International and Intra national Real Exchange Rates: Theory and ...

International and Intra-national Real Exchange Rates: Theory and Evidence Michael Devereux

Viktoria Hnatkovska

University of British Columbia

University of British Columbia

[email protected]

[email protected]

September 2009 Preliminary

Abstract In this paper we study two long-standing puzzles in the International Finance literature: the fact that the real exchange rate (RER) is very volatile (RER volatility puzzle) and that it covaries negatively with domestic consumption relative to foreign consumption (Backus-Smith puzzle). To understand the two puzzles we depart from the existing literature by focusing on a disaggregated analysis of consumption and RER. First, using region-level data for a large number of developed and developing economies we document the characteristics of the two puzzles in cross-country and within-country data. We then develop a combined model of inter-regional and international trade. The model exhibits variations in inter-regional as well as in inter-national consumption and real exchange rates. We show that with a combination of within country and across country shocks, and endogenous tradability the model can rationalize the two puzzles and does so in both international and intra-national dimensions.

JEL Classi…cation: F3, F4 Keywords: Backus-Smith puzzle, real exchange rate volatility, endogenous non-tradability, regional heterogeneity We are grateful to Matias Cortes for excellent research assistance and to the participants of the Bank of Canada-UBC Macroeconomics Workshop, Midwest Macro meeting 2009, SED 2009 and EEA 2009 annual meetings, Brussels Trade Workshop, seminar participants at UBC, Kyiv School of Economics, and Minneapolis FED for helpful comments and suggestions.

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Introduction Despite dramatic changes a¤ecting international asset markets, there exists ample evi-

dence that international risk-sharing is far from perfect. One of the well-known corroborations of this fact is the Backus-Smith puzzle, due to Backus and Smith (1993) and Kollmann (1995). The puzzle is the result of a striking disagreement between theory and data. In particular, existing theoretical models with complete asset markets imply a high positive correlation between the cross-country consumption growth rates and the real exchange rate (RER). In the data, however, such positive correlation is forcefully rejected. A related though distinct puzzle in the international …nance literature is known as the real exchange rate volatility puzzle. This puzzle says that the volatility of the real exchange rate is very high and signi…cantly exceeds the volatility of consumption and GDP, while the corresponding statistic predicted by the theory is small. Both these puzzles pose signi…cant challenge to the standard open-economy models with complete markets and representative agents. In this paper we propose a novel explanation for the Backus-Smith and volatility puzzles. Our explanation relies on two central elements: within country (regional) heterogeneity and endogenous non-tradability. We show that each feature, when introduced in an incomplete markets setting, implies a lower international consumption-RER correlation, but is not su¢ cient by itself. Only a combination of regional heterogeneity and endogenous tradability resolves the Backus-Smith puzzle and generates realistic volatility of RER relative to consumption. Our theoretical analysis is supplemented by a novel empirical study of the Backus-Smith correlation and real exchange rate volatility. In particular, we depart from the existing studies of the consumption-RER relationship that focus on cross-country analysis and instead develop a more disaggregated analysis using region-level data for a large sample of economies. This analysis helps us to establish the key properties of the real exchange rate and consumption in regional data, a line of research not explored in the existing literature. We then show that our theoretical model can rationalize these regional facts as well. For this purpose we construct a new dataset of historical series for consumption and prices for a large number of regions in a number of developed and developing countries. To date we have collected regional data for Canada, Japan, Spain, Germany, India and China. Two sets of our empirical …ndings are particularly interesting. First, the data indicates that the Backus-Smith puzzle is not a purely cross-country phenomenon. In fact, we …nd that the departures from perfect risk-sharing are signi…cant within countries as well. For within country data, we normally …nd a positive correlation between relative consumption

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and real exchange rate. But this correlation is very small, much less than predicted by the theory. For instance, the average correlation between relative consumption and the regional real exchange rate across Canadian provinces is equal to 0.12, a signi…cant departure from a coe¢ cient of 1 predicted by the theory. Similar results characterize the degree of interregional risk-sharing in the other countries in our sample. We also …nd that the degree of risk-sharing varies substantially across regions and over time in a given country. For instance, in Canada, the correlation ranges from -0.3 (between Alberta and Ontario) to 0.55 (between Quebec and Newfoundland). Figure 1 illustrates these …ndings. For each country in our sample it presents a scatterplot of all bilateral regional average CPI growth and consumption growth pairs over a corresponding sample period. Japan and China exhibit practically no correlation between the two variables, while the correlation is slightly positive for Canada, Spain, Germany and India.

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Figure 1. Backus-Smith correlation across regions Consumption growth and CPI growt h in Japan 1990-2005 .02

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Consumption growth and CPI growth in Canada 1981-2007

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m icToyam GMunm a MY e iG YO amaguchi m a e ihi Tokushi S hi ziunm uoka m iykayam agi S hi ihi ifa u S hi m ane NM ar aamanashi IiyG bar aki K um am ot o ochi Tokushi mo a AkiAtice N aahi ar aicIhiwat O itaa Nar N ar K um Fukuoka am ot A om or i C hi b a I bar M i y azaki N ar a Wakayam t M or e i i a Fukuoka Fukushim a I wat YeFukushi amagat aN aitar SToyam hiIzNwat uoka S hi m ane Tokyo K yot Wakayam o Tot t or a i O kayam a K um am ot amagat a aga M i f e i u Y amanashi S hi m Y amanashi G ane NY ar Y amaguchi N agasaki Wakayam a a S m ane K S ochi hi g Tot a t or i H okkai Y amagat d o a E hi m e A i c hi M Y a amanashi e i O ki n awa A ki t a N ar a M i y azaki Fukui S ai Tot thiam thi a oriGS unm Tot thi aifK or K iazaki agaw aot ae Y Ktamanashi agaw Yamaguchi AG ic M Tokushi maoa aAom orKagaw i Wakayam Yamanashi Sbar gaS a K akayam MY iamagat Toyam agaw Nzar a M Ae iki aaKOum agaw aS uum e i ar Kochi I wat e KIagaw tane or iyagaw uoka Y amanashi ita aK am hi m ane Tokushi m a Saki hi m auoka K mKam O aYamaguchi a iyaazaki a Nar aShiK am oaK IITot aki Nane agasaki N a aNSar IzA bar Sot hie agaw Fukuoka a m ShiFukuoka ga G unm Fukushim aFukushim a hi icM hi Fukuoka bar hiwat IA mbar aki aki a Yamagat M e iagaw Toyam ki tWakayam aam Fukuoka IKbar aki Fukuoka Kane M agaw iaK aane Tot t agoshi or i or M I shi IM bar kzawa Fukuoka eK Iar bar Fukuoka K ochi Y S ane Tot tFukuoka or A om iEhi ic hiYFukushi ochi Maki e ia Y amanashi Tokushi m a Tot t or i aar S hi Kamanashi agaw GS ifhi Ieibar Yamanashi NO ar N M a e iaki e i uoka agaw auN ita um K agaw am ot a o IS bar aki Fukuoka Nar a N Wakayam aa amagatmaFukushi m aaa Tokushi IFukuoka bar m aki a Fukuoka hihi zm uoka G unm a aki NA agano M e i Y amaguchi N ar NarIabar m ane S hi g K um am ot o Toyam a S aga A i c hi A ki t a S hi m g i ane at a M i y azaki Tot t or i I bar aki I bar Fukuoka aki I wat e Wakayam a K agaw a S hi m ane Fukuoka K Y agaw amanashi a Y amagat a ar aki Tokushi m a FukushiYm aMMe aK Tot t oram ia ot i Kagaw um oiyagi Sot hizoauoka Kochi Aic hi amanashi ie agaw a K agaw Tokushi m amagat KWakayam ochi Me i KYiochi ISbar KM um Iaki bar aki Fukuoka m ane NWakayam Aic hiFukushi Naaraa Yaamagat Kochi K agaw aFukuoka a Nar I barYamanashi akibar a Fukuoka Tokyo K yot oam K ochi Tot tagaw ara ane aki Fukuoka ITot bar Hokkai dSohi Tokushi moa A c hi Fukuoka m S uoka e i S tKor am Kaum am ot agaw aki Fukuoka t aki orhiYm iFukui Wakayam a Kagawa amagat K ahiizM aaN Yamanashi Me i I Ibar N ar aakiane icS hi hi m Kai ochi Kochi tKor kayam i Tot Yamanashi agasaki Nar a IA Ehior mTot e Tokushi ma Fukuoka I bar Fukuoka him ane um am ot toaor i i agaw a bari aki Fukuoka Yamanashi Aom i KO WakayamTot a tI or YSamagat abaraki G if u Yamaguchi Kagaw a OAM iticahie Shim ane Nar a Me i G unm a iyazaki I bar aki Yamanashi aochi Kochi Fukuoka Shi m ane Kochi Toyam aochi K Nar a A kit eK a K ochi M I wat M i KochiM e I bar aki K Fukuoka ochi agaw ae i Kochi ShigY Kochi ar aaki KKFukuoka agaw Tot t or S i him ane KK ochi amanashiM e Nar aaKochi IN bar i Kochi Shizuoka K agaw a ochi Fukushim I bar a aki Fukuoka N Kagawa Shim ane I barK aki Kbar ochiaki Kar ochi Kochi MaFukuoka e i Kagawa a IFukuoka Tokushi ma K ochi ochi Kum amI ot ar a Wakayam Yamagat a KN ochi baroaki Aic hi Kagaw a Fukuoka Kochi Kochi Kochi I bar akiKochi Fukuoka Tot t or i Yamanashi Kochi Shim ane Kochi Me i Kochi Nar a Kochi Kochi Kagawa I bar aki Fukuoka Kochi Kochi

relative C growth 0 .01

_bc

_nwtnun _mb

_nwtnun _bc _nwtnun _nwtnun _nwtnun _bc

_mb

_pei _yt _yt

_qc _ns _pei

_pei _nb _nfl

_nfl

_nfl

_mb _nwtnun _bc _pei

_qc _ns _on _mb

_mb _qc _ns _mb_mb _sk _nb _on _on _on _ab _pei _qc _ns _sk _on _mb _nwtnun _ns _qc _ns _qc _sk _sk _on _bc _ab _ab _nb _pei _pei

_yt

_yt

_bc _yt

_nwtnun _nwtnun _bc _bc

_bc

_nwtnun _yt _nwtnun _bc _bc _yt

_yt _yt

-.01

-.01


Tochigi

_yt_nfl _yt _nb

_on

_pei _mb _nb

_qc _ns _ns _qc _nfl _pei

_ab

_ab

_mb _on

_ab

_ab _ab _qc

_sk _qc _ns _sk _nfl _nb _qc _ns _nb _nb _sk _pei _nb _nfl _nfl

_on

_sk _ab

_sk

_ns

_sk

_mb _on _on

_ab

_ab

_sk

_pei _nb _nb

_nfl _nfl

_nfl

-.02

-.02

Kochi

-.01

-.005

0 relative CPI growth

.005

-.005

.01

Consumption growth and CPI growth in Spain 1995-2004


.05

.02

Consumption growth and CPI growth in China 1993-2006 Tibet Xinjiang Heilongjiang

Heilongjiang Heilongjiang Heilongjiang

MURCIA MURCIA (Región (La) de) RIOJA (La) de) MURCIA (Región de) (Región RIOJA (La) RIOJA MURCIA (Región RIOJA (La) de) MURCIA (Región RIOJA (La) de)

-.01

relative C growth 0


MADRID (Com.Y de) CEUTA Y MELILLA MADRID (Com. de) CEUTA Y MELILLA NAVARRA (C. Foral de) MADRID (Com. de) CEUTA MELILLA MURCIA (Región de) NAVARRA (C. Foral de) RIOJA (La) MADRID NAVARRA (Com. de) (C. CATALUÑA Foral de) CEUTA Y MELILLA CATALUÑA NAVARRA (C. Foral de) CATALUÑA CATALUÑA MADRID (Com. de) MURCIA (Región de) CEUTA Y MELILLA RIOJA (La)(Región NAVARRA (C. PAIS VASCO MURCIA de) de) PAIS VASCO CATALUÑA RIOJA (La)Foral PAIS VASCO CASTILLA-LA MANCHA CASTILLA-LA MANCHAPAIS VASCO CASTILLA-LA CASTILLA-LA MANCHA MADRID (Com.MANCHA de) CEUTA Y MELILLA COMUNIDAD VALENCIANA PAIS VASCO NAVARRA COMUNIDAD (C. Foral VALENCIANA de) COMUNIDAD VALENCIANA CATALUÑA COMUNIDAD VALENCIANA MANCHA MURCIA (Región de) (C. Foral MADRID (Com.CASTILLA-LA de) CEUTA Y MELILLA BALEARS RIOJA (La) (Illes) BALEARS (Illes) NAVARRA MADRID (Com. BALEARS de) (Illes) CEUTA Y MELILLA COMUNIDAD VALENCIANA CATALUÑA NAVARRA BALEARS (C. de) Foral (Illes) de) PAIS VASCO CATALUÑA MURCIA (Región RIOJA (La) de) CASTILLA-LA MANCHA BALEARS (Illes) MURCIA (Región de) PAIS VASCO RIOJA (La) COMUNIDAD VALENCIANA PAIS CANTABRIA MURCIA (Región de)VASCO CASTILLA-LA MANCHA CANTABRIA RIOJA (La) de) MADRID (Com. CANTABRIA de) CEUTA Y MELILLA ARAGON CASTILLA-LA MANCHA ARAGON NAVARRA CANTABRIA (C. Foral ARAGON BALEARS (Illes) CATALUÑA COMUNIDAD VALENCIANA ARAGON COMUNIDAD VALENCIANA MADRID (Com. MURCIA de) (Región de) CEUTA Y MELILLA RIOJA (La) CANTABRIA ASTURIAS (Ppdo. NAVARRA Foral de) MURCIA (Región de) de) ASTURIAS (Ppdo. de) RIOJA (La) MURCIA (Región de) ASTURIAS (Ppdo. de) ARAGON CATALUÑA MURCIA (Región BALEARS (Illes) RIOJA (La)(C. (La) de) ASTURIAS (Ppdo. de) PAIS(Com. VASCO MADRID de) CEUTA Y MELILLA BALEARS (Illes) NAVARRA (C.RIOJA Foral de) CATALUÑA CASTILLA-LA MANCHA MADRID (Com. de) CEUTA Y MELILLA ASTURIAS (Ppdo. de) NAVARRA (C. Foral CANTABRIA ANDALUCIA PAISde) VASCO ANDALUCIA CATALUÑA ANDALUCIA COMUNIDADARAGON VALENCIANA ANDALUCIA CASTILLA-LA MANCHA PAIS VASCO MADRID (Com. de) EXTREMADURA CEUTA Y MELILLA EXTREMADURA CANTABRIA NAVARRA (C.GALICIA Foral CEUTA de) CASTILLA MADRID EXTREMADURA (Com. de) CEUTA Y(Com. MELILLA MURCIA (Región de) GALICIA MADRID de) Y MELILLA Y LEON RIOJA (La) NAVARRA (C. Foral de) CANARIASCANARIAS NAVARRA (C. Foral de) CASTILLA Y LEON CATALUÑA ARAGON CANARIAS CATALUÑA YGALICIA LEON CATALUÑA CASTILLA-LA MANCHA COMUNIDAD CASTILLA VALENCIANA EXTREMADURA BALEARS ASTURIAS (Ppdo. de) (Illes) PAIS VASCO CANARIAS COMUNIDAD VALENCIANA CASTILLA ASTURIAS YGALICIA LEON (Ppdo. de) VASCO ANDALUCIA PAIS VASCO PAIS VASCO BALEARS PAIS (Illes) CASTILLA-LA MANCHA CASTILLA-LA MANCHA CANTABRIA CASTILLA-LA MANCHA CASTILLA-LA MANCHA BALEARS (Illes) MADRID (Com.(Com. de) EXTREMADURA CEUTA Y MELILLA ARAGON COMUNIDAD VALENCIANA MADRID de) ANDALUCIA CEUTA YVALENCIANA MELILLA NAVARRA (C. Foral de) de) CANARIAS COMUNIDAD NAVARRA (C. Foral CASTILLA CATALUÑA ANDALUCIA YGALICIA LEON COMUNIDAD VALENCIANA COMUNIDAD VALENCIANA CATALUÑA CANTABRIA ARAGON EXTREMADURA BALEARS (Illes) ASTURIAS (Ppdo. EXTREMADURA BALEARS (Illes)de) CANARIAS CANTABRIA BALEARS (Illes) BALEARS (Illes) CASTILLA YGALICIA LEON PAIS VASCO CANARIAS PAIS VASCO ARAGON CASTILLA YGALICIA LEON CANTABRIA ASTURIAS (Ppdo. de) CASTILLA-LA MANCHA ARAGON CASTILLA-LA MANCHA ANDALUCIA ASTURIAS (Ppdo. de) COMUNIDAD VALENCIANA COMUNIDAD VALENCIANA CANTABRIA ASTURIAS (Ppdo. de) EXTREMADURA CANTABRIA ANDALUCIA CANTABRIA ARAGON CANARIAS ARAGON BALEARS (Illes) ARAGON CASTILLA YGALICIA LEON ARAGON BALEARS (Illes) CANTABRIA ANDALUCIA EXTREMADURA ASTURIAS (Ppdo. de)de) CANARIAS ANDALUCIA ASTURIAS (Ppdo. CASTILLA(Ppdo. YGALICIA LEON ASTURIAS de) ASTURIAS (Ppdo. de) EXTREMADURA GALICIA CANARIAS CASTILLA Y LEON EXTREMADURA CANTABRIA ANDALUCIA CANARIAS CANTABRIA YGALICIA LEON ANDALUCIA ANDALUCIA ARAGON ANDALUCIA ARAGONCASTILLA EXTREMADURA EXTREMADURACASTILLA CANARIAS EXTREMADURA ASTURIAS YGALICIA (Ppdo. LEON de) EXTREMADURA GALICIA CANARIAS ASTURIAS (Ppdo. de) CASTILLA YGALICIA LEON CANARIAS CANARIAS CASTILLA Y LEON CASTILLA YGALICIA LEON

-.02

-.05

ANDALUCIA ANDALUCIA EXTREMADURA EXTREMADURA CANARIAS CASTILLA YGALICIA LEON CANARIAS CASTILLA YGALICIA LEON

-.01

-.005


.005

.01

Xinjiang Tibet Xinjiang Tibet Qinghai Xinjiang Tibet Qinghai Xinjiang Tibet Tibet

Xinjiang Tibet Xinjiang

Tibet

Qinghai Qinghai

Tibet XinjiangTibet Heilongjiang Liaoning Xinjiang Heilongjiang Qinghai Liaoning Tibet Xinjiang Xinjiang Qinghai Tibet Heilongjiang Tibet Heilongjiang Tibet Xinjiang Qinghai Heilongjiang Tibet Heilongjiang Xinjiang Qinghai Jilin Heilongjiang Liaoning Liaoning Qinghai Xinjiang Tibet Jilin Qinghai Jilin Heilongjiang Qinghai Xinjiang Yunnan Yunnan Guangxi Liaoning Hainan Heilongjiang Xinjiang Yunnan Tibet Qinghai Guangxi Guangdong Heilongjiang Tibet Fujian Hainan Qinghai Heilongjiang Guizhou Tibet JilinLiaoning Guangdong TibetQinghai Fujian Hainan Heilongjiang XinjiangQinghai Xinjiang Hunan Tibet Beijing Guizhou Yunnan Guangxi Liaoning Xinjiang Jilin Guangxi Fujian Hainan JiangxiLiaoning Tibet Xinjiang Hunan Qinghai Guangdong Heilongjiang Beijing Xinjiang Yunnan Beijing Tibet Guangxi Heilongjiang Liaoning Jiangxi Tibet Hainan Tibet Guizhou Hunan Heilongjiang Xinjiang Tianjin Qinghai Liaoning Jilin Qinghai Hubei HeilongjiangQinghai Fujian Guangdong Liaoning Guizhou Xinjiang Tibet Jilin HunanYunnan Heilongjiang Xinjiang Xinjiang Guizhou Tianjin Hubei Qinghai Beijing Guangxi Liaoning Shanxi JiangxiJiangxi Heilongjiang Fujian Hainan Qinghai Sichuan JilinGuangdong Tianjin Liaoning Yunnan Guangxi Xinjiang Hebei Hunan Jilin Gansu Heilongjiang Qinghai Hainan Shanxi Liaoning Guangdong Beijing Heilongjiang Jiangxi YunnanHubei Guangxi Liaoning Heilongjiang Jilin Ningxia Hebei Guizhou Tibet Qinghai Guangdong Jilin Yunnan Hainan Hainan Guangxi Qinghai Qinghai Inner Mongolia NingxiaHeilongjiang Fujian Guizhou Liaoning Shanghai Hubei Guangdong Yunnan Hunan Guangxi Fujian Tianjin Yunnan Guangxi Guangdong ShaanxiJilinGansu Inner Mongolia Hubei Shanxi Beijing Qinghai Hainan Guizhou Shanghai Hainan Jilin Jiangxi Hunan Xinjiang Guizhou Yunnan HebeiShaanxiJiangxi Beijing Guangdong Fujian Jilin Guangxi Tianjin Guangdong Liaoning Hainan Fujian Ningxia Liaoning Yunnan JilinHebei Guizhou Hunan ShanxiHunan Guangxi Guizhou Gansu Anhui Liaoning Hainan Beijing Shanghai Hebei Guangdong Jiangxi Fujian Guangxi Hubei Jilin Heilongjiang Inner Mongolia BeijingYunnan Jiangsu Liaoning Yunnan Fujian Shanxi Jiangxi Shanghai Hainan Anhui Guangdong Guangxi Ningxia Hunan Guizhou Hunan Hubei Liaoning Tianjin Hainan Beijing Qinghai Guizhou Guangdong Beijing Fujian Jiangxi JiangsuFujian Shaanxi Yunnan Guangxi Jiangxi Shanxi Inner Liaoning Mongolia Tianjin Gansu Guangdong Shanghai Hunan Hainan Guizhou Hubei JilinHebei Shanxi Jilin Guangdong Hubei Beijing Liaoning Henan Guizhou Hunan Jiangxi Shaanxi Jilin Fujian Tianjin Guangxi Anhui Jiangsu Ningxia Liaoning Liaoning Hebei Beijing TibetFujian Jiangxi Hunan Gansu Tianjin Shanxi Hubei Guizhou Yunnan Jilin Shaanxi Henan Hubei Yunnan Jiangsu Ningxia Guangxi Shanxi Hunan Hainan Jiangxi InnerBeijing Mongolia Fujian Hebei Yunnan Shanghai Hainan Guangxi JilinZhejiang Liaoning Tianjin Beijing Anhui Tianjin Xinjiang Hebei Jiangxi Hainan Hubei Hunan Inner MongoliaTibet Yunnan Guangdong Ningxia Shanxi Guangxi JilinGansu Shanghai Guangdong Zhejiang Shanxi Jiangsu Shaanxi Hainan Ningxia Beijing Hubei Jiangxi Yunnan Guizhou Inner Mongolia Guangdong Guangxi Hebei TianjinChongqing Xinjiang Guizhou Jilin Hebei Chongqing Inner Mongolia Hainan Shaanxi Guangdong Hubei Henan Tibet Shanghai Shanxi Yunnan Fujian Ningxia Guizhou Tianjin Jilin Guangxi Jilin Inner Mongolia Gansu Fujian Shandong Ningxia HeilongjiangFujian Shanghai Hainan Hubei Anhui Shanxi Hunan Guizhou Guangdong Hebei Tianjin Yunnan Hunan Guangxi Shaanxi Anhui Inner Mongolia Shandong Beijing Henan Chongqing Guangxi Fujian Chongqing Shanghai Jiangxi Chongqing Hainan Hainan Beijing Hunan Yunnan Shanxi Hebei Ningxia Guizhou Jiangxi JiangsuGansu Guangdong Tianjin Qinghai Shaanxi Sichuan Beijing Inner Mongolia Anhui Shaanxi Guangdong Hebei Shanghai Jiangxi Ningxia Guangxi Hunan Guangdong Shanxi Gansu Fujian Xinjiang Zhejiang Guizhou Anhui Ningxia Gansu Zhejiang HainanHeilongjiang Inner MongoliaYunnan Beijing Jiangsu Ningxia Shanghai Guizhou Hunan Jiangxi Chongqing Guizhou Hubei Jiangxi Hebei Shaanxi Fujian Hubei Jiangsu Henan Inner Mongolia Shandong Beijing Anhui Guangdong Shanghai Anhui Fujian GansuShaanxi Fujian Hubei Hunan Tianjin Qinghai Inner Mongolia Henan Tianjin Heilongjiang Shanghai Guizhou Hunan Jiangsu Ningxia JilinZhejiang Gansu Hubei Beijing Hunan Sichuan Jiangsu Shanxi Jiangxi Shaanxi TianjinShanxi Beijing Anhui Inner Mongolia Shandong Beijing Jiangxi Gansu Jiangxi Chongqing Fujian Gansu Hubei Shanghai Henan Shanxi Hebei Tianjin Shaanxi Qinghai Hebei Anhui Chongqing Yunnan Guangxi Jiangsu Zhejiang Henan ShanxiHunan Hubei Beijing Ningxia Hebei Tianjin Sichuan Hainan Ningxia Shaanxi Anhui Jiangsu Jiangxi Inner Shanxi Gansu Ningxia Tianjin Hubei Zhejiang Henan AnhuiHebei Henan Guangdong Jiangsu Mongolia Guizhou HubeiTianjin Inner Mongolia Hubei Zhejiang Shanghai Ningxia Shanxi Hebei Shandong Liaoning Shanghai Jiangsu Tianjin Inner Mongolia Ningxia Shanghai Henan Hebei TianjinShaanxi Anhui Shandong Shanxi Chongqing Zhejiang Sichuan Inner Mongolia Zhejiang Fujian Hubei Jiangsu LiaoningShaanxi Shanghai Shanxi Shanxi Shandong Henan Gansu Ningxia Hebei Chongqing Gansu Hunan Shaanxi Inner Mongolia Sichuan Shanghai Tianjin Hebei Henan Gansu Beijing Zhejiang Shaanxi Ningxia Shandong Jiangxi Inner Mongolia Anhui Chongqing Shanxi Gansu Henan Shanghai AnhuiHebei Ningxia Shaanxi Zhejiang Sichuan Ningxia Liaoning Anhui Jiangsu Inner Mongolia Shandong Hebei Gansu Sichuan Jilin Jiangsu Shanghai Zhejiang Henan Shandong Shaanxi Anhui Inner Mongolia Inner Mongolia Shanghai Jiangsu Ningxia Hubei Chongqing Shanghai Gansu Zhejiang Sichuan Anhui Shaanxi Gansu Jiangsu Liaoning Shandong Sichuan Yunnan JilinTianjin Guangxi Shaanxi Shaanxi Zhejiang Inner Gansu Mongolia Hainan Anhui Shanghai Jiangsu Shandong GansuGuangxi Henan Yunnan Sichuan Henan Chongqing Shanxi Shandong Jiangsu Anhui Guangdong Hainan Chongqing Shaanxi Sichuan Henan Anhui Anhui Jilin Shandong Hebei Gansu Chongqing Guizhou Jiangsu Henan Guangdong Sichuan Zhejiang Zhejiang Ningxia Jiangsu Jiangsu Henan Sichuan Chongqing Guizhou Guangxi Zhejiang Anhui Yunnan Jilin Hunan Shandong Hainan Fujian Henan Chongqing Zhejiang Inner Mongolia Fujian Sichuan Shanghai Beijing Jiangsu Jiangxi Chongqing Chongqing Hunan Guangdong Zhejiang Yunnan Henan Guangxi Shandong Shaanxi Beijing Henan Hainan Henan Chongqing Shandong Guizhou Jiangxi Zhejiang GansuSichuan Shandong Guangdong Fujian Zhejiang Chongqing Sichuan Henan Hunan Anhui Zhejiang Zhejiang Hubei Sichuan BeijingShandong Shandong Guizhou Tianjin Hubei Sichuan Jiangsu Fujian Shandong Shanxi Jiangxi Zhejiang Hunan Tianjin Chongqing Sichuan Beijing Hebei Shandong Shanxi Sichuan Shandong Ningxia Jiangxi Shandong Hubei Hebei Chongqing Sichuan Henan Sichuan Chongqing Tianjin Sichuan NingxiaShandong Inner Mongolia Shanghai Chongqing HubeiShanxi Chongqing Chongqing Zhejiang Inner MongoliaSichuan Shanghai Shaanxi HebeiGansu Tianjin Chongqing Ningxia Shanxi Shaanxi Gansu Chongqing Anhui Hebei Inner Mongolia Shanghai Ningxia Shandong Jiangsu Anhui Shaanxi Sichuan Inner Mongolia Gansu Jiangsu Shanghai AnhuiShaanxi Gansu Henan Jiangsu Henan Zhejiang Anhui Jiangsu Zhejiang Henan Shandong Henan Zhejiang Sichuan Shandong Zhejiang Sichuan Shandong Sichuan ChongqingShandong Sichuan

-.02

Consumption growth and CPI growth in India 1960-2000

-.01


.01

.02

.02

Consumption growth and CPI growth in Germany 1995-2007 .02

_be

_assam_rajasthan

_assam_rajasthan

_mp _up

_assam_rajasthan _rajasthan _rajasthan _assam _rajasthan _assam _assam _wb_gujurat _assam_rajasthan _rajasthan _maharashtra _mp_assam _assam_rajasthan _karnataka _ap _punjab _mp _mp _mp _tn _assam_rajasthan _rajasthan_mp _mp _haryana _up _assam _mp _haryana _up _up _wb_gujurat _up _orissa _haryana _mp _up _haryana _haryana _rajasthan _maharashtra _wb_gujurat _up _assam _mp _haryana _karnataka _ap _up _haryana _maharashtra _punjab _haryana _wb _karnataka _ap _wb _wb _gujurat _punjab _gujurat _gujurat _maharashtra _tn _wb _up _maharashtra _gujurat _karnataka _maharashtra _wb _ap _rajasthan _karnataka _ap _assam _gujurat _punjab _karnataka _punjab _maharashtra _ap _haryana _maharashtra _kerala _punjab _karnataka _ap_ap _karnataka _punjab _tn_tn _punjab_tn _wb_gujurat _kerala _tn _kerala _kerala _wb_maharashtra _up _mp _mp_tn_tn _gujurat _kerala _karnataka _ap _maharashtra _haryana _kerala _punjab _kerala _bihar _karnataka _ap _punjab _tn _wb_gujurat _tn _up _kerala _up _maharashtra _haryana _orissa _kerala _haryana _karnataka _ap _punjab _orissa _orissa _wb_gujurat _tn _wb_gujurat _orissa _orissa _maharashtra _orissa _kerala _maharashtra _orissa _karnataka _ap _karnataka _ap _punjab _punjab _bihar _tn _tn _bihar _orissa _bihar _bihar _orissa _kerala _kerala _bihar _bihar _bihar

_be

_kerala

_mp

_bihar

_bihar _bihar

_kerala

_be _be _be

_th

_th _th

_sl _sl

_sl _sl _sl _sl _th

_bw _ni _by

_he

_he

_be _be

_bb_bb _st _sn _bw _bw _ni _sl _bw _rp _nw _bw _bw _by _ni _sn _rp _nw _ni _ni _rp _nw _rp _st _by _nw _by _by _he _ni _bb _nw _he _rp _he _he _by _he _mv_mv _st _sn _st _sn

_th _th

_th _th

_rp _nw

_bw _ni _bw _sl _rp _nw _by _ni _bb _rp _nw _by _he

_be

_mv _sl _sl _bb _bb

_bb _mv _bb _bb

_sl _bb

_st _sn

_rp _nw

_bw _ni _by

_mv

_he

_bb

_th _th

_st _sn _st _sn _st _sn _st _sn _st _sn _st _sn _sl _mv

_th

_mv _mv _mv

_th

-.02

_orissa _orissa

_bihar _bihar

_be

_bw _ni _by

_bw_bw _ni _ni _nw_nw _rp _rp _by _by _he_he

_orissa

_bihar

-.02

_rp _nw _he

_be

_up

_orissa

_be_be

_haryana

_wb_gujurat _maharashtra _up _karnataka _ap _punjab _haryana _tn

_assam_rajasthan

relative C growth -.01 0 .01


_mp

-.01

.005

_mv _mv

_bb

_st _sn

_mv

-.02

-.01


.01

-.01

.02

-.005


.005

.01

Notes: Each point in the scatterplots above represents the sample average of the RER growth versus the sample average of the relative consumption growth for a pair of regions within a given country over a corresponding sample period. Details on our data sources and coverage are provided in Appendix A.1.

In terms of volatilities we …nd that real exchange rate is signi…cantly less volatile than consumption in the regional data and signi…cantly more volatile than consumption in the international data. These empirical …ndings discipline our theoretical work as they provide us with the additional testable implication that our model should comply with. In particular, any explanation for the Backus-Smith puzzle and the real exchange rate volatility puzzle in the international data must also comply with the corresponding properties of the RER and consumption in the regional data. Providing such an explanation is the main objective of 4

our paper. Our model is a relatively canonical model of international business cycles with incomplete asset markets. We consider a world economy composed of two symmetric countries. Each country consists of two regions. Each region is represented by a continuum of households and a continuum of monopolistically competitive …rms who choose whether to sell their goods in the domestic regional market, foreign regional market within the same country, or to export to the foreign country. The split is endogenously determined by a combination of regional and international transportation costs and …xed costs of exporting. Each …rm is subject to a sector-speci…c total factor productivity (TFP) shock and a …rm-speci…c productivity shock. As TFP varies, the mass of …rms producing in each sector ‡uctuates over time, inducing changes in the supply of goods and relative prices. We …nd that allowing for regional heterogeneity and endogenous international and regional non-tradability generates a negative correlation between RER and relative consumption across countries and a positive correlation across regions within a country, thus allowing us to account for the Backus-Smith puzzle in both international and regional dimensions. The two features also generate realistic volatility of RER relative to consumption in the regional dimension, but underpredict it in the international dimension. TFP shocks in our model originate in the internationally and regionally traded and regionally nontraded sectors. Internationally traded goods shocks trigger a wealth e¤ect and a Balassa-Samuelson e¤ect and thus imply a negative correlation between international RER and relative consumption. Internationally nontraded shocks have the opposite e¤ect. By introducing within country heterogeneity and endogenous tradability, one at a time, we show that the international Backus-Smith correlation can be reduced. Regional heterogeneity is important because agents within a given country can share some of their internationally uninsurable country-speci…c shocks with their countrymen living in the other regions. This reduces the e¤ects of these shocks on RER and relative consumption. Endogenous tradability is important because it makes the size of each sector cyclical, as new varieties become traded across regions and countries. This ampli…es the e¤ects of internationally traded shocks and suppresses the e¤ects of internationally-nontraded shocks. A combination of the two features generates the Backus-Smith correlation consistent with both the intra-national and international data. This paper contributes to the existing literature along several dimensions. First, it contributes to the literature studying Backus-Smith and real exchange rate volatility puzzles in the context of open economy models. A number of papers have tried to explain the Backus-Smith puzzle using models with incomplete …nancial market structure. Benigno and 5

Thoenissen (2008), introduce a production economy with non-traded and traded goods sectors. Under the assumption that productivity shocks to the traded goods sector are more persistent and more volatile than those in the non-traded goods sector, they can obtain a negative correlation between relative consumption and the real exchange rate. Corsetti, Dedola, and Leduc (2008) show that with a very low elasticity of substitution between domestic and foreign traded goods, they can account for the Backus-Smith puzzle.1 By contrast, our model can account for the Backus-Smith puzzle while using stationary sectoral productivity processes and standard values for substitution elasticities. Further, the existing work focuses only on international risk-sharing and does not study the intra-national risk-sharing. They also cannot account for the RER volatility puzzle. Bodenstein (2006) introduces limited contract enforcement into a two-country complete markets model with the production structure of Corsetti, Dedola, and Leduc (2008) to rationalize both the Backus-Smith and RER volatility puzzles. However, he too does not study intranational risk-sharing. Engel and Wang (2008) introduce durable goods into an otherwise standard business cycle model and show that they can replicate the Backus-Smith correlation in the data under the parametrization similar to that in Benigno and Thoenissen (2008). Our contribution is to extend the existing work by departing from a representative agent framework and incorporating regional heterogeneity within each country. Furthermore, our approach allows sectoral tradability to be endogenously determined as a result of pro…t maximizing decisions by heterogenous …rms. Studies that do allow for inter-regional risk-sharing are primarily empirical and focus entirely on the comovement between consumption and output growth rates (Sorensen and Yosha, 1998; Crucini and Hess, 1999; Labhard and Sawicki, 2006). Our contribution to this empirical literature is to focus on a measure of risk-sharing due to Backus and Smith (1993) and provide a set of stylized facts about the intra-national Backus-Smith correlation and real exchange rate volatility for a relatively large sample of developed and developing countries. Furthermore, we show that these regional and international facts can be rationalized jointly. Finally, several recent papers have studied the role of endogenous non-tradability for the international business cycles. Ghironi and Melitz (2005) incorporate Melitz (2003) model of trade into a dynamic two-country model and show that it improves the model performance in terms of several business cycle moments. Alessandria and Choi (2007) …nd that endogenous …rms export decisions have negligible e¤ects on the properties of international business cycles. In a similar framework, Naknoi (2008) studies the dynamics of RER under di¤erent exchange 1

When elasticity of substition between domestic and imported tradable goods is high, the model requires very persistent productivity shocks (approaching unit root) to generate negative Backus-Smith correlation.

6

rate regimes. Corsetti, Martin, and Pesenti (2008) use endogenous tradability model to study the dynamics of current account adjustment. Bergin and Glick (2005) and Bergin, Glick, and Taylor (2006) use a model of endogenous tradability to understand price dispersion and the Balassa-Samuelson e¤ect. Our approach embeds the sectoral tradability decisions into a dynamic international business cycles framework and allows for both international and regional export decisions in the …rm’s problem. The rest of the paper is structured as follows. The next section describes the BackusSmith and real exchange rate volatility puzzles. Section 3 presents our empirical …ndings on the two puzzles in the international and intra-national data. Section 4 develops the model that we propose to rationalize our empirical results. Section 5 describes how the model is calibrated and solved. The results are presented in Section 6. The last section concludes.

2

Stating the Puzzles

To illustrate the main idea behind the Backus-Smith puzzle, consider a two country international business cycle model. A problem facing a representative household in country j = 1; ::; J is as follows: max Et

1 X

i

j j ; Xt+i ); U (Ct+i

i=0

where is the subjective discount factor, Ct denotes a composite consumption good in period t; Xt denotes factors other than consumption that can a¤ect utility (e.g. preference shocks, labor choice, etc.). Ct is de…ned as a CES aggregator over the consumption of traded, Ctt , and nontraded, Ctn ; goods. Let Ptj denote consumption price index in country j: Every period households receive stochastic endowments of traded and nontraded goods: Ytt and Ytn ; respectively. Further, suppose that in this world a complete set of statecontingent securities is available to households in all countries. In this case, the equilibrium can be obtained as a solution to a social planner’s problem. In particular,

max Et

J X

j

j=1

1 X

i

j j U (Ct+i ; Xt+i )

i=0

subject to the resource constraints X

t Cjt =

j

n Cjt =

X

j Yjtn ;

7

Yjtt ;

8t 8j; 8t

The resulting key optimality condition from an equally-weighted planning problem is Uc (Ci;t ; Xi;t ) =

Pi;t Uc (Cj;t ; Xj;t ); Pj;t

where Uc (Ci;t ; Xi;t ) denotes the marginal utility of aggregate consumption in country i: Further, let ei;j t = Pi;t =Pj;t denote the real exchange rate between country i and j: Then the condition above can be written as Uc (Ci;t ; Xi;t ) = ei;j t Uc (Cj;t ; Xj;t ):

(1)

This equation must hold in every date and state of the world, between any two countries i and j. It says that the social planner allocates consumption between households in countries i and j in such that its marginal utility (converted into the same units using the real exchange rate) is equalized across countries. If utility is of a constant relative risk aversion (CRRA) form, with the coe¢ cient of relative risk aversion !; equation (1) becomes Ci;t Cj;t

!

= ei;j t ;

or equivalently in logs ! (ln Ci;t

ln Cj;t ) = ln ei;j t :

These expressions establish the close relationship between the real exchange rate and relative consumption in countries i and j: They imply that during the times when the real exchange rate in country i appreciates, the consumption in that country should fall relative to country j’s consumption. The expression above must also hold in growth rates: ! ( ln Ci;t where e;cj =ci

ln Cj;t ) =

ln ei;j t ;

ln Xi;t = ln Xi;t

ln Xi;t 1 . Therefore, if markets are complete, the correlation,

= corr( ln ei;j t ;!

ln Cj;t ) should be equal to 1, as pointed out by Backus and Smith i;t

C

(1993). In the data, a number of authors have found this correlation to be very low and often negative (see Backus and Smith, 1993; Kollmann, 1995; Chari, Kehoe, and McGrattan, 2002; Corsetti, Dedola, and Leduc, 2008). Ravn (2001) and Lewis (1996) study the role of preference nonseparabilities in explaining Backus-Smith puzzle and …nd that the puzzle is robust to them. A related puzzle is the real exchange rate volatility puzzle. It relates the volatility of the real exchange rate to the volatility of macroeconomic aggregates, such as consumption or 8

output. In the context of the complete markets model presented above, the relative volatility of the real exchange rate and consumption can be written as 2 e ci

cj

V( ln ei;j t ) V( ln Ci;t )V( ln Cj;t ) V( ln Ci;t ) + V( ln Cj;t ) 2CV ( ln Ci;t ; ln Cj;t ) p = !2 V( ln Ci;t )V( ln Cj;t ) s s ! V( ln C ) V( ln C ) i;t j;t + 2 ci ;cj ; = !2 V( ln Cj;t ) V( ln Ci;t )

= p

where V(.) and CV(.) denote the unconditional variance and covariance; and ci ;cj

= corr( ln Ci;t ;

ln Cj;t ): The complete markets model above predicts

2 e ci

cj

to be

very small, unless ! is very large. In the data, this relative volatility is large (see Chari, Kehoe, and McGrattan, 2002; Bodenstein, 2006 and our estimates in the next section).

3

Empirical analysis

In this section we study the properties of real exchange rate and relative consumption empirically. We start by documenting these properties in the cross-country data, and then develop a more disaggregated analysis that relies on the region-level data for a large sample of economies. To conduct this analysis we construct a novel dataset of historical series for consumption and prices for a large number of regions in a number of developed and developing countries. Our dataset contains regional data for Canada, Japan, Spain, Germany, India and China and covers at least ten years of annual data. A detailed description of the data and its sources is provided in Appendix A.1. Following Backus and Smith (1993), a number of papers have studied the correlation between consumption and real exchange rates in the international data. Chari, Kehoe, and McGrattan (2002) report the median value for the correlation between real exchange rate and relative consumption in quarterly data between 1973-1994 for a sample of OECD countries to be -0.07. Corsetti, Dedola, and Leduc (2008) in a sample of OECD economies during 1970-2001 …nd that this correlation is well below one and often is negative when measured in levels or growth rates. Benigno and Thoenissen (2008) re-estimate the correlations between the two variables (both logged and Hodrick-Prescott …ltered) in levels and …rst-di¤erences using the US as a reference country. In their sample of annual data during 1970-2000 they …nd that the median correlation is -0.288 in levels and -0.168 in the …rst-di¤erences. We con…rm these …ndings for our sample of countries in annual data during 1970-2004. 9

Table 1 summarizes the results. Columns (i)-(iii) report the average, maximum and minimum of the bilateral correlations between the growth rates of real exchange rate and relative consumption for 6 di¤erent countries measured relative to OECD economies.2 For all countries, on average, the correlation is negative. Table 1. International C-RER correlation and relative volatility q 2 e

e;cj =ci

Canada Japan Spain India China Germany

ci

cj

mean (i)

max (ii)

min (iii)

mean (iv)

-0.100 -0.045 -0.152 -0.150 -0.134 -0.101

0.185 0.323 0.435 0.567 0.345 0.486

-0.568 -0.656 -0.570 -0.608 -0.616 -0.674

3.132 3.252 2.267 2.895 4.611 1.954

e;cj =ci heading report summary statistics on the cross-sectional distribution of the correlation coe¢ cient between the growth rate of RER and the growth rate of relative consumption Notes: Columns under

for a set of all bilateral pairs of a given country with other OECD countries calculated over the sample period. Mean, max, min q refer to the average, maximum and minimum of that distribution. In the same spirit, column under

2 e

ci

cj

heading reports the sample mean of the cross-sectional distribution of

standard deviation of RER growth between countries

i and j relative to the square root of the product

of standard deviations of consumption growth in those countries. Details on the sample coverage and data sources are provided in Appendix A.1.

Column (iv) reports the volatility of the growth rate in the real exchange rate relative to the volatility of consumption growth in country i and OECD economies (j). For all countries in our sample the volatility of real exchange rate exceeds the volatility of consumption. The real exchange rate in China exhibits the most volatile relative consumption among the countries in our sample. These numbers are consistent with the available estimates in the literature. For instance, in the quarterly data for US relative to 9 European economies during 1972:1-1993:3, Chari, Kehoe, and McGrattan (2002) …nd that the volatility of real exchange rate is 4.4 times that of US GDP. For the period of 1978:4-2003:4 Bodenstein (2006) computes that real exchange rate in US relative to the other G7 countries is 3.4 times as volatile as US consumption. Similarly, Johri and Lahiri (2008) report a number for this volatility equal to 5.5. Our …ndings are for the annual data and are consistent with these numbers. Next, we re-estimate the correlation and volatility properties of real exchange rate in the 2

For each OECD economy in our sample, we used the remaining OECD countries as a reference sample.

10

intra-national data. Such analysis also allows us to control for the movements in the nominal exchange rate. Table 2 presents the results. Table 2. Regional C-RER correlation and relative volatility q 2 e

e;cj =ci

ci

cj

mean (i)

max (ii)

min (iii)

mean (iv)

0.123 0.121 0.196 0.656 0.117 0.108

0.55 0.85 0.80 0.85 0.88 0.89

-0.35 -0.69 -0.70 0.34 -0.84 -0.63

0.47 0.47 0.25 0.73 0.36 0.23

Canada Japan Spain India China Germany Notes: Columns under

e;cj =ci heading report summary statistics on the cross-sectional distribution of the correlation coe¢ cient between the growth rate of regional RER and the growth rate of relative consumption for a set of all bilateral regional pairs in a given country calculated over the sample period. Mean, max, min qrefer to the average, maximum and minimum of that distribution. In the same spirit, column under

2 e

ci

cj

heading reports the sample mean of the cross-sectional distribution of standard

deviation of RER growth between regions

i and j relative to the square root of the product of standard

deviations of consumption growth in those regions. Details on the sample coverage and data sources are provided in Appendix A.1.

As in the international data, columns (i)-(iii) report the summary statistics on the bilateral correlation of the growth rate in real exchange rate and relative consumption between two regions within a given country. All of the average correlations are well below 1, with the minimum values well into the negative range. The numbers clearly indicate a failure of full RER-adjusted risk sharing within countries, but provide some evidence of more risk sharing within countries than across countries. Column (iv) reports the relative volatility of real exchange rate growth between two regions within a given country to their consumption growth. All numbers are well below one, suggesting that regional real exchange rate is signi…cantly less volatile than regional consumption. Overall, two results stand out from our empirical study. First, the Backus-Smith puzzle is not a purely cross-country phenomenon. The puzzle remains in the region/state-level data. Further, the Backus-Smith correlation varies substantially across regions/states in a given country. Second, the RER is more volatile than consumption in the international data and is less volatile than consumption in the intra-national data. We next develop a model to rationalize these …ndings.

11

4

Model economy

We work with a conventional two-country model with incomplete asset markets. This type of model has been used extensively in the literature to study the properties of international business cycles, including the Backus-Smith correlation and the volatility puzzle. First, we extend this model to allow for regional heterogeneity within each country and show this model can not account for the properties of international data. We then introduce endogenous non-tradability into this model and show that it too can not produce a realistic Backus-Smith correlation and RER volatility. Only when both regional heterogeneity and endogenous tradability are added simultaneously, can our model generate negative correlation between real exchange rate and relative consumption (in both levels and growth rates) in the international dimension, while predicting a positive correlation between the two variables in the intra-national dimension. Furthermore, the model’s predictions for the volatility of real exchange rate and relative consumption across regions and across countries are in accord with the data. We next present our full model that allows for both regional heterogeneity and endogenous non-tradability.

4.1

Firms

We consider a world economy consisting of two symmetric economies, home (h) and foreign (f). Each country is populated by a continuum of …rms and households who reside in two regions, a and b, within each country. Each region in our economy specializes in the production of a continuum of goods, indexed by i 2 [0; 1]: Each di¤erentiated good i is produced using constant returns to scale technology in labor input, li : yit = Xt Ai lit : Here Xt is the total factor productivity (TFP), and Ai is the good-speci…c productivity. Firms can sell their output in three markets: the domestic market, in another region within the same country, and abroad. Exporting to a foreign country is costly. There is a …xed cost of beginning to export, denoted by fx ; and iceberg transportation costs,

I.

In order

to sell goods in another region within the same country producers have to incur iceberg transportation costs

R;

where

R

0 is the elasticity of substitution between tradable goods. Here

h (ij )

h (ij )

and

denote the weights that households in country h assign to the consumption

of h and f-produced tradable goods. Again, these weights are endogenously linked to the shares of non-traded goods in the two countries as: h (ij )

=

1 2

ij ij

and

^{

f (ij )

=

1 2

^{ ij

^{

;

where ^{ = 0:5(^{a + ^{b ) is the average nontraded share in country f. Nontradable consumption of households living in region j = fa,bg is given by a CES ag-

gregator over consumption of tradables produced in regions a and b, Zja and Zjb ; respectively, and consumption of nontradables produced in region j within country h, Zjn : Cjn where

t

and

=

n

1 t

h

=1

1

a (ij )

t

Zja

+

1

b (ij )

Zjb

i

1

1 n

+

Zjn

;

represent the weights that households in region j assign to the

consumption of regionally tradable and nontradable goods. As mentioned before in the …rm’s problem, we assume that

n

is exogenously given. The elasticity of substitution between

regionally tradable and nontradable goods is given by (1

16

)

1

> 0: Further,

a (ij )

and

b (ij )

=1

a (ij )

denote the weights that households in region j assign to the consumption

of a- and b produced tradable goods. These weights are also functions of ij as a (ij )

=

ia ia + i b

and

b (ij )

ib : i a + ib

=

The elasticity of substitution between two regionally tradable goods is given by (1

)

1

>

0: Each consumption good is a CES aggregate of individual goods. For instance, consumption aggregates in region j; country h are given by: Zjn

=

Z

n ij

0

Zja

=

Zjb

=

Z

Z

1

1 n ij

ij

1

1 (1

n ij

ij +(1

=

Z

n )i j

ij

Cjf

=

Z

1

di

j

i 2 rt

1

1 1

i 2 it

cij di

ij

2 ^{

zi;

n) i j

1

1 1

1

1 (1

1

i 2 rt

zij di

n ) ij

ij

Cjh

i 2 rn

zij di

cîj

^{

i 2 it

di

See Appendix A.2 for derivations. For simplicity, we assume that households treat internationallytraded goods originated in the two regions of the foreign country as perfect substitutes. Preferences of f households living in region j = fa,bg are similarly de…ned in terms of tradable consumption basket, C^jt ; and a nontradable consumption basket, C^jn : The tradable consumption bundle in f country is de…ned symmetrically in terms of h tradables, C^jh ; and f tradables, C^jf ; as i1 h : C^ t = ^ h (^{ )1 (C^ h ) + ^ f (^{ )1 (C^ f ) j

j

j

j

j

Nontradable consumption bundle of f households living in region j is de…ned in terms of tradables produced in the regions a and b, Z^ja and Z^jb ; respectively, and consumption of region j nontradables, Z^jn :

C^jn =

"

1 t

{j )1 a (^

Z^ja

+

{j )1 b (^

Z^jb

+

1 n

Z^jn

#1

:

Note that by de…ning consumption aggregates in this way we rule our "love for variety" e¤ects, which are characteristic of Dixit-Stiglitz aggregators. In doing so we follow the tradi-

17

tion of the standard international business cycle models. Households in each country …nance their consumption expenditures with wage income and pro…ts,

j;

received from domestic

…rms. Households also have access to international borrowing and lending at interest rate Rt .3 As a result, asset markets in our model are incomplete both across regions within a country, and across countries. We use this simple asset structure in our model to preserve comparability with the existing models of international business cycles, and to maintain the symmetry across regions and across countries in the asset structure dimension. As noted before, we assume that internationally-traded goods produced in both regions within the same country are perfect substitutes. In combination with no barriers to regional trade in these goods, we get for h country Pa,t = Pb,t country Pâ,t = P^b,t P^t ; with = fh,fg:

Pt ; with

= fh,fg; and for f

Therefore, the period t budget constraint of households living in country h, region j is

given by h f a a b b n n Pth Cj;t + Ptf Cj;t + Rj;t Zj;t + Rj;t Zj;t + Rj;t Zj;t +

+ Lhj;t + Ljj;t + Lnj;t + fx (1

1 h P Bj;t Rt t

ij ) wj;t +

Pth Bj;t

1

(5)

j;t ;

where Bj;t denotes period t holdings of international bond, Pt is the h country price of internationally traded goods produced in country , with

= fh,fg; and Rt{ is the price

of regional good { = fa,b,ng as de…ned in (2)-(3): Lhj;t ; Ljj;t ; Lnj;t denote aggregate labor employed in the production of internationally tradable, regionally tradable and non-tradable

goods, respectively, in region j. Households in the f country, living in region j face an analogous constraint h f ^j;t ^ a Z^ a + R ^ b Z^ b + R ^ n Z^ n + 1 P h B P^th C^j;t + P^tf C^j;t +R j;t j;t j;t j;t j;t j;t Rt t ^ jj;t + L ^ nj;t + fx (1 ^{j ) w^j;t + ^ j;t ; ^ fj;t + L + L

^j;t Pth B

1

(6)

^j;t denotes the bond holdings of f households in region j. Finally, we choose interwhere B nationally traded goods produced in country h to be a numeraire in our economy. 3

We assume that bonds are denominated in units of internationally tradable goods produced by country

h.

18

4.3

Equilibrium

The …rst-order conditions for h households living in region j are given by f @Ut =@Cj;t h @Ut =@Cj;t a @Ut =@Zj;t h @Ut =@Cj;t b @Ut =@Zj;t h @Ut =@Cj;t n @Ut =@Zj;t h @Ut =@Cj;t

Ptf ; Pth a Rj;t = Pth b Rj;t = Pth n Rj;t = : Pth

(7a)

=

(7b) (7c) (7d)

These equations de…ne the relative prices of f traded goods, region a and b-originated traded goods, and region j nontraded goods in terms of h international tradables as ratios of their respective marginal utilities to the marginal utility of h tradables. Household’s consumption demand in region j for individual goods is given by zij =Z n = zij =Zja = zi; j =Zjb = cij =Cjh = cîj =Cjf =

1 1 [rij =Rn ] 1 ; n ij 1 1 rij =Rja 1 ; (1 n )ij 1 1 ri; j =Rjb 1 ; (1 n )i j 1 1 [pij =P h ] 1 ; 1 ij 1 1 pîj =P f 1 ; 1 ^{

i 2 rn i 2 rt i 2 rt i 2 it i 2 it.

In the bond economy, the …rst-order conditions also include h h h @Ut =@Cj;t+1 ; Pth @Ut =@Cj;t = Rt Et Pt+1

(8)

which is the standard pricing equations for the bond. The …rst-order conditions for f households are symmetric. In equilibrium, households and …rms decisions must also be consistent with the market clearing conditions. The market clearing conditions in the nontraded goods sector in region j = fa,bg are

zijt = yijt

and

zîjt = yîjt ;

19

i 2 rn:

For the regional traded goods originated within country h, the market clearing requires ziat + ziat

1 1

= yiat

and

zibt + zibt

R

1 1

= yibt ; R

i 2 rt:

while for those originated within country f, the market clearing is zîat + zîat

1 1

= yîat

and

zîbt + zîbt

R

1 1

= yîbt ; R

i 2 rt:

Here zij (^ zij ); i 2 rt is consumption demand for h (f) country, region j produced regionallytraded goods sold in the other region within h (f) country, as before.

Finally, in equilibrium, the world demand for each internationally-traded good must be equal to its corresponding supply. The internationally traded goods in each country originate in the two regions within the country. Therefore, for the internationally-tradable good produced in the h country, market clearing requires that ciat + cibt + (ciat + cibt )

1 1

i 2 it

= yiat + yibt ; I

while for the international tradables originated in the f country, market clearing requires cîat + cîbt + (^ ciat + cîbt )

1 1

= yîat + yîbt ; I

i 2 it:

Here cij (^ cij ); i 2 it is consumption demand for h (f) country, region j produced internationallytraded goods sold in the f (h) country, as de…ned before.

Labor market clearing in each region within country h requires Z

n ij

lijt di +

0

Z

ij

lijt di + n ij

Z

1

lijt di + fx (1

ij ) = Lsjt ;

ij

j = fa,bg;

where Lsj is exogenously given labor supply in region j: Similarly, for each region in country f, labor market clearing implies Z

0

{j n^

^lijt di +

Z

^{j {j n^

^lijt di +

Z

1

^lijt di + fx (1

^{j

^s ; ^{j ) = L jt

j = fa,bg;

^ s is labor supply in region j in country f. where L j An equilibrium in this economy consists of a sequence of goods prices a b â ; R ^ b ; Rn ; R ^ n } and an interest rate Rt ; such that house{Pth ; Ptf ; P^th ; P^tf ; Rj;t ; Rj;t ;R j;t j;t j;t j;t holds in both countries make their consumption and bond allocation decisions optimally,

20

taking prices as given; …rms in both countries make their pro…t maximizing decisions; and all markets clear. We also require an asset market clearing condition. We assume that bonds â;t + B ^b;t = 0. are in zero net supply, so that bond market clearing condition is Ba;t + Bb;t + B

4.4

Variables of interest

In our economy there is a sequence of price indices that comprise regional and international real exchange rates. As before, we use letter R to denote regional price indices; and letter P to denote international prices indices. To simplify the notation, we omit explicit references to ij in the consumption weights,

a;

b;

h;

f;

n;

t:

t Recall that Rj;t denotes the price of regionally traded basket in region j = fa, bg: This

price index in h country is given by t Ra;t

=

h

a

a Ra;t

1

+

b b Ra;t

i

1

h

1

t Rb;t

and

a Rb;t

= â

1

+

b ^ b Rb;t

1

i

1

: (9)

n Next, recall that Pj;t denotes the price of aggregate internationally nontraded consumption

basket in region j: It is a composite of prices of regionally traded and regionally nontraded goods: n Pa;t =

h

t

t Ra;t

1

+

n

n Ra;t

1

i

h n t Pb;t = ^ t Rb;t

1

and

1

n + ^ n Rb;t

1

i

1

(10)

Finally, recall that Ptt denotes the price of aggregate internationally traded consumption basket in regions a and b in country h. It is composed of prices of internationally traded goods in country h: Ptt =

h

h h (Pt )

h

t t (Pt )

1

+

f f (Pt )

1

i

1

(11)

:

The aggregate price index in region j in country h, therefore, is given by Pj;t =

1

+

n

n Pj;t

1

i

1

:

(12)

n Notice that only the price of internationally nontraded goods, Pj;t ; is region-speci…c. The

price of internationally traded goods, Ptt ; is common across the regions within the same country. The price indices in the foreign country are symmetrically de…ned. Thus, the prices of

21

:

regional traded baskets in regions a and b in country f are given by 1

t ^ a;t R

=

a

a ^ a;t R

1

+

b

1

1

b ^ a;t R

and

t ^ b;t R

a ^ b;t R

= â

1

+ ^b

b ^ b;t R

1

:

(13) Prices of aggregate internationally nontraded consumption baskets in regions a and b in country f are: 1

n Pâ;t

=

t

t ^ a;t R

1

+

n

1

1

n ^ a;t R

and

n P^b;t

t ^ b;t R

= ^t

1

+ ^n

n ^ b;t R

1

(14) Price of aggregate internationally traded consumption basket in regions a and b in country f is given by

"

1

P^tt = ^ h P^th

+ ^ f P^tf

1

#

1

;

(15)

:

(16)

while the aggregate price index in region j in country f is 1 1

P^j;t = ^ t P^tt

n + ^ n P^j;t

1

Now we are ready to de…ne RERs. The regional real exchange rate is de…ned as the price of consumption basket in region b relative to the price of consumption basket in region a. Thus, the regional real exchange rates in country h, RERth ; and country f, RERtf ; are given by RERth =

Pb;t Pa;t

and

RERtf =

P^b;t : Pâ;t

(17)

International real exchange rate in our model, RERti ; is given by the ratio of f to h aggregate price indices: RERti

=

1 ^ P 2 a;t 1 P 2 a;t

+ 12 P^b;t ; + 12 Pb;t

(18)

where we exploited the symmetry of the regions within a given country to construct the country-wide aggregate price index as the average of price indices across regions within that country. The terms-of-trade in the model are de…ned as a relative price of foreign to domestic internationally-traded goods and are given by T OTt = P^tf =Pth :

22

:

5

Parameter values and computations

Parameter values for the calibration of our benchmark model are summarized in Table 3. We consider the world economy as consisting of two symmetric countries, matching the properties of US economy in annual data. Most of the preference parameter values are standard in the literature and, in particular, follow closely those adopted by Stockman and Tesar (1995). In particular,

is set to 0.96 to obtain the steady-state real interest rate of 4%

per annum. Coe¢ cient of relative risk aversion, !; is set to 2. The values for substitution elasticities are chosen as follows. First, the value for

is set, following Mendoza (1995), to

obtain the elasticity of substitution between tradable and nontradable consumption equal to 0.74. Second, the elasticity of substitution between h and f traded goods is set to equal 6 to obtain a 20% mark-up of price over marginal costs, a value commonly used in the literature (Obstfeld and Rogo¤, 2000). Table 3. Benchmark Model Parameters preferences Subjective discount factor Risk-aversion share of nontraded goods elasticity of substitution b/n traded and nontraded goods h and f traded goods Productivity persistence of traded shocks persistence of nontraded shocks volatility of traded innovations (std.dev.) nontraded innovations (std.dev.)

0.96 ! 2 International goods:

1=(1 1= (1

) )

0.74 6

International goods: 0.9 ahii = afii h e

=

f e

0.01 0.005

Regional goods: 0.55 n = ^n 1=(1 1= (1

) 0.74 ) 6

Regional goods: 0.9 aaii = abii 0.9 anii a e n e

=

b e

0.01 0.01

We parameterize the …xed cost of exporting parameter, fx ; in both countries to obtain the share of nontradables in aggregate consumption expenditure, n and ^ n ; equal to 0.55 in the steady state. This number is calculated using OECD STructural ANalysis (STAN) database.4 We set the shares of home goods in the internationally-traded consumption basket in both countries, h and ^ f ; to 0.5 in the steady state, so that there is no consumption 4

These numbers are similar to the estimates in the literature. For instance, Corsetti et al. (2008) and Dotsey and Duarte (2006) use n = 0:55, Stockman and Tesar (1995) report n close to 0.5; Pesenti and van Wincoop (2002) also argue that 0.5 of consumers budget is allocated to nontradables; Benigno and Thoenissen (2008) assume n = 0:45.

23

home bias built in exogenously in the model. Instead we calibrate the international iceberg transportation costs to match the share of international imports to be equal to 10% of output in the steady state. The regional dimension of the model is more di¢ cult to parametrize as the estimates are not readily available in the literature. Under our benchmark calibration, we assume that preference parameters over regional consumption are symmetric to the preference parameters in the international dimension. That is, we assume that the value of

is chosen to set the

elasticity of substitution between regional tradable and nontradable goods to 0.74, while the share of nontraded regional goods, costs,

R;

n

= ^ n ; is set to 0.55. Regional iceberg transportation

are calibrated to obtain regional trade volume to be 1.25 times the international

trade volume in the steady state. The latter number was obtained as an average using interprovincial and international trade data for Canada during 1984-2005. State-level trade data for US is not available, to the best of our knowledge, so we use the trade numbers for Canada to calibrate

R

parameter in our model. The elasticity of substitution between two

regional goods is set to 6 in both countries, which pins down the value for . The available estimates for sectoral productivity processes in the literature are very dissimilar (see, for instance, Corsetti, Dedola, and Leduc, 2008; Benigno and Thoenissen, 2008; Tesar, 1993; Stockman and Tesar, 1995), thus we assume independent productivity processes across sectors, across regions and across countries. Each of the productivity processes follows an AR(1) process. The AR(1) coe¢ cients are all set to 0.9. Innovations to internationally traded productivity have standard deviation of 0.01, while innovations to regionally traded and nontraded productivity are half that size. These numbers are consistent with the empirical …ndings that traded productivity exhibits more volatility than nontraded productivity. We also assume that international shocks are common across the regions within the same country, while all other shocks are region-speci…c. We parameterize good/…rm-speci…c productivity following Bergin and Glick (2005) as Ai = (1 + i); with

= 1:

The model is solved by linearizing the system of equilibrium conditions and solving the resulting system of linear di¤erence equations. To make our bond economy stationary, we introduce small quadratic costs on bond holdings. We study the properties of the model’s equilibrium by simulating it over 100 periods. The statistics reported in the next section are derived from 200 simulations.

24

6

Results

This section presents the main …ndings from the numerical solution to the model. We start by characterizing the properties of RER in a production economy with no …rm heterogeneity due to …rm-speci…c productivity. Such simpli…cation eliminates the endogenous non-tradability feature in our model and reduces the setting to a standard international business cycle economy with a representative …rm in each region. Then we allow for …rm heterogeneity and consider a production economy with endogenous tradability. Under each scenario we also evaluate the role of within country heterogeneity by studying a version of the model with and without regions. We use an economy without endogenous tradability and with no regional heterogeneity as a benchmark.

6.1

No …rm heterogeneity

The properties of real exchange rate and consumption generated by the production bond economy with homogeneous and perfectly competitive …rms are summarized in Table 4. Column (i) reports the volatility of the two variables and the correlation between them across regions, while column (ii) presents the corresponding statistics across countries. We study two versions of this model: without regional heterogeneity, the results for which are reported in the top panel; and with regional heterogeneity, the results for which are summarized in the bottom panel of Table 4. The latter exercise allows us to isolate the contribution of each new feature that we introduce. Without the regional dimension, our model has been used extensively in the literature to study international business cycles, terms of trade and real exchange rate movements (see Tesar, 1993; Corsetti, Dedola, and Leduc, 2008; Benigno and Thoenissen, 2008). The international business cycle properties of our model under the benchmark parametrization, therefore, are standard. As expected, it predicts international RER that is positively correlated with consumption across countries, even though asset markets are incomplete. This correlation is positive and high when we consider both levels and growth rates of the two variables (0.3 in levels and in growth rates). When regional heterogeneity is re-introduced, the international correlation between RER and relative consumption declines slightly, but remains positive. Across regions, the correlation between RER and consumption is also positive, consistent with the data, but is signi…cantly above its empirical counterpart. These results re-con…rm the consumption-RER puzzle in the international data, and show that within country heterogeneity is not su¢ cient to resolve the Backus-Smith puzzle. The model also fails to account for the volatility of RER relative to that of consumption. 25

The implied relative volatility of international RER is well above one, but below its value in the data. At the same time, the model predicts too much volatility for the regional RER. Table 4: RER and relative consumption: representative …rm Across regions Across countries No heterogeneity (i) (ii) q regional 2 e 1.57 c c j

i

Correlation b=n RER and C C^ growth rate of RER and C

0.30 0.30

C^

Regional heterogeneity q 2 e

ci

cj


C^

0.98

1.57

0.99 0.99

0.29 0.29

Notes: Statistics reported in the table are based on 200 simulations of annual series, each 100 periods long.

Overall, our benchmark model with heterogenous regions and a single non-state-contingent international bond is not able to account for the di¤erences in consumption-RER relationship across regions and across countries. We next explore the role of endogenous tradability in capturing the link between the two variables.

6.2

Endogenous tradability

Next, we present the results obtained in the production economy with heterogeneous …rms. As before, we consider two versions of the model. The top panel of Table 5 reports the statistics obtained in the version of our model without regional heterogeneity, while the bottom panel summarizes the corresponding statistics in the version of the model with regional heterogeneity. The contribution of endogenous non-tradability to the properties of RER and relative consumption can be isolated by comparing the statistics in the top panels of Tables 4 and 5. Introducing endogenous tradability into a generic international business cycle model (with no regions) increases the relative volatility of RER, and reduces consumption-RER correlation both in levels and growth rates. This correlation, however, remains positive (0.03 in levels, 0.04 in growth rates). Therefore, endogenous non-tradability per se is not su¢ cient to overturn the Backus-Smith puzzle.

26

Table 5: RER and relative consumption: endogenous tradability Across regions (i)

No heterogeneity q regional 2

Across countries (ii) 1.67

e

ci

cj


0.03 0.04

C^

Regional heterogeneity q 2 e

ci

cj


C^

0.51

1.25

0.62 0.62

-0.10 -0.10

Notes: Statistics reported in the table are based on 200 simulations of annual series, each 100 periods long.

The results from our full model that incorporates both regional heterogeneity and endogenous non-tradability are presented in the bottom panel of Table 5. This model provides the best match to the data. It is able to generate a positive correlation between RER and relative consumption across regions and a corresponding negative correlation across countries. It also implies relative volatility of RER to consumption that is well above one across countries and well below one across regions, consistent with the data. We next develop the intuition for these results.

6.3

Discussion

To understand the results above it is useful to decompose the international and regional real exchange rates into their components. In particular, in Appendix A.3 we show that the log international real exchange rate (in deviation from the steady state), can be expressed as i rera;t =

h

f

1

1 1

I

pft 1 (^

pht ) + ^ n

2

p^na;t

p^ta;t

n 2

pna;t

pta;t ;

(19)

where lowercase letters denote the log transformations for all variables in deviations from 1

their steady state values (e.g., pht Pân =Pâ

ln Pth

ln P h ; etc.); and

1

1

=

P^ f =P t ;

2

=

= (Pan =Pa ) are coe¢ cients that depend on the steady state values of relative

prices. For analytical tractability, in this decomposition we focused on the aggregate price indices between regions a in countries h and f. Clearly, the international real exchange rate will depend on the weighted average of the aggregate price indices in regions a and b in the 27

two countries. The expression in (19) decomposes international real exchange rate into two components: (i) a component associated with the international terms of trade movements, h

f

1

1 1

I

pft 1 (^

pht ); (ii) a component arising due to variations in the relative

prices of internationally-nontraded goods (the last two terms in the expression above).5 If there is consumption home bias in households’ preferences,

h

f

1

1 1

I

> 0,

then the improvements in the terms of trade will be associated with the real exchange rate appreciation. Furthermore, any variations in the relative price of nontradable goods in the two countries will also contribute to real exchange rate movements to the extent of the weight of nontradable consumption in the aggregate consumption basket of the two countries, and ^ n 2 . 6.3.1

n 2

Role of regional heterogeneity

This decomposition allows us to develop the intuition for the mechanism underlying the model without regions and no endogenous non-tradability. In that model the productivity shocks can originate in two sectors: internationally tradable and nontradable. In response to a positive shock in the internationally tradable sector of country h, its terms of trade deteriorates, which will tend to depreciate the RER. This e¤ect, however, may be counterbalanced by the movements in the relative prices of nontraded goods. The latter will arise from two sources: …rst is due to a standard Balassa-Samuelson e¤ect, where positive productivity shock in the tradable sector will trigger an increase in the real wage, thus driving up relative prices of nontraded goods; second is due to a wealth e¤ect in country h, resulting from productivity improvement in that country. In our model these two e¤ects dominate and, as a result, positive productivity shocks in the internationally-tradable sector are associated with RER appreciation and an increase in relative consumption. Positive shocks that originate in the internationally-nontradable sector have the opposite e¤ect on the RER. Such shocks lead to a fall in the relative prices of nontraded goods, thus depreciating the RER. The RER depreciation is accompanied by an increase in relative consumption, thus working against the resolution of the Backus-Smith puzzle. As the top panel of Table 4 illustrates, a model with no regions and no endogenous non-tradability implies that productivity shocks to internationally nontraded sector dominate the consumption-RER correlation, predicting large positive numbers for that correlation. When heterogenous regions are introduced, the positive association between relative consumption and RER becomes 5

A similar decomposition is also derived in Benigno and Thoenissen (2008) in a two-country two-sector model with no iceberg trade costs.

28

weaker. Much of the decline is due to attenuated e¤ects of the shocks in the internationally nontraded sector. To understand this result, we must deduce how the relative prices of nontradable goods are determined. Due to the presence of regional heterogeneity in our model, we can write the price of nontraded goods in region a in country h as a pna;t = ra;t +

1

1 b

1

b ra;t

3

R

a ra;t +

n 4

n ra;t

t ra;t ;

1

where

3

= (Rbb =Rat ) =

^ at ^ bb =R R

= (Rbb =Rbt ) =

^ bt ^ bb =R R

(20)

1

and

4

= (Ran =Pan ) =

^ n =P^ n = (Rn =P n ) = R ^ n =P^ n : The relative price of nontraded goods contains two comR a a b b b b ponents: (i) a component associated with the movements in the relative prices of regionallyb produced goods, rta ; and regional terms of trade movements, ra;t

a ra;t ; (ii) a component

arising due to variations in the relative prices of regionally-nontraded goods (the last term). When regional heterogeneity is introduced, the internationally nontraded shocks can originate in the regionally-traded sector and in the regionally-nontraded sector. These two shocks will have an e¤ect on pna;t analogous to those on pa;t in the model with no regions. For instance, a positive regionally nontraded shock in region a will tend to depreciate pna;t n , but this e¤ect will be partially o¤set by an increase in the price of through lower ra;t a regionally traded goods from that region, ra;t . Similarly, a positive regionally traded shock a in region a will depreciate pna;t through a drop in ra;t ; but again, this e¤ect will be partially n counterbalanced by higher prices of regionally-nontraded goods, ra;t : As a result, in response

to these shocks, pna;t will adjust by less, thus leading to smaller movements in the international RER. Splitting the internationally nontraded sector into its regionally-traded and regionallynontraded components allows agents to share these shocks across regions within the same country, thus moderating their contribution to the relative consumption-RER correlation. As a result, the Backus-Smith correlation falls when regions are present. Next, we derive the regional RER. In region b in country h, the price of nontraded goods is given by the same two components as in (20): a pnb;t = rb;t + ^b

3

b rb;t

a rb;t + ^n

4

n rb;t

t rb;t :

Using these decompositions, the regional real exchange rate in country h can be written as rerth =

n 2

pnb;t

6

pna;t :6

Note that the international terms of trade component of the RER is common to the two regions within the same country and thus cancels out.

29

From the results in (2)-(3), the prices of regional goods in country h in deviations from the symmetric, we get ^ b =

a;

rtb : Further, since regions are

b b = rb;t rta and ra;t

a a = rb;t steady state are related as ra;t

which allows to write the regional real exchange rate in country

h as: rerth =

n 2

"

1

1 a

b

1

R

!

3

(rtb

rta ) + ^ n

4

n rb;t

t rb;t

n 4

n ra;t

t ra;t

#

:

(21) These derivations show that the regional real exchange rate, like its international counterpart, ! is a composite of two components: (i) regional terms of trade,

1

1

a

b

1

3

R

(rtb

which depends on the degree of regional consumption home bias; and (ii) relative price of regionally-nontraded goods in regions a and b, given by ^ n

4

n rb;t

t rb;t

n 4

n ra;t

t : ra;t

In terms of their e¤ect on the regional RER, regionally traded and nontraded productivity shocks act very much like the internationally traded and nontraded shocks on the international RER. For instance, positive regionally nontraded shocks will tend to depreciate regional RER and increase regional consumption, implying a positive correlation between the two. Positive regionally traded shocks will worsen regional terms of trade, an e¤ect that will be partially o¤set by increasing prices of regionally nontraded goods. However, contrary to internationally-traded shocks, the increase in regionally nontraded goods prices is not strong enough to appreciate the regional RER. The reason is the smaller degree of home bias in regional consumption, which results from smaller trade costs across regions than across countries. With smaller regional home bias, the terms of trade moves less in response to shocks. The resulting spillovers into the regionally nontraded sectors (through regional Balassa-Samuelson e¤ects or wealth e¤ects) become more symmetric across regions, leading to similar increases in prices of regionally nontraded goods. As a result, two last terms in (21) will tend to cancel out, leading to regional RER depreciation (through terms of trade e¤ect) and an increase in regional consumption. Therefore, regional RER and relative regional consumption are always positively correlated in our model, consistent with the data. Next, we illustrate these e¤ects on regional real exchange rate, rerth ; and international i real exchange rate, rera;t in more details using impulse response analysis. We start by

analyzing the bond economy with a representative …rm. The responses of international and regional real exchange rates and relative consumption, as well as their decomposition into the components derived in (19) and (21) are presented in Figures 2a

2c. Figure 2a presents

the e¤ects of a 1% positive productivity shock in the h internationally-traded sector.7 7

As before, in the model calibration we assume that shocks to internationally-traded goods productivity

30

rta ) ;

Figure 2a. Int’l and Regional RER and C

C^ with representative …rm: Positive 1% Int’l shock

The left-hand-side panel of Figure 2a presents the impulse responses of RER and relative consumption across countries and across regions within each country, h and f. The correlation between the two variables is clearly negative across countries. Regional RER and relative consumption are una¤ected by international shocks, because both regions within the country are subject to that shock. The reason for the negative international correlation is twofold: First, there is a strong wealth e¤ect that is triggered by the supply shocks. Second, there is also a Balassa-Samuelson e¤ect that results from higher wages in the sector experiencing the shock. The right-hand-side panel presents the decomposition of the international and regional RERs. In response to a positive shock to productivity of internationallytraded goods, terms of trade depreciate. Because households’ preferences are such that consumption of internationally-traded goods is close to complementary with the consumption of internationally-nontraded goods, the demand for regional goods, both traded and nontraded, goes up. As a result, their relative prices increase, which drives P n (pictured in the top right chart of Figure 2a) up. Productivity changes in country h also spill over to country f through their e¤ects on the international terms of trade. As the relative price of experienced by the two regions within each country are perfectly positively correlated.

31

foreign internationally-traded goods goes up, households residing in the f country increase their consumption of regional goods. This drives up the relative price of these goods (P^ n increases). This increase, however, is smaller than that in h country. While the depreciation of terms-of-trade and an increase in P^ n tend to depreciate the international real exchange rate, and increase in P n tends to appreciate it. If the supply of domestic internationally-nontraded goods is relatively inelastic, the latter e¤ect will dominate, leading to international RER appreciation. At the same time, h country relative to f country consumption will increase, implying a negative correlation between the two variables. Next, we look at the e¤ects of regional shocks. Figure 2b presents the responses of RERs and relative consumption, as well as the components of RERs to a 1% positive shock to productivity of regionally-traded goods in region a of country h.

Figure 2b. Int’l and Regional RER and C

C^ with representative …rm: Positive 1% Reg. T shock

Here both international RER and relative consumption increase in response to the shock, implying a positive correlation between the two variables. The reason for such dynamics can be seen from the right-hand-side panel. In response to the shock, the international terms-oftrade and the relative price of foreign internationally-nontraded goods do not change much. 32

The dynamics of the international RER are therefore dominated by the drop in the relative price of domestic internationally-nontraded goods, P n ; which leads to RER depreciation. The regional RER follows similar dynamics following the shock. When the supply of regionally-traded goods in region a of country h increases, the price of these goods, Ra ; drops, while the relative price of regionally-traded goods originated in region b, Rb ; goes up. This depreciates regional terms of trade in region a. Households also demand more of regionally-nontraded goods, driving their price, Rn up. Overall, the fall in Ra dominates the other e¤ects, and causes P n to fall. In region b of country h, the e¤ects of the shock are similar, but less pronounced, also leading to a fall in the relative price P n in that region. In the presence of risk-free bonds, regions a and b are able to share regional traded shocks quite well. The regional RER, which is the sum of the regional terms of trade and the log-di¤erence of the internationally-nontraded prices in regions b and a, therefore, increases (depreciates). Finally, Figure 2c presents the impulse responses of international and regional RERs and relative consumption following a 1% positive shock to productivity of the regionallynontraded good in region a, country h.

33

Figure 2c. Int’l and Regional RER and C

C^ with representative …rm: Positive 1% Reg. N shock

Both international and regional RERs are positively correlated with relative consumptions. An increase in the supply of regionally-nontraded goods, depresses their prices in h country, but does so more in region a than in region b. The regional RER, therefore, depreciates. The drop in the relative price of internationally-nontraded goods also leads to a depreciation of the international RER. At the same time, relative consumption rises both across countries and across regions.8 To sum up, international traded shocks imply a negative correlation between international RER and relative consumption and have no e¤ect on the corresponding correlation across re8

Following internationally-nontraded shocks (originating in either regionally-traded or regionally nontraded-sector), international TOT depreciates in this version of the model. This happens due to several e¤ects. First, following the shock, households demand more of internationally traded goods, both home and foreign. Under the chosen values for inter- and intra-temporal elasticities, internationally traded and nontraded goods are weak substitutes. Coupled with the assumed degree of international consumption home bias, this calibration implies that international TOT depreciate in the model following productivity shocks in internationally-nontraded sector. The second e¤ect is driven by an increase in the production of internationally-traded goods in that country. This increase in production is due to a sectoral reallocation of labor from n to t sector. Finally, in the response to the shock, world interest rate declines, generating an additional spillover channel for foreign households. With lower interest rates, foreign households demand more consumption. Due to consumption home bias, this e¤ect also leads to TOT depreciation for H households.

34

gions. The shocks to productivity of internationally-nontraded goods (both regionally-traded and regionally-nontraded) generate a positive Backus-Smith correlation in both international dimension and regional dimension. As a result, regional consumption-RER correlation is always positive in our model. The net international consumption-RER correlation depends on which shocks dominate. Under our benchmark calibration the e¤ects of regional shocks dominate, resulting in a positive Backus-Smith correlation. This is because regions have access to an international bond, so they are able to share shocks to productivity of internationallytraded goods with the other country reasonably well. The e¤ects of those shocks, therefore, are moderated. The shocks to regional productivity are harder to insure using international asset markets, implying that those shocks have profound e¤ects on real exchange rates and relative consumption. However, when regions are heterogenous, they can also use international bonds to share some of the regionally-traded shocks with their countrymen in another region. This limits the e¤ects of internationally-nontraded shocks. The e¤ect, however, is not strong enough to overturn the Backus-Smith puzzle. Next we study the role played by endogenous non-tradability in the model. 6.3.2

Role of endogenous non-tradability

Next, consider the impulse responses arising in the model with endogenous tradability, but no regional heterogeneity. Shutting down regional heterogeneity allows us to isolate the contribution of …rm heterogeneity and their export entry-exit decisions on the risk-sharing. Figures 3a

3b summarize the responses.

35

Figure 3a: International RER and C

C^ with endog. tradability: Positive 1% Int’l T shock

The top panel of Figure 3a shows the responses of relative consumption, international RER and its components to a 1% positive productivity innovation in the internationally traded sector. These responses are qualitatively similar to those in the model with no endogenous tradability presented in Figure 2a; however, they are quantitatively larger. The ampli…cations is due to an endogenous change in the size of the internationally traded sector as new varieties become exported after a positive TFP shock. The bottom panel of Figure 3a con…rms this intuition. It show the impulse responses of nontraded, n and ^ n ; consumption shares as well as domestic and foreign traded consumption shares in country h. Recall that n

is also equal to the threshold index i of the marginal …rm that will export, thus controlling

the size of nontraded sector in country h. Following a positive shock, more …rms enter the export market in country h, while the reverse happens in country f. The expansion in the size of the h internationally traded sector and a corresponding decline in the size of the nontraded sector leads to a larger increase in the relative price of domestic internationallynontraded goods, P n ; implying a greater appreciation of the RER, as a result. Figure 3b present impulse responses following a 1% positive productivity shock in the internationally nontraded sector. 36

Figure 3b: International RER and C

C^ with endog. tradability: Positive 1% Int’l N shock

As in Figure 2c, relative consumption in country h goes up, and RER depreciates following the shock. Much of the RER depreciation is driven by the fall in relative prices of nontraded goods in country h. In the economy with endogenous tradability, a positive shock to n sector productivity leads to a contraction in the size of internationally non-traded sector. As initially nontraded …rms become more productive they enter the export market. This effect partially o¤sets the drop in prices of n goods in country h. As a result, RER depreciates by less, weakening the positive correlation between RER and relative consumption. Overall, endogenous tradability enforces the e¤ects of t shocks and counteracts the e¤ects of n shocks on the RER, thus lowering the international Backus-Smith correlation in the model. Why does the international tradability of goods changes the way it does? To answer this question, we rewrite the marginal trading condition facing h country …rms residing in region j as

h

1

1

1 1 Xjt Aij 1

I

i

"

^h C jt 1 ij

37

~it;j Xjt A Xjt Aij

1 1

#

= fx :

(22)

This expression implies that the pro…ts of the marginal …rm that will export, among other factors, depend on the productivity of that …rm relative to the aggregate productivity in the internationally-traded sector, as well as aggregate foreign demand for internationallytraded goods that h …rms in region j face. Note that it is an increase in the trade at the h ) that will trigger trade at the extensive margin. With no intensive margin (thought C^jt changes in …rm-speci…c productivity, TFP shocks to nontraded goods sector productivity h term; TFP shocks to traded sector will will only a¤ect individual …rm’s pro…t through C^jt h and a¤ect …rm’s pro…ts through both C^jt

1 Xjt Aij

terms.9

As argued above, positive nontraded shocks a¤ect foreign demand for h goods through the terms of trade movements. In particular, following a positive n shock, h terms of trade h ; therefore, rises, worsens thus improving the purchasing power of foreign households. C^jt and pro…ts of h exporters increase. To restore the equality in the marginal trading condition above, ij must fall, that is the share of exporters must go up. This implies lower

n

ij

following positive shocks in the nontraded sector: Similar logic applies when a region experiences a positive shock to TFP in the internationallytraded sector. In this case, Xjt will go up, which in turn will lower …rm’s pro…ts. This decline, however, will be more than o¤set by higher foreign demand for h-produced internationallytraded goods. The higher demand is again driven by the risk-sharing motive that operates through the terms of trade. With high international consumption home bias the adjustment in the terms of trade is large, leading to signi…cant spillover e¤ects of t productivity across countries. Therefore, following such shocks, …rm’s pro…ts increase on net, leading to a fall in ij and a corresponding decline in 6.3.3

n

ij :

Full model

Finally, we explore the properties of the full model that allows for both within-country heterogeneity and endogenous tradability. We …nd that the dynamics of this model in response to internationally traded shocks and regionally-nontraded shocks are very similar to those in the two versions of the model presented above. The responses of consumption and RER, however, are quite distinct from before when we consider shocks to regionally-traded productivity. We present these responses in Figures 4 9

5 below.

As in Bergin and Glick (2005), we can interpret the …rst term on the left-hand-side of equation (22) as a per unit export pro…t, and the second term as export value. Then internationally-traded shocks a¤ect aggregate …rm’s pro…ts through both per unit export pro…t term and export value term, while internationallynontraded shocks only through export value term.

38

Figure 4. Int’l and Regional RER and C

C^ with endog. tradability: Positive 1% Reg. T shock

The main di¤erence between Figure 2b and Figure 4 is in the dynamics of international RER and relative consumption. In particular, with endogenous tradability, the correlation between the two variables becomes negative following the shock; that is RER appreciates, while relative consumption increases. This is in contrast to a positive correlation between the two variables in the model with a representative …rm. A decomposition of the international RER in the top right …gure helps us understand what drives the results. The key di¤erence arises from the adjustment in the terms of trade. With endogenous tradability, international terms of trade improves following the shock. The improvement is large enough to o¤set the RER depreciation pressure coming from a fall in the relative price of nontraded goods in country h and an increase in the corresponding price in country f. The intuition for this result can be obtained from Figure 5, which plots the impulse responses of consumption shares in the two regions (a and b) in the two countries (h and f).

39

Table 5. Consumption shares: Positive 1% Reg. T shock

As the top left …gure indicates, the share of internationally non-traded sector in country h increases in both regions. Since the share of regionally non-traded sector is …xed to our calibration, all of the expansion in

n

n

in

is due to …rms’exist from the international export

sector into the regional export sector (which has experienced an improvement in TFP). The internationally-traded sector thus contracts in country h. The reverse takes place in country f. With the decline in the supply of internationally-traded goods from h country, both regions in country f expand their supply of internationally traded goods ( ^ n declines in both regions). These adjustments lead to a drop in relative price of f traded goods, which is the appreciation in the international terms of trade. This latter e¤ect reduces the purchasing power of foreign households, causing them to reduce their demand for internationally-traded goods produced in country h. This, in turn, lowers pro…ts of h …rms operating in the export sector and triggers an increase in

n:

Overall, the model predicts that productivity improvements in the sectors that produce tradable goods, either regionally-traded or internationally-traded, are accompanied by an entry of new …rms/varieties into those sectors as …rms try to take advantage of favorable conditions. This productivity improvements can then be shared with households in other regions and/or countries. Productivity shocks to regionally-nontraded sector can not be 40

shared, as only households living in that region consume the goods. Therefore, such shocks are associated with an exit of …rms from the nontraded sector.10 In summary, a combination of regionally traded shocks and endogenous tradability strengthen the negative correlation between international RER and relative consumption, thus helping us to rationalize the Backus-Smith puzzle in the international data. At the same time, the regional consumption-RER correlation remains positive, consistent with the empirical evidence.

7

Conclusion

This paper studies two well-known puzzles in the International Finance: Backus-Smith puzzle and RER volatility puzzle. First, we document the two puzzles in the international data. We complement this empirical analysis with a set of stylized facts on RER and relative consumption using regional/state level data for a large sample of countries. Second, we rationalize the two puzzles in a two-country real business cycles model, and do so in both international and intra-national dimensions. Our explanation relies on two key elements: within country heterogeneity and endogenous non-tradability. Each feature individually allows us to reduce the Backus-Smith correlation, but is not su¢ cient by itself. Only a combination of regional heterogeneity and endogenous tradability resolves the Backus-Smith puzzle in the international and regional dimensions, and qualitatively can account for the di¤erences in volatility of RER relative to consumption across countries and regions.

10

While …rm entry and exit emphasizes the extensive margin of adjustment following productivity shocks, similar dynamics also take place at the intensive margin. In particular, labor moves into tradable sectors following productivity improvements there and moves out of regionally nontraded sector after productivity shocks hit that sector. The two e¤ects reinforce each other.

41

References Alessandria, G., and H. Choi (2007): “Do Sunk Costs of Exporting Matter for Net Export Dynamics?,”The Quarterly Journal of Economics, 122(1), 289–336. Backus, D. K., and G. W. Smith (1993): “Consumption and Real Exchange Rates in Dynamic Economies with Non-traded Goods,”Journal of International Economics, 35(34), 297–316. Benigno, G., and C. Thoenissen (2008): “Consumption and Real Exchange Rates with Incomplete Markets and Non-traded Goods,”Journal of International Money and Finance, 27(6), 926–948. Bergin, P. R., and R. Glick (2005): “Tradability, Productivity, and Understanding International Economic Integration,” NBER Working Papers 11637, National Bureau of Economic Research, Inc. Bergin, P. R., R. Glick, and A. M. Taylor (2006): “Productivity, tradability, and the long-run price puzzle,”Journal of Monetary Economics, 53(8), 2041–2066. Bodenstein, M. (2006): “International Asset Markets and Real Exchange Rate Volatility,” International Finance Discussion Papers 884, Board of Governors of the Federal Reserve System (U.S.). Chari, V. V., P. J. Kehoe, and E. R. McGrattan (2002): “Can Sticky Price Models Generate Volatile and Persistent Real Exchange Rates?,” Review of Economic Studies, 69(3), 533–63. Corsetti, G., L. Dedola, and S. Leduc (2008): “International Risk Sharing and the Transmission of Productivity Shocks,”Review of Economic Studies, 75(2), 443–473. Corsetti, G., P. Martin, and P. Pesenti (2008): “Varieties and the Transfer Problem: The Extensive Margin of Current Account Adjustment,” NBER Working Papers 13795, National Bureau of Economic Research, Inc. Crucini, M. J., and G. D. Hess (1999): “International and Intranational Risk Sharing,” CESifo Working Paper Series 227, CESifo GmbH. Engel, C., and J. Wang (2008): “International Trade in Durable Goods: Understanding Volatility, Cyclicality, and Elasticities,” NBER Working Papers 13814, National Bureau of Economic Research, Inc. 42

Ghironi, F., and M. J. Melitz (2005): “International Trade and Macroeconomic Dynamics with Heterogeneous Firms,”The Quarterly Journal of Economics, 120(3), 865–915. Heston, A., R. Summers, and B. Aten (2006): “Penn World Table Version 6.2,”Database, Center for International Comparisons of Production, Income and Prices at the University of Pennsylvania. Johri, A., and A. Lahiri (2008): “Persistent Real Exchange Rates,”Department of Economics Working Papers 2008-04, McMaster University. Kollmann, R. (1995): “Consumption, Real Exchange Rates and the Structure of International Asset Markets,”Journal of International Money and Finance, 14(2), 191–211. Labhard, V., and M. Sawicki (2006): “International and Intranational Consumption Risk Sharing: The Evidence for the United Kingdom and OECD,” Bank of England working papers 302, Bank of England. Lewis, K. K. (1996): “What Can Explain the Apparent Lack of International Consumption Risk Sharing?,”Journal of Political Economy, 104(2), 267–97. Melitz, M. J. (2003): “The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity,”Econometrica, 71(6), 1695–1725. Mendoza, E. G. (1995): “The Terms of Trade, the Real Exchange Rate, and Economic Fluctuations,”International Economic Review, 36(1), 101–37. Naknoi, K. (2008): “Real exchange rate ‡uctuations, endogenous tradability and exchange rate regimes,”Journal of Monetary Economics, 55(3), 645–663. Obstfeld, M., and K. Rogoff (2000): “The Six Major Puzzles in International Macroeconomics: Is There a Common Cause?,”in NBER Macroeconomics Annual 2000, Volume 15, NBER Chapters, pp. 339–412. National Bureau of Economic Research, Inc. Ravn, M. O. (2001): “Consumption Dynamics and Real Exchange Rate,”CEPR Discussion Papers 2940, C.E.P.R. Discussion Papers. Sorensen, B. E., and O. Yosha (1998): “International Risk Sharing and European Monetary Uni…cation,”Journal of International Economics, 45(2), 211–238. Stockman, A. C., and L. L. Tesar (1995): “Tastes and Technology in a Two-Country Model of the Business Cycle: Explaining International Comovements,” American Economic Review, 85(1), 168–185. 43

Tesar, L. L. (1993): “International Risk-Sharing and Non-Traded goods,” Journal of International Economics, 35(1-2), 69–89.

44

A A.1 A.1.1

Appendix Data description and sources International data

Our sample includes 29 OECD countries (no data is available for Korea) plus China and India over 1970-2004 period. The aggregate series for consumption are taken from Penn World Table 6.2 (Heston, Summers, and Aten, 2006). In particular, we used "Real gross domestic product per capita: CGDP" series in combination with "Consumption Share of CGDP: CC" series to obtain our measure of real per capita consumption for each country. The components of the RER are obtained from the following sources: (i) nominal exchange rate series are period averages calculated as local currency units per US $, and are from IMF’s International Financial Statistics (IFS) database; (ii) prices for all OECD countries in our sample and India are obtained from SourceOECD database and are consumer price index (CPI) series with base year of 2005. CPI for China is from China Statistical Yearbooks published by the Department of National Accounts, National Bureau of Statistics of China. A.1.2

Regional data

To be completed.

A.2

Derivation of consumption aggregates

In our setup households have access to three types of consumption goods: regionallynontraded goods, goods that can be traded between regions but not internationally, and internationally-traded goods. Let’s consider …rst the …nal consumption aggregate. It consists of internationally nontraded goods and a basket of internationally traded goods. The latter can be produced by local …rms or imported from the foreign country. Recall that all local …rms are located on [0; 1] interval. Firms that produce internationally nontraded goods occupy [1; i ] interval, where i is the threshold export index in a given region. Domestic internationally-traded goods are produced by …rms located on (i ; 1] interval. We append this continuum of local …rms by a [1; 2

^{ ] measure of foreign …rms that can export their

goods to the local market. As a result, the consumption basket of households residing in

A1

region a in country h becomes

C

1

= (i )

Z

i

0

1

= (i )

Z

1

1 i

1

i

0

1 2 ^{ i

1 i

1

(1

ci di

+ (1

ci di

Z

i)

(C n ) + (1

i )1

= (i )1

(C n ) + (1

i )1

h

1

ci di +

i

Z

2 ^{

cî di

1

i )1

1

i

Z

1

1 2 ^{ i

i)

+ (1 1

= (i )1

1

1

1 1 i

Z

1

ci di + (1

^{ )

2 ^{ 1 1 ^{

1

1

1 i 2 ^{ i

(C h ) +

1 ^{ 2 ^{ i

1

(C f )

(C t ) ;

1

cî di

i

where n

(C )

=

Z

i

0

h

(C )

=

Z

1 1 1 i

i

f

(C )

=

Z

1

1 i

ci di; 1

2 ^{ 1 1 ^{

1

ci di; 1

cî di:

From above, consumption shares are endogenously linked to the threshold export index of …rms: n (i h

)=i

(i ) =

1 i 2 ^{ i

and

t (i

and

f

)=1

i;

(i ) =

1 ^{ 2 ^{ i

:

Next, consider the internationally nontraded consumption aggregate.

It consists of

regionally-nontraded goods (i.e. haircuts, real estate services, etc); regionally traded goods that can be produced by the local …rms or by the …rms residing in another region within the same country (i.e. utility and government services, agricultural produce, many retail services, like those provided by Future Shop in Canada - a retail chain that does not operate in the US, etc.). Firms residing in region a that do not export to region b and thus produce regionally nontraded goods are located on [1;

n]

interval. Firms operating in region a that

produce goods that are regionally tradable occupy [ of …rms with the [ia + ib (1

n )]

n ; ia ]

interval. Appending this measure

interval of …rms residing in region b who supply goods to

the local market. Therefore, internationally nontraded consumption aggregate in region a

A2

can be written as 1

(ia )

n

(C )

= (

n ia )

" = (

n ia

n )+ib (1

Z

n ia )

n)

n ia

1 n ia )

Z

1

1 n

zi di +

Z

ia +ib (1

ia

1 ia (1

n ia

zib di 1 n ))

+ (ia (1

1

zi di +

n)

1 n ))

ia ia +ib

1 n)

(Z n ) + (1

!#

n)

ia

n ia

zi di

(Z n ) + (ia (1

= (ia )1

ia

1

1 n ia

0

ia ia +ib

Z

1 n ))

+ (ia (1

zi di

1

1 ia (1

1

1 n ia

0

1

" = (

Z

1

(Z t )

ib ia +ib 1

1

Z

ia +ib (1

n)

ia

(Z a ) +

ib ia +ib

1

1 ib (1

1 n)

(Z b )

;

where n

(Z )

=

Z

n ia

1 n ia

0

a

(Z )

=

b

(Z )

=

Z

Z

ia

1 ia (1

n ia

ia +ib (1

n)

ia

A.3

1

zi di; 1

zi di;

n)

1 ib (1

1

zib di:

n)

International and intra-national RER

In order to derive the decomposition for international RER, we log-linearize aggregate consumption price indices as follows. The log-pice of aggregate price index in region j in country h can be written as pj;t =

t

Pjt Pj

= ptt +

1

n

ptt +

Pjn Pj

1

1

pht +

n

Pjn Pj

pnj;t

1

pnj;t

ptt ;

(A1)

with ptt =

h

Pjh Pjt

= pht +

f

Pjf Pjt

A3

1

f

(pft

Pjf Pjt

pht )

1

pft (A2)

!#

zib di

and pnj;t =

Rjt Pjn

t

1

t + rj;t

Rjn Pjn

n

1

n : rj;t

(A3)

Finally, the aggregate log-price of regionally traded goods in region j is t = rj;t

Rja Rjt

a

1

a + rj;t

Rja Rjt

b

1

b : rj;t

(A4)

A similar set of conditions applies to the foreign country. For analytical tractability, we assume that international RER is a relative price of foreign to domestic consumption basket in regions a in countries h and f. In the numerical analysis of the model we use the de…nition of RER that relies on the average price in regions a and b as de…ned in (18). Now RER can be written as i rera;t = pâ;t

pa;t :

Substituting in the de…nitions above, we get

i rera;t =

h

f

1

1 1

pft 1 (^

I

pht ) + ^ n

2

p^na;t

p^ta;t

n 2

pna;t

pta;t ;

where lowercase letters, as before, denote the log transformations for all variables in devia1

1

tions from their steady state values (e.g.,

ln Pth

pht

h

ln P ; etc.); and

= P^ f =P t ;

1

2

Pân =Pâ = (Pan =Pa ) : This is expression (19) in the text. Having regional dimension in the model allows us to better understand the dynamics of nontradable goods prices. In particular, substituting equation (A4) into equation (A3), regionally nontraded log-price index can be written as: a pna;t = ra;t +

1

1 b

1

3

R

b ra;t

a ra;t +

n 4

n ra;t

t ra;t ;

1

where

3

= (Rbb =Rat ) =

^ bb =R ^ at R

= (Rbb =Rbt ) =

^ bb =R ^ bt R

1

and

4

= (Ran =Pan ) =

^ n =P^ n = (Rn =P n ) = R ^ n =P^ n : This is equation (20) in the text. In region b, country R a a b b b b h, the corresponding expression is a + ^b pnb;t = rb;t

3

b rb;t

a + ^n rb;t

4

n rb;t

t rb;t :

In country f the prices of internationally-nontraded goods can be approximated in an

A4

=

analogous way. For instance, in region a it is given by a p^na;t = râ;t +

1

1 b

1

b râ;t

3

R

a râ;t +

n 4

n râ;t

t râ;t ;

while in region b it is given by a pnb;t = r^b;t + ^b

3

b r^b;t

a r^b;t + ^n

4

n r^b;t

t r^b;t :

Using these decompositions, the regional real exchange rate in country h can be written as rerth =

n 2

pnb;t

pna;t :

a a = rb;t We can simplify the derivations by recognizing that in our model ra;t b b ra;t = rb;t

rtb : Furthermore, because regions are symmetric, we get ^ b =

a:

rta and

Substituting

for pnb;t and pna;t in the expression for rerth and combining with the simpli…cations above allows to write rerth as rerth =

n 2

"

1

1 a

b

1

R

!

3

(rtb

rta ) + ^ n

which is equation (21) in the text.

A5

4

n rb;t

t rb;t

n 4

n ra;t

t ra;t

#

;