATIONAL TRADE 11 endowment and negatively related to the bilateral trade barrier d between themi with the strength of the latter effect depending on the elasticity of substitution 039 However there are more terms in the gravity equation Trade flows turns out to additionally depend on the endogenous product of price indices of the two countries PHP 7 a term referred to by Anderson and Wincoop as gravitas Our next task is to link this term to the primitives in the model Gravity with Gravitas Let s first take a look at the formula for the price level in country n Pu Z 041dmpn1ia i1N and think what makes a country price level high Since all countries face the same pii s we observe that high P can arise as consequence of high overall level of dm s andor ii high positive correlation of dm s with pii s Thus if we think of the iceberg cost dm s in terms of distance between countries in some space means that a country is distant from all other countries and ii means that a country is distant from the countries that are least distant from the rest of the world5 Clearly both and ii are an indication of isolation Figure 11 illustrates an example of such situation which will naturally arise when we are dealing with regions of a large country and regions of a small country An obvious example would the case of the states of the US large country and the provinces of Canada small country Upon closer inspection of equation 17 we note the following 1 P1 P77 0 Observation 1 The multilateral resistance term in the gravity equation makes the small country two isolated dots in Figure 1 to trade relatively more with each other 5Because the price 17 is high the good produced by the country must be in high demand This happens when the country is close to all the other countries rest of the world CHAPTER 1 INTERNATIONAL TRADE 12 Region of CA Mg National border Region of U3 P710112 Figure 11 An example of isolated small country and implications for aggregate price level 0 Observation 2 The multilateral resistance term makes the large country dots that are not isolated in Figure l to trade relatively more with the small country Formally we can derive the above two observations as follows For the sake of argument let s simply denote the two isolated regions provinces of Canada and rest of the regions the states of the US d stands for the cost of crossing the national border and set the following notation for their underlying price levels PcA high PUS low Simplifying also the notation for iceberg transportations chlfs d gt 0 ch cA dUS US l we obtain from gravity equation 170 XOAUs i XUS highxlaw i XUS 17 low XCAUS XoA i gil H XCA high XOAOA 7 X 1 X CA highxhigh XCACA 170 XUS CA XUS CA 739 i XCA highxlaw i XCA X XUsUs 1 0 i X UsUs XUS XUS US high dlia low 071 lawxlaw 7 X As we can see the additional endogenous term 1 7 1 does make OA US higher high XOAOA XUsOA X lower as claimed above UsUs and it does make CHAPTER 1 INTERNATIONAL TRADE 13 The Economics Behind Gravitas As we explain below in the context of the example illustrated in Figure 11 the key feature that the asymmetric size between the small country and the large country is that the demand for imported goods is more elastic in the large country than in the small country and that the supply of foreign goods is more elastic in the small country than in the large country To see this conclusion simply note that the households from the larger country can more effectively shift their expenditures from the foreign goods towards the domestic goods For instance in the context of the example considered above Figure 1 when the US consumers cut spending on each Canadian good by 1 they must shift only 2 of spending on 12 home goods However when the Canadian consumers cut spending by 1 on each US good they must shift as much as 12 on only 2 Canadian goods Now because the marginal utility from consumption of each good declines the immediate consequence of this property is a more elastic demand for Canadian goods in the US than the demand for American goods in Canada6 The opposite conclusion applies to the supply side because the US producers face a much smaller decline in the price as move their sales from Canada ie one unit from each province to US than the Canadian producers face as they move their sales from the states ie one unit from each state to Canada Exercise 7 To formalize the above argument solve for the demand from the following problem Ll Li H 11mm Nq3 m subject to pAQA NquB 1 Speci cally derive the demand for good B and show that for large N the price index will be a ected by the price p B 7 implying a lower measured price elasticity of demand Bln simple words when all Canadian goods become more expensive in the US the US households can shift to a wide variety of home goods but if all American goods become more expensive in Canada Canadians have to take the hit CHAPTER 1 INTERNATIONAL TRADE 14 The burden oficeberg cost falls on The burden oficeberg cost falls on Canadian consumers Canadian producers A Camdzmi marketZr US Goadr A US marketZr Canadiangaadr P Comma Pm quot Comamec m Pmdam Pm Produnr quot PM Figure 12 Canadian Producers and Consumers Pay the Iceberg Cost HINT Derive an equation analogous to 111 Calculate the price index when N is in nite Given the described above implication of relative size on elasticities it should not surprise that an increase in the iceberg transportation cost d might have a very di erent effect on the two countries These elasticities determine who bears the burden of this cost As illustrated in Figure 12 in this case these are the consumers and producers of the small isolated country who will pay for it Thus isolation implies that the terms of trade price of imports in terms of exports of the small country worsens relative to the large country and the worsened terms of trade makes the small country shift relatively more spending towards the home goods than the large country This is the economic intuition why the two endogenous terms that appear in the theoretical gravity equation convey and it is a beautiful example how general equilibrium considerations sometimes matter To formalized the above idea let s push the previously used argument to the limit and make US arbitrarily large relative to Canada the inelastic demand and supply lines become vertical ln such case assuming each region of the same size normalized CHAPTER 1 INTERNATIONAL TRADE 15 to l L Lj l as number of regions in the large country goes to in nity we have P PCACA 7 CA 7 d 7 pUSUS PUS and thus XCAUS 7 XUSd17L7PUS 01 7 pvsys d1 PUSyH 7 dildliadlia d172a 7 7 7 7 XCACA XCA PCA pCACA PCA X d 170 UsOA X XUSCA 7 XUs 7 CA Mahxlowgt 7 ddliadail 7 d X XUsUs 1 1 39 US US XUS XUS lawxlaw lf however we did not have the gravitas we would have obtained instead XCAUS 7 XUsdka 7 PUSUS d1 7 d71d17a 7 div 7 7 7 7 7 XCACA XCA pCACA X d 1 UsOA X XUSCA 7 XUs 7 CA WWW 7 ddka 7 d27a X XUsUs 1 1 39 US US XUS XUS lawxlaw Does the Model Fit the Data The simplest test of the model is to look at the predicted asymmetric e ect of trade costs on trade between a small country like Canada and a large country like US that are somewhat isolated from the rest of the world In such case the Armington model predicts that if crossing the national border involves a cost tari and non tari barriers then the impact of this cost should be asymmetric In particular such cost should drastically reduce trade of each Canadian province with the US but should not reduce as much the trade of each US state with the Canadian provinces By running 2 simple regressions we can check if this is the case The empirical speci cation that we are going to adopt will simply assume that the iceberg cost of transportation is a function of distance and the national border We will do the same for gravitas Formally we are going to have d exp7bn6f PHP expvbm where I a border dummy 1 if there is national borders between CHAPTER 1 INTERNATIONAL TRADE 16 Table 11 Comparing CA US and US CA gravity equations Parameter Regression from CA side Regression from US side a 280 12 41 05 A 122 04 113 03 B 98 03 98 02 C 138 07 108 04 D 164 20 15 08 R2 76 85 regions n and i 0 otherwise and on is the distance between n and i 739 is border cost a ecting bilateral trade barrier and U is the border e ect operating through the multilateral resistance term Given the speci cation of the iceberg cost and gravitas plugging into 17 we thus need to estimate the following equation long Ii 1410an B logX Clog 6m Dbm where C pl 7 039 and D 1 039TU The equation takes into account the multilateral resistance term HP in the form of a border dummy Based on our previous analysis we should expect this term to capture well the notion of isolation when regressed from the US side and the Canadian side separately In particular based on our discussion we should expect to nd that D is much higher when we run the regression from the Canadian side CA CA and CA US observations than when we run it from the US side US US and US CA observations The results are as follows Replicated Table 1 from Anderson and Wincoop 2003 Exercise 8 Go to Anderson s website Download the zip le with the dataset sup porting the paper Replicate the above regressions using this dataset As we can see the estimated values do exhibit strong asymmetry In fact D is CHAPTER 1 INTERNATIONAL TRADE 17 by far more negative in the regression from the Canadian side than in the regression from the US side Each Canadian province controlling for income and distance trades 1600 more with another Canadian province than with a US state Given such huge asymmetry in the regression our next question should be whether the model is quantitatively capable of generating it Exercise 9 Numerical experiment with the model Consider the Armington model with the following parameter setting N 100 04 lN 039 11 L Lj 1 all ij 1N Assume that the rst 90 of the N regions are in a large country US and the last 10 are in a small country Canada which roughly corresponds to the ratio of Canadian GDP to the US GDP Furthermore assume that the transportation cost between the regions within the same country is zero i e dmv 1 whenever in E US or in 6 CA and assume that the iceberg transportation cost between the regions within two di erent countries is 20 ie dmv 12 wheneueri E US n 6 CA or i39 6 CA n e U S a Use the following equilibrium relation from the model piiLi Zan Xn Zaldfh1 men to construct an iterative algorithm that solves the model in IMATLAB7 Using the algorithm compute the overall price level of a representative US region and Canadian region and the prices of the corresponding goods HINT The algorithm may be unstable unless you slow down the updates a bit To be on the safe side I suggest to divide both sides by Mpg compute pa and use the updating rule that puts 5 weight on the old value and only 5 weight on the newly solued ualue8 le 1p 9p where i is the iteration number and p is used to solve for the vector p in iteration i Don t forget to evaluate the convergence and the residuals of equilibrium conditions 7If it is a contraction then it will converge to the xed point 8This way you enlarge the domain on which out mapping is contraction You then do not need a very precise guess for convergence to the xed point to occur CHAPTER 1 INTERNATIONAL TRADE 18 at the end Remember that pNN is the numeraire Print out the code and hand in with the b What is the homebias from the US side de ned as XU5U5XUSCAN and from the Canadian side de ned as XcA CAXcAps c Using data generated by the model suppose you run the following regression of trade flows on the border dummy referred to as the McCallum regression Xni XXn 1n Ii A gtlt borderdummy e where n E CA7 i 6 CA or US What is the value of the regression coe icient on the border dummy d Suppose you run the same regression as in point c but from the US side ie n E US7 i 6 CA or US What is the value of the regression coe icient on the border dummy e How do your answers to c and d compare to the coe lcients that Anderson and Wincoop found in the data by running McCallum s regression separately from the US side and the Canadian side Eaplain briefly the implications of your ndings f Redo points c and d with 039 8 g Comparing the answers in e andf what fraction of the border e ect is accounted for by the endogenous multilateral resistance term h What is the average share of trade with the US for a representative Canadian province in the model measure it by 90X0A U510X0ACA90X0AU5 Consider two levels of trade cost dUS CA 12 same as before and dUS CA 1175 Given that the median and average value of this object in the data is aboth 45 which level of the border cost accounts better for this number i Would the answers to bg change instead you had 100 Canadian regions and 900 US regions Explain your answer analytically 9Pulled out form the data available from Anderson s website CHAPTER 1 INTERNATIONAL TRADE 19 Structural Estimation Anderson and Wincoop 2003 structurally estimate the model using a set of 10 provinces 30 states of US and 20 OECD countries Their exercise is meant to address the question whether the model can quantitatively t the data for plausible parameter values An alternative approach to theirs would be a detailed calibration of the model in the spirit of the numerical example you solved above Speci cation To structurally estimate the model Anderson and Wincoop use the following speci cation for the iceberg transportation cost d exp bm X 6 M7 117 where I is the national border dummy 1 if there is a national border between region 71 and region i 0 otherwise 6 is the distance between regions 71 and i in miles p is the impact parameter of distance between on the implied iceberg transportation cost and 6 is the impact parameter of the national border on the implied iceberg transportation cost Substituting out d in the theoretical gravity equation they obtain the following empirical speci cation of the model Xni Ylyb If 11 121 7 7 ln P 7 ln Pf 5quot ln where 11 l 7 0 a2 l 7 76 and the vector of aggregate prices 3 solves to the xed point problem given by 11610 2 170 7 PH 7 Hail 7 n 1N71 119 i1N PNl 10lncome data is assumed to be nomralized so that 7 X7 l CHAPTER 1 INTERNATIONAL TRADE 20 Note that the observable data includes distance matrix 6Hm border dummy Xm K 77 matrix bmm multilateral expenditure shares m and income vector The price vector 1 is unobservable and so we must use theory to solve for it Numerical Algorithm to Estimate the Model 118 0 Set 039 6 in consistency with the estimates of the long run impact of a change in tari rates on trade from the literature11 Set the values of a1 a2 and solve for the vector of prices 1 from the xed point problem given by 119 Plug in the price vector 1 into the regression equation 118 and nd the constant If that minimizes the squared sum of regression residuals Given resid ual minimizing value of Ii evaluate the squared sum of residuals 7 8 Repeat steps 2 4 above by choosing ahag to minimize the residual 7 calculated in step 3 Results The results of estimating the structural model are presented in the table reproduced below Table 2 in the paper As we can see the model does an magnif icent job in account for the asymmetry and the border puzzle In the two country case second column of Table 2 it underpredicts trade between Canadian province on average by only 17 and overpredicts trade of US states with other US states by 6 Given that Canadian provinces trade 1600 more with each other than with US states this is a huge success In addition Anderson and Wincoop show that when the model is extended to include other countries it does an even better job see last column in Table 2 11Note that a39 can not be identi ed separately from the other parameters CHAPTER 1 INTERNATIONAL TRADE 21 Twquounn39 Multi Cuunrly Model Model Parameters l 7 rquotJ 70 9 003 ll rYlnbl 7165 L 5 H 1005 i Ml bLs saw 17 r1er baa aou 1 397 b ou aou USUS 006 U06 C ArC A 7017 7002 URCA UU5 gt004 Table 2 Eslimntlun Results ofss The able reporta parameler estimates from the wwounny model and the rust at nd Th t b 5 mumCountry model R01 aherrorsare parent 3325 9 a e also repms awrage error terms for Huerstme mlerprovincla and smmmoxince lratle Table 2 from the paper A Note on the Literature This part Was based on tWo in uential papers McCallum 1995 and Anderson and Wincoop 2003 McCallum s paper shows that an ad hoc gravity equation on trade between Canada and US from Canadian side yields puzzling results Namely af ter controlling for distance and income Canadian provinces trade 2200 more With another Canadian province than With US state The original paper interprets this nding as possibly suggesting an enormous cost of crossing the border and is referred to as the border puzzle Anderson and Wincoop 2003 is a response to this nding Anderson and Wincoop show that according to the theory the speci cation of the ad hoc gravity model in most applications s incomplete and so the results may be CHAPTER 1 INTERNATIONAL TRADE 22 biased 14 Dornbusch Fisher and Samuelson Model Dornbusch Fisher and Samuelson model is the most general version of the Ricardian model DES model hereafter for the case of two countries The key idea is to span goods on a unit interval and thus summarize the endogenous equilibrium specializa tion pattern by two cutoff values pivotal goods de ning the set of goods that are produced only by country 1 and the set of goods that are produced only by country 2 The DES model nests a two country Armington model it also nests the two coun try Eaton and Kortum model discussed in the next section lnterestingly the DES model is particularly dif cult to extend to a multicountry framework in full generality and it wasn t until Eaton and Kortum parameterization that this framework took off as a basis for any quantitative analysis The exercise below will walk you through a simple symmetric version of the DES model ln particular you will establish here its relation to the Armington model and solve for the cutoff values Later we will nd all these results useful to understand the intuition behind the Eaton and Kortum 2002 model Exercise 10 Dornbusch Fisher and Samuelson 1977 Consider a world with two symmetric countries and a continuum of goods indexed on a unit interval Prefer ences in each country are identical and given by 1 U lncwdw i 12 0 and all markets are perfectly competitive Assume each country has access to a linear technology to produce each good using labor law Ziwliw7 CHAPTER 1 INTERNATIONAL TRADE 23 where is the e iciency level in producing good w in country i and liw is the labor input Assume that the labor endowment of the standin household in each country is one7 and the production e iciency schedules are given by the following functions 21w 61 120 In addition assume there is a positive tari rate T between the two countries that amounts to 10 of the value of the transported goods across the border The revenue from the tari is lumpsum rebated to the households a De ne competitiue equilibrium for this economy b Refers to point a above Compute the competitive equilibrium you have de ned in a HINT Find 2 cuto s that divide the space of goods into 3 categories traded and produced in country 1 traded and produced in country 2 and not traded both countries produce them for home market only Exploit symmetry to say that wages must be 1 in both countries Use the fact that in the case of log utility the share of expenditures on each good is always a constant fraction of total expenditures on all goods c Apply NIPA rules to compute the GDP of each economy What happens to the GDP in equilibrium when the tari s are increased HINT Read handbook of NIPA accounting available from BEA website12 d How would you have to modify the assumed e iciency schedules stated in 120 to e ectiuely obtain a symmetric twocountry Armington model Based on your answer what is the key qualitatiue di erence between the Armington model and the DES model 12See httpWWWbeagovnationalpdfNIPAhandbookch1 4pdf CHAPTER 1 INTERNATIONAL TRADE 24 15 HecksherOhlin Model The following exercise will walk you through the setup of the 2X2 Hecksher Ohlin Model ln this version of the Hecksher Ohlin model countries have access to the same technologies to produce 2 goods but differ in factor endowment of capital and labor Because technologies to produce each good use these two factor at different intensities in equilibrium countries partially specialize in the production of the good more intensive in the abundant factor The specialization leads to a very peculiar result Despite the fact that factors are immobile across countries trade in goods leads to factor price equalization across countries wages and interest rates are the same13 Exercise 11 Hecksher Ohlin model Consider the world with 2 countries and 2 trad able goods Preferences of the standin household in each country are U Ziogog39 i12 j12 where C denotes consumption in countryi of good j The standin household in coun try 1 has 2 units of labor L and 3 units of capital K and the standin household in country 2 has 3 units of labor and 2 units of capital Firms in each country have access to the same CRS technology to produce both goods The technology to produce good 1 is Y KL sector 1 and good 2 is Y KgL sector 2 For simplicity assume there is no transportation cost a Assume factors are perfectly mobile across countries De ne the competitive equi librium b Refers to equilibrium de ned in a It can be shown using the First and the Second Welfare Theorems that the competitive equilibrium is unique up to the unde termined allocation of capital and labor across countries within sectors 13These two results are referred to as the Hecksher Ohlin Theorem and the Factor Price Equal ization Theorem CHAPTER 1 INTERNATIONAL TRADE 25 and it solves the following planning problem for u leongHP O ZlogCi CZ KZ LZMJ39QVZ 1512 j12 subject to ZKg39 5 2135 ij12 ij12 0 2mm i12 i 03 ZW WW i12 i Compute the competitive equilibrium you de ned in a HINT Remember that allo cation is undetermined wrt to allocation of production across countries who produces what Exploit symmetry to argue that the relative price between goods and factors must be 1 Then introduce an aggregate rm that produces the entire world output max output and compute KL from its problem c Assume factors are immobile across countries De ne the competitive equilibrium d Refers to equilibrium de ned in c It can be shown that there is a unique competitiue equilibrium and it solves the following planning problem for u leongHP O ZlogCi CZ KZ Lzlw39 J 1512 j12 CHAPTER 1 INTERNATIONAL TRADE 26 subject to 37 Eli27 ZKg27 ZLg37 j12 ij12 j12 j12 ZKg39 5 2135 ij12 ij12 0 Dem i12 i 03 2022mm i12 i Compute the competitive equilibrium you de ned in point c HINT Guess that the solution to the planning problem above solves a relaxed problem with constraints omitted like in the planning problem in point b Verify the guess by showing that factor markets clear 7 use in combination with the factor demand functions you derived in problem 2 to nd market clearing production pattern write it in matrix form will be easier e What is the pattern of trade in the competitive equilibrium you found in d More precisely in which good the labor abundant country is a net exporter f Note that the trick you used in d to solve for the equilibrium would not work in general i e for an arbitrary distribution of factor endowment levels Show which step of your solution in e would break down this was not true and explain why HINT Recall that there are nonnegatiuity constraints on all the variables 16 Eaton and Kortum Model Essentially Eaton and Kortum 2002 model is a versatile and tractable probabilistic parameterization of the Ricardian model with a continuum of goods due to Dornbusch Fisher and Samelson 1977 EK model extends the DFS framework to a multicountry 14The range of endowment vectors for which the trick7 works is referred to as the cone of diver si cation CHAPTER 1 INTERNATIONAL TRADE 27 context and allows for an explicit derivation of the gravity equation Model Economy Goods are indexed on a unit interval w 6 01 and the world is comprised of N countries regions Every country can produce every good from the continuum but the labor requirement to produce each good di ers Unlike in the DES model the productivity schedules are described probabilistically Namely it is assumed that the ef ciency of producing a good in country n is a realization of an iid Frechet distributed random variable Zn FnZn S 2 expiTnZ 9 121 where Tu and 0 are parameters governing the mean and the dispersion15 and n is the country index As before geography is modeled by an iceberg transportation cost obeying three standard properties symmetry d d 2 1 ii no cost within the country d l and iii the triangle inequality Probabilistic Notion of Comparative Advantage Figure 13 illustrates the plots of the Frechet density function The moments of this distribution are given by meanTl l 7 ii coef cient of variation standard deviationmean acm Because 0 unambiguously determines the coef cient of variation the two parameters have a natural interpretation T characterizes the overall level of technology of a country absolute advantage and ii 0 a parame ter common to all countries characterizes the dispersion of ef ciency across goods 15We will use the convention of denoting a random variable by a caligraphic capital letter By the law of large numbers note that F is also the fraction of goods produced at ef ciency 2 or lower CHAPTER 1 INTERNATIONAL TRADE 28 Probability density Figure 13 Frechet density function comparative advantage Frechet distribution or in general any exponential distribution has the following four properties that will greatly simplify our analysis of the model 0 Property 1 Mechet distributed extreme values Let 211N be a vector of Frechet distributed random variables with parameters Ti Then Z miaXZv 123 is Frechet distributed with parameter T Ti and d 0 Property 2 Mean determined winning probability Let 21N be a vector of Frechet distributed random variables with parameters d Then T P 25gt Z 5 r iglfgf l ET 124 0 Property 3 Memorylessness Let Z be a Frechet distributed random vari ables with parameters T d Then the conditional distribution is equal to the CHAPTER 1 INTERNATIONAL TRADE 29 unconditional distribution S 22 S 21 67722279 0 Property 4 Scale invariant dispersion Let Z be a Frechet distributed random variables with parameters T Then the distribution of a random variable 12 a6 13 is Frechet with parameters 079710 Proof Property 1 Prm xz z H P42 z a Property 2 PrZS gt O H P42 mamas O H P42 25F 25d25 0 0 O 92579717267 217 T zgge T zggdZS 0 O 0257971726721T zggd25 0 fl 2 T049465 ng i1N A Ti E ZJQZZT 2 Property 3 follows directly from the Bayes rule E ZiTi Property 4 follows by rearranging the formula for the Frechet distribution l Households To state the household s problem formally we need to transform this problem to guarantee integrability of the utility function and the budget constraint To this end CHAPTER 1 INTERNATIONAL TRADE 30 given the equilibrium distribution of prices in country n we exploit the fact that the distribution of prices tells us the measure of goods that are available at price p Since all variables of the model as functions of w typically take identical value as long as the price is the same without loss of generality we will index goods by their underlying prices rather than the type of good w Under such reformulation the preferences of the stand in household from country n can be described by the following utility function U momm ompn 125 where 039 denotes the elasticity of substitution between the goods 039 gt l7 is the price identical consumption level of goods at price p7 and d0 p is the measure weightfraction of goods at price p16 The problem of the household is thus to choose an integrable function that maximizes 125 subject to the budget constraint p0npdGnp wnLn Hquot 126 0 where H are the pro ts paid out by the local rms in equilibrium U will be zero and wnLn is compensation of labor Ln is endowment of labor From the household s problem we calculate that the ideal CPI price index is given by P wp damm 127 A Note on Optimization with Integrals Note that the utility maximization problem above involves a choice of an optimal function that maximizes the integral Taking rst conditions in such case may be 1BNote that the above formulation restricts attention to allocations in which the household chooses the same consumption of all goods that have the same price This would be the case endogenously7 but here it is build into the problem This trick allows us to use the coarser indexation by price and guarantee integrability
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