Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Essays on international economics and industrial organization Lu, Min 2007

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_2007-318805.pdf [ 7.96MB ]
Metadata
JSON: 831-1.0100661.json
JSON-LD: 831-1.0100661-ld.json
RDF/XML (Pretty): 831-1.0100661-rdf.xml
RDF/JSON: 831-1.0100661-rdf.json
Turtle: 831-1.0100661-turtle.txt
N-Triples: 831-1.0100661-rdf-ntriples.txt
Original Record: 831-1.0100661-source.json
Full Text
831-1.0100661-fulltext.txt
Citation
831-1.0100661.ris

Full Text

Essays on International Economics and Industrial Organization by M i n L u B . A . (Hons.), Eas t C h i n a N o r m a l Universi ty, 1995 M . A . , S imon Fraser Universi ty, 2000 ' • . . . A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y in . T H E F A C U L T Y O F G R A D U A T E S T U D I E S (Economics) T h e Univers i ty of B r i t i s h C o l u m b i a September 2007 © M i n L u , 2007 Abstract T h i s dissertation addresses two issues i n international economics and one issue i n industr ia l organization. T h e first two chapters use st icky-price inter temporal op-t imiz ing models w i t h incomplete financial markets to analyze the dynamics of the current account after technology shocks and the effects of the op t ima l monetary pol -icy on current account movements. T h e th i rd chapter models the upgrade behavior of exist ing software users and new software users under two market structures. T h e first chapter studies a smal l open economy w i t h two sectors. In a perfect foresight, ra t ional expectation general equi l ibr ium model , w i t h s t icky prices i n the non-traded goods sector, the evolution of the current account following a positive technology shock is efficient even wi thout t ime consistent op t imal monetary policy. The second chapter extends the general equi l ibr ium model to a two-country economy and ana-lyzes the effects of the op t imal monetary pol icy on current account dynamics. T h e welfare gain for home households from the ind iv idua l op t imal expansionary mone-tary pol icy main ly comes from the home country 's terms of trade improvement when most firms price the export prices i n buyer currency. T h e th i rd chapter finds that, for the new software version, the.software vendor should offer a price discount to existing users and charge a higher price to new users if software users are sufficiently heterogeneous. For the old software version, a price discount should be applied to exist ing users w i t h a higher price to new users if the price of the co-existing new version is high. M i n L u . mlmlml@interchange.ubc.ca Contents Abstract .'. i i Contents i i i List of Tables vi List of Figures vi i Acknowledgements • i x Summary x 1 Current Account Dynamics and Optimal Monetary Policy in a Small-Open Economy 1 1.1 Introduction . . : 1 1.2 The Model 5 1.2.1 Households 5 1.2.2 Firms and the Structure of Goods Markets 8 1.2.3 Government 10 1.2.4 Debt Elastic Interest Rate 10 1.3 Equilibrium . 11 1.4 Calibration and Solution 12 1.4.1 Calibration 12 1.4.2 Impulse Response Analysis 13 Current Account Dynamics with Flexible Prices 14 Current Account Dynamics with Sticky Prices 16 1.4.3 Current Account Dynamics with Optimal Monetary Policy . . 20 i i i Discret ionary Mone ta ry Po l i cy 20 O p t i m a l Mone ta ry Po l i cy w i th Commi tmen t 22 1.4.4 Current Account Dynamics w i t h Interest Ra te R u l e . . . . . . 25 1.5 Conclusions 26 2 Current Account Dynamics and Optimal Monetary Policy in a Two-Country Economy 41 2.1 Introduct ion . . 41 2.2 Structure of the M o d e l 45 2.2.1 General Features 45 2.2.2 . Marke t Structure 46 2.2.3 F i r m Behavior . : 48 2.2.4 Household Behavior 49 2.3 Baseline Ca l ib ra t ion 53 2.4 Impulse Response Ana lys i s for the Current Account 55 2.4.1 Intertemporal and Intratemporal Elas t ic i ty of Subst i tu t ion . . . 5 5 2.4.2 Home and Foreign Goods Elas t ic i ty of Subst i tu t ion . . . . . . 57 2.4.3 L o c a l Currency P r i c i n g 58 2.4.4 M a r k u p 59 2.4.5 Non-Zero Ini t ia l Deb t /Asse t Posit ions 60 2.5 O p t i m a l Mone ta ry Policies . . '. . 60 2.5.1 O p t i m a l Moneta ry Policies W i t h M a r k u p Dis to r t ion 61 2.5.2 O p t i m a l Mone ta ry Policies W i t h N o M a r k u p Dis tor t ion . . . . 64 2.5.3 Discussions 66 2.6 Conclusions . 66 3 Software Upgrade, Consumer Behavior and Software Vendor's Choice 76 3.1 Introduct ion 76 3.2 Li terature Review : 79 iv 3.3 M o d e l i n g Software U s e r s ' B e h a v i o r for Software Upgrade 82 3.3.1 Ex i s t i ng Users ' Upgrade Demand Under Structure 1 . . . . . . 82 3.3.2 Ex i s t i ng Users ' Upgrade Demand Under Structure 2 90 3.3.3 New Consumer 's Demand of Software Versions . . . . . . . . . 91 Case One: New User 's Demand under Structure 1 . 91 Case Two: New User 's Demand under Structure 2 93 3.4 Choice of Software Vendor 95 3.5 Discussions and L imi ta t ions . . . 100 3.6 Conclusions 103 B i b l i o g r a p h y 113 A Append i ce s to C h a p t e r 3 119 A . l P roo f of Proposi t ions 119 v List of Tables 1.1 Baseline Ca l ib ra t ion Parameters for Chapter 1 28 2.1 Baseline Ca l ib ra t ion for Chapter 2 67 3.1 Ex i s t i ng Users' Opt ions for Software Upgrade 105 3.2 Demand for Software Upgrade Under Marke t Structure 1 (Exis t ing User) 106 3.3 Compara t ive Static Analys i s for Upgrade Demands of Ex i s t i ng Users 107 3.4 New User ' s Choice Under Structure 1 108 3.5 Compara t ive Stat ic Ana lys i s for Software Demands of New Consumers 108 3.6 Types of Software Users i n Simplif ied 2x2x2 model 108 3.7 Different Demands Under T w o Marke t Structures 109 A . l Notat ions Used in the Order of Appearance 121 A . 2 O p t i m a l Prices of the New Version for Ex i s t i ng and New Users . . . . 122 A . 3 T h e O p t i m a l Prices of the O l d Vers ion to Different Users 122 v i List of Figures 1.1 A 1% Temporary W o r l d Interest Rate Increase w i t h Different In i t ia l Net Debt Levels (Flexible Prices) 28 1.2 A Posi t ive 1% Technology Shock i n the Traded. G o o d Sector w i t h Different Intertemporal Elast ici t ies of Subst i tu t ion (Flexible Prices) . 29 1.3 A n Unexpected 1% M o n e y Supply Increase w i t h Different Intertem-poral Elast ic i t ies of Subst i tu t ion (Sticky Prices) 30 1.4 A n Unexpected 1% M o n e y Supply Increase w i t h Different Elast ici t ies of Labor Supply [a — 2, p = 0.75] (St icky Prices) 31 1.5 A n Unexpected 1% M o n e y Supply Increase w i t h Different F i r m M a r k u p s [a = 2, p = 0.75] (St icky Prices) 32 1.6 A n Unexpected 1% Money Supply Increase w i t h Different F i r m M a r k u p s [a = 0.5, p = 0.75] (St icky Prices) . 33 1.7 A 1% Posi t ive Technology Shock i n the Traded G o o d Sector w i t h Basic Ca l ib ra t i on (St icky Prices) 34 1.8 O p t i m a l Mone ta ry Po l i cy W i t h N o F i r m Subsidy [a = 2, p = 0.75] . 35 1.9 O p t i m a l Mone ta ry Po l i cy W i t h N o F i r m Subsidy [a = 0.5, p = 0.75] 36 1.10 O p t i m a l Mone ta ry Po l i cy W i t h F i r m Subsidies (Basic Cal ibra t ion) . . 37 1.11 A Posi t ive 1%.Technology Shock in the Traded G o o d Sector w i th Interest Rate Rule (Sticky Prices) 38 1.12 A n Unexpected 50% Interest Ra te Decrease (St icky Prices) 39 1.13 A n Unexpected 50%/200% Interest Rate Increase (St icky Prices) . . . 40 v i i 2.1 A n Unexpected Posi t ive 1% Technology Shock in the Traded G o o d Sector w i t h Different Consumpt ion Smoothing 68 2.2 A n Unexpected Posit ive 1% Mone ta ry Shock w i t h Different Consump-t ion Smooth ing . 69 2.3 A n Unexpected Posit ive 1% Moneta ry Shock w i t h Different Exchange Rate Pass-Through [b = 0.5; 6 = 0.75] 70 2.4 A n Unexpected Posi t ive 1% Mone ta ry Shock wi th Different Exchange Rate Pass-Through [b = 0.7; 0 = 0.75]. 71 2;5 A n Unexpected Posit ive 1% Mone ta ry Shock W i t h Different Markups [a = 2] 72 2.6 A n Unexpected Posi t ive 1% Mone ta ry Shock W i t h Different Markups [a = 0.5] . . 73 2.7 Home 1% Technology Shock W i t h and W i t h o u t the O p t i m a l Mone ta ry Po l i cy 74 2.8 Home 1% Technology Shock W i t h and W i t h o u t the O p t i m a l Mone ta ry Po l i cy W i t h F i r m Subsidies 75 3.1 Upgrade Software Releasing Sequence Under Structure 1 110 3.2 Dis t r ibu t ion of Various Demands w i t h O l d Version 110 3.3 Dis t r ibu t ion of Various Demands without O l d Vers ion . . . . . . . . . 110 3.4 Upgrade Software Releasing Sequence Under Structure 2 I l l 3.5 Dis t r ibu t ion of the New Consumers Under Structure 1 I l l 3.6 Dis t r ibu t ion of the New Consumers Under Structure 2 I l l 3.7 Demand of Ex i s t i ng Users 112 3.8 Demand of New Users 112 v m Acknowledgements I am extremely grateful to my supervisor, Professor Michae l B . Devereux for advice, guidance and encouragement. I am also greatly indebted to other members of my supervisory committee, Profes-sor P a u l Beaudry and Professor Ange l a Redish for research direct ion and discussion. I would like to thank Professor Pa t r ick Francois, Professor A m a r t y a L a h i r i and Professor Juany i X u for helpful suggestions and discussions at various stages of this dissertation. I am also grateful to K a n g Shi and Y a n b i n T u for helpful suggestions, discus-sions, and support . I thank the seminar part icipants at U B C macro-lunch, the 2004 and 2005 meetings of the Canad ian Economics Associa t ion and the U B C Target Workshop for comments and helpful discussions. I thank the Univers i ty of B r i t i s h C o l u m b i a Fellowship for financial assistance. I am solely responsible for any errors and misinterpretations. ix Summary This dissertation consists of two essays in international economics and one essay i n industr ia l organization. T h e first two essays use st icky-price inter temporal op t imiz ing models w i t h incomplete financial markets to analyze the dynamics of the current account after technology shocks and the effects of the op t imal monetary pol icy on current account movements. T h e th i rd chapter studies the upgrade behavior of exist ing software users and new software users under two market structures. The first essay studies a smal l open economy w i t h two sectors. In a perfect fore-sight, r a t iona lexpec ta t ion general equi l ibr ium model , w i t h st icky prices i n the non-traded goods sector, the current account responses to monetary shocks depend on the elasticity of subst i tut ion between consumption of non-traded and traded goods (the in t ra temporal effect) and risk aversion (the intertemporal consumption smoothing effect), the country's in i t i a l net foreign asset posi t ion, and the degree of monopolis-tic compet i t ion. In this analysis, the current account can go into either deficit or surplus in response to an expansionary monetary policy. W i t h flexible traded good prices and only slow price adjustment i n the non-traded goods sector, the current account reacts quite efficiently to technological shocks i n a smal l open economy. T h e welfare gain for households from adopting op t imal monetary pol icy i n contrast to constant money growth rule is quant i ta t ively smal l . T h e second essay extends the general equi l ibr ium model to a two-country econ-omy and analyzes the effects of the op t imal monetary pol icy on current account dy-namics. W i t h sluggish price adjustment among the firms, and w i t h most firms pr ic ing their exports i n the buyers' currency, the home monetary authori ty w i l l choose an expansionary monetary pol icy when facing a home technological improvement. T h e expansionary monetary policy, i n turn, w i l l improve the welfare of home households at the expense of the welfare of foreign households. T h e welfare gain for home house-x holds, from this op t imal expansionary monetary policy, main ly comes from the home country's terms of trade improvement. If a supranational monetary authori ty is to choose an opt imal monetary pol icy for both countries, the welfare gain from the expansionary monetary pol icy for both home and foreign counties is quant i ta t ively smal l . In this environment, the current account response to the op t imal monetary pol icy of the supranat ional monetary authori ty w i l l also be quanti tat ively smal l . T h e th i rd essay investigates the upgrade behavior of existing software users and new software users under two market structures: Structure 1 - where both the old and new versions coexist i n the market; and Structure 2 - where only the new version is available i n the market. We find that keeping the old version in the market is an effective tool to salvage the "disappearance" demand of software upgrade. T h e "cannibal izat ion" effect between the old version and new version, is different for exist ing users, compared to new users. We explore the possibi l i ty of software vendors u t i l i z ing price discriminations between exist ing users and new users as a method to reap higher profit. We find that, for the new software version, the software vendor should offer a price discount to exist ing users and charge a higher price to new users if the users are sufficiently heterogeneous. For the o ld software version, a price discount should be offered to exist ing users w i t h a higher price for new users, if the price of the co-existing new version is high. x i Chapter 1 Current Account Dynamics and Optimal Monetary Policy in a Small-Open Economy .1.1 I n t r o d u c t i o n A n open economy can run current account imbalances, thus, enabling different directions of movement of consumption and product ion in the short-run. For the last two decades, most of the current account li terature has emphasized the inter temporal approach. 1 Emphas iz ing the importance of consumption smoothing and investment, this literature focuses on a country's inter temporal budget constraint: a current account deficit now means trade surpluses i n the future. T h i s line of research has its at tractive feature that the analysis is based on op t imiz ing models, where explici t preferences, technology and capi tal market access are present. Mos t papers, however, have focused on non-monetary environments or flexible-price economies. In such models, money is always neutral , so that there is no room to analyze the effects of monetary policies on the current account. In recent years, researchers have been developing intertemporal op t imiz ing sticky-price open economy macroeconomics models. The new wave of research was in i t ia ted by Obstfeld and Rogoff (1995a)'s redux mode l . 2 T h e y introduced imperfect compe-1Please see Obstfeld and Rogoff (1995b) for a survey on this literature. 2See Lane (2001b) for a comprehensive survey of'the recent literature in this area. 1 t i t ion and st icky prices into the dynamic general equi l ibr ium macroeconomic frame-work. In these models, monetary shocks have real effects, thus providing a potential role for monetary policies. Obstfeld and Rogoff (1995a) show that w i t h s t icky prices and purchasing power par i ty ( P P P ) , a positive home money shock generates a long-run improvement in the home current account i n their two-country model . Bet ts and Devereux (2000a) also show that in a two-country wor ld w i t h pricing-to-market, a domestic monetary expansion w i l l in general improve the home country 's current account as long as not a l l goods are priced under local currency pricing. In this paper, we propose a smal l open economy dynamic general equi l ibr ium model w i t h incomplete financial markets to analyze the dynamics of the current account after different shocks and the effects of the op t imal monetary pol icy on current account. There are many factors that affect the current account dynamics. Lane (2001a) demonstrates that i n the short-run the response of the current account to a monetary shock depends on the parameter values of the elasticity of subst i tut ion between the consumption of traded and non-traded goods and the risk aversion measure. Devereux (2000) studies the impact of a devaluation on the current account i n a model w i t h pricing-to-market and shows that the impact depends cr i t ica l ly on the extent of pricing-to-market, and the infratemporal and inter temporal effects. Thoenissen (2003) claims that a key determinant of the current account dynamics is the in i t i a l net foreign asset posit ion. Lombardo (2002) shows that the degree of compet i t ion qual i ta t ively affects the current account response to nomina l shocks i n a two-country world. We show that, in our model w i t h st icky prices i n the non-traded goods sector, the current account responses to monetary shocks depend on the values of the elasticity of subst i tut ion between consumption of non-traded and traded goods, the "infratempo-ral effect" and the risk aversion measure, the "intertemporal consumption smoothing effect", the country's in i t i a l net foreign asset posi t ion and the degree of monopolis t ic compet i t ion. In this analysis, the current account can go into either deficit or surplus i n response to an expansionary monetary policy. O u r results confirm the studies of the above-mentioned research. O u r m a i n contr ibut ion is to show how the op t imal monetary pol icy w i l l affect 2 the current account dynamics. T h e opt imal monetary pol icy maximizes the welfare of a representative agent, given frictions in the economic environment. We con-struct a smal l open general equi l ibr ium model w i th two sets of frictions - costly price adjustments by imperfectly competi t ive non-traded goods firms and monopolis t ic compet i t ion. For evaluating the welfare of the representative agent, we have an ex-pl ici t u t i l i ty function and the dynamics of the u t i l i ty of the agent. In our model , we investigate the passive op t imal response of the monetary authori ty to a posi-tive technology improvement i n the traded good sector. We find that the op t imal monetary pol icy has two different implicat ions. W i t h both frictions - costly price adjustments by imperfectly competi t ive non-traded goods firms and monopolis t ic compet i t ion, the op t imal monetary authori ty takes the firms' price setting behavior as given,. Th i s gives a big incentive to the monetary authori ty to inflate and raise output and hence consumption - the discretionary result. T h e opt imal monetary pol -icy can expand the economy above the natural rate of output using an expansionary monetary policy. In this case, due to the monopolis t ic compet i t ion i n the non-traded goods sector, the non-traded goods' output is sub-opt imally low. Domest ic output is demand-determined i n the short-run because of the slow price adjustment in the non-traded goods sector. W i t h slow price adjustments, the monetary expansion can boost current demand for the non-traded goods consumption above the natural level of output i n the short-run. T h e op t imal monetary pol icy can increase the household's welfare compared to the case i n which the monetary authori ty s imply responds to the unexpected technology improvement by a passive increase of money supply. T h e expansionary op t imal monetary pol icy has a great impac t .on the in i t i a l current ac-count response. In this case, the current account responses to the positive technology shock w i t h the op t imal monetary pol icy are very different from the current account responses wi thout the op t imal monetary policy. In the discretionary case, al though the monetary authori ty has substantial lever-age over real ac t iv i ty in our model economy, the discretionary monetary pol icy is not time-consistent. The monetary authori ty cannot systematically move the smal l open economy away from the natural rate of output i n the long-run. To have a t ime consistent op t imal monetary policy, the monetary authori ty has to follow a commit-3 merit policy. Because people understand the monetary authori ty 's incentives in our perfect foresight model , the surprise monetary expansion and the resulting benefits cannot arise systematically. To have the time-consistent monetary policy, the mon-etary authori ty should not take advantage of the firm's price setting behavior. In our model , we use firm subsidies for the non-traded goods sector to eliminate the monopoly distort ion. T h e firms s t i l l set the prices over a markup, but get a lump-sum transfer from the government to compensate for the monopolis t ic power. After we get r id of the monopolis t ic compet i t ion distort ion, the output is already op t imal i n the non-traded goods sector. We have flexible prices in the traded good sector. T h e only distort ion in the economy is the sluggish price adjustment in the non-traded goods sector. W i t h the technology improvement i n the traded good sector, the real economy is expanding, an expansionary monetary pol icy by the monetary authori ty w i l l be welfare improving theoretically. Nevertheless, in our model , as the technology improvement in the traded good sector is temporary, and the sluggish price adjustment is only i n the non-traded goods sector, the welfare gain from the expansionary monetary increase in response to the technology shock, w i l l be smal l . In this case, without the markup dis tor t ion i n the smal l open economy, the opt i -mal monetary pol icy w i l l increase the money supply by a negligible amount and be time-consistent. T h e passive expansionary monetary policy, in fact, w i l l decrease the household's welfare since the economy is already at the long-run op t imal out-put level. The welfare gain for the household, from adopting the op t imal monetary policy, w i l l be quant i ta t ively smal l . T h e current account response to the op t imal monetary pol icy is quant i ta t ively small , too. Thus , i n this case, once we remove the monopolis t ic dis tor t ion i n the welfare analysis, the current account responds fairly efficiently to the unexpected technology improvement. For the smal l open economy in our model , the time-consistent op t imal monetary pol icy to a positive technology shock only improves the household's welfare by a negligible amount. T h e time-consistent op t ima l monetary pol icy is to expand the economy by a quanti tat ively smal l amount. The response of the current account to the technology shock w i l l not change quanti tat ively w i t h the op t imal monetary policy. In other words, the evolution of the current account to the technology shock 4 is already efficient, even wi thout the op t imal monetary policy. For a smal l open economy, the op t imal monetary pol icy has an insignificant effect on the welfare of the economy and current account dynamics. T h e detailed structure of the work is as follows: Section 1.2 presents the model . Section 1,3 presents the equi l ibr ium of the model . Section 1.4 discusses the cal ibra-t ion and solution w i t h flexible prices and then presents the more general case where the prices are st icky in the non-traded goods sector. F ina l ly , Section 1.5 concludes the paper. 1.2 T h e M o d e l This .paper assumes a perfect foresight smal l open economy wi th a representative household, firms and a domestic government. T h e economy has two different types of goods - a homogeneous traded good and differentiated non-traded goods. T h e non-traded goods sector is monopol is t ical ly competi t ive. T h e price of the traded good is covered by the law of one pr ice . 3 1.2.1 Households T h e economy is populated by a cont inuum of consumers/households of measure unity. T h e representative household is endowed w i t h a certain amount of t ime, which is d iv ided between leisure and work. She consumes two types of goods - traded and differentiated non-traded goods. T h e consumer can hold two types of nominal assets: non-interest bearing home money M, a one-period noncontingent foreign debt D denominated in foreign currency. She gets income from labor income, profits from domestic firms and lump-sum government transfers and pays back interests on the foreign debt. Preferences T h e lifetime u t i l i ty of the home representative household is: oo max ^2 ft t=o l - a l - € \ P t ) V+ip 1 (1.2.1) 3 In the following, we normalize the foreign currency price of the traded good to unity and consider a flexible exchange rate regime so that the domestic currency price of the traded good and the nominal exchange rate are the same: Pxt = et, where et is the nominal exchange rate, defined as the number of domestic currency units per unit of foreign currency. 5 where B £ (0,1) is the discount rate, a is the inverse of intertemporal elasticity of aggregate consumption. T h e household's instantaneous u t i l i ty depends posi t ively on consumption, Ct, and real money b a l a n c e s , ^ , where Mt is nominal balances held at the beginning of per iod t and Pt is a consumption based price index for period t. Ht is the labor effort at t ime t. T h e consumption index Ct is a C E S aggregate of traded and non-traded goods Ct = a^CN"t +(l-a)-PCTl p-i where p > 0 is the constant elasticity of subst i tut ion between traded and non-traded goods. T h e non-traded good is i n tu rn defined over the consumption of differentiated goods, so that i r-l Nt CNt(iy *di where Cjvt is an index of consumption of the non-traded goods, Crt is the consump-t ion of traded good. There exists a cont inuum of home produced non-traded goods indexed by i 6 [0,1] and a homogeneous traded good. A l l consumption goods are perishable. CWt(i) denotes date t consumption of the non-traded good i and A > 1 denotes the elasticity of subst i tut ion among non-traded goods. T h e non-separabili ty between traded and non-traded goods consumption means that shocks to the non-traded goods sector have spillover effects on the traded good consumption and hence the current account. For instance, i n the case that traded good consumption rises together w i th non-traded goods consumption, a boom in the non-traded sector w i l l cause an increase i n demand for imports and a current account deficit. T h e consumption price indices are defined as Pt = [aP1-/ + (1 - a)Pk>] 1"> (1.2.2) PNt — Pmi^di i l - A Assume the law of one price holds for the traded good, P t * is the wor ld price of the traded good and et is the current exchange rate: Prt = e t P t *. 6 O p t i m a l consumption behavior implies that the ind iv idua l demands for non-traded and traded good are given as: CTt = (l-a)(^)-"Ct C m = a(^)-'Ct Cm(i) C, Nt T h e household can provide different types of labor services. There exists a con-t inuum of labor types, indexed by i € [0,1]. Let Ht(i) denote the number of hours of type i labor. T h e variable Ht that appears in the u t i l i ty function is defined as a sum of labor services in non-traded and traded goods sectors: Ht= f HNt{i)di + HT Jo Household Budget Constraint T h e household can hold two financial assets: lo-cal money and nominal debts denominated i n foreign currency. T h e debts have a matur i ty of one period. T h e household's budget constraint in period t is: Mt + et(l + it-{)Dt-.1 + PtCt = Mt^ + etDt + Tt + WtLt + Ut (1.2.3) where Dt is the household's stock of foreign currency debts that become due i n per iod t. it-\ is the nominal interest rate on the foreign debt. T h e household owns the firms and II( is the profit from the traded and non-traded goods firms. T h e representative household's intertemporal consumption decisions and her de-mand for money are determined by max imiz ing the life-time u t i l i ty specified i n (1.2.1) subject to the restrict ion that the budget constraint (1.2.3) holds i n al l periods and for a l l states of the world . R u l i n g out the Ponz i schemes, we can get the following first-order conditions: ^ = vC?Ht (1.2.4) 1 = ^(1 + 0 ^ ^ - ^ (1-2-5) MtV nCr+i Pt x{ft) = c r [ t ~ p c r p ^ ] ( L 2 - 6 ) where equation (1.2.4) is the infratemporal op t imal labor supply schedule, equation (1.2.5) is the inter temporal Euler condi t ion and equation (1.2.6) is the op t imal money demand schedule. 7 1.2.2 Firms and the Structure of Goods Markets T w o types of firms exist i n the country: (i) producers of non-traded consumption goods; and (ii) producers of the traded consumption good. Domest ic producers use domestic labor as the only input , and labor is immobi le internationally. T h e per iod t product ion functions of the firms producing non-traded good i and traded good are concave and given by: Yxt(i) A.xtH.xtdr YTt = ATtWTt .' where Y^t and YTt are the firms' outputs and a < 1,7 < 1, while Am and An are period t labor productivi t ies . A^t and Art are exogenous random variables. 4 There are competi t ive profit max imiz ing firms i n the traded good sector. T h i s implies that the price is equal to marginal cost: Wt = 1PTtATtWT-1 (1.2.7) where Wt is the wage rate. In the non-traded goods sector, each product ion firm has monopolis t ic market power, and sets its price as a markup over the marginal cost. If non-traded goods prices were perfectly flexible, then the profit max imiz ing decision for firm i would imply : j^jWt = aPmWAmHNtli)"-1 (1.2.8) The ma in objective of this paper is to address the current account dynamics w i t h the op t imal monetary policy. In this respect, we need to have price r igidi ty i n our model . We assume that non-traded goods firms adjust their prices w i t h an adjustment cost. We follow Sheshinski and Weiss (1977) and Rotemberg (1983) and introduce sluggish price adjustment by assuming that the non-traded good firm faces a resource cost that is quadratic in the inflat ion rate of the goods it produces: 7rjvt(0 = Pm(i)YNt{i) - WtHNt(i) - S-2 Yfft(i)- T h e parameter S measures the degree of price stickiness. T h e higher is <5 the more sluggish is the adjustment of nomina l prices. If 5 = 0, then prices are flexible. 4 I n this small open economy model, A^t and Art are assumed to be unity at steady-state. 8 •Pm(i) Pi Nt-l (0 - 1 Pm(i)Ym(i where price adjustment cost is Pmd) P j v t - i ( i ) The producer of non-traded good i maximizes oo TlNt(i) = ^ Q t : t + j T T N t ( i ) where pt 1t+j '•-'i is, the pr ic ing kernel used to value date t + j pay-offs. A s firms are owned by the representative household, it is assumed that firms value future payoffs according to the household's intertemporal marginal rate of subst i tut ion i n consumption. T h e firms' pr ic ing decision is given by: {(1 - X)YNt(i) + - ~ -p-^TS)Pm(i)YNt{i) a Pmw Pm-iW pm-iW 1 1 Nt-l(i) + P 5 F ^ c r { { ^ ^ - -^w)Pm+i(l)YNt+l(z)} = 0 (1.2.9) W i t h s t icky prices i n the non-traded goods sector, the price of good i equals the product of the shadow value of one extra unit of output (the marginal cost), t imes a markup. Symmetr ic non-traded goods firms make identical choices i n equi l ibr ium. T h e markup ^t ( i ) depends on the output demand as well as on the impact of today's pr ic ing decision on today's and tomorrow's costs of adjusting the output price: *t(i) = XYNt{i)l(\-l)YNtfi) + 6 .•Pm-wr Pm-iW • + ^ ( ^ - i ) 2 ^ ( 0 1 ^Nt-\(i) Rct~+i Pt (Pm+i{i)2 Pm+i(}),p r ^ v r ^ (1.2.10) Pt+iCf Pm(i)3 Pm(i)7 If 5 — 0, i.e., if prices are fully flexible, ^t{i) = JZT is the familiar constant-elasticity markup. If 8 ^ 0, price r ig idi ty generates endogenous fluctuations of the markup. T h e markup * t ( z ) depends on P N t + 1 ( i ) , PNt(i), P j v t ^ i ( i ) , YNt(i) and Y N t + 1 ( i ) . T h e non-traded goods firms react to p ^ ' ^ ) dynamics in their pr ic ing decisions. Changes i n monetary pol icy generate changes i n p ^ t ( ^ dynamics. Hence, they affect the non-traded goods prices and the markup. T h r o u g h this channel, they generate different 9 dynamics of relative non-traded goods prices and the current account. If A approaches infinity, firms have no monopoly power, and the markup reduced to 1, the competi t ive level. Under perfect compet i t ion, the presence of a cost of adjusting the price level is irrelevant to the firms decision. Some degree of monopoly power is necessary for the nominal r ig id i ty to matter. 1.2.3 Government T h e government issues local currency, has no expenditures, and runs a balanced budget every period. T h e nominal lump-sum transfers of seignorage revenues Tt are given by: Tt = Mt-Mt_1 1.2.4 Debt Elastic Interest Rate This set-up of our model implies incomplete asset markets. A well-known conse-quence is that this model w i l l display non-stationary dynamics, so that l inearizing the model around the in i t ia l steady-state could yield a poor approximat ion of the nonlinear model . A s Schmit t -Grohe and Ur ibe (2003) point out, the uncondi t ional variances of endogenous variables are infinite, even i f exogenous shocks are bounded. In our rat ional expectation model, we use a nonlinear solution to avoid the approx-imat ion error of log-linearization, for evaluating welfare and also to accommodate large shocks. We follow Schmit t -Grohe and Ur ibe (2001) and Berg in (2006) and impose a "premium" on the asset return which is propor t ional to the outs tanding stock of foreign debts. The nominal return of the debt is closely related to its wor ld nominal interest rate i\. it = i*t + MeDt~d ~ 1) (1-2.11) where J is a constant - the in i t i a l aggregate level of foreign debt. The nominal interest rate of the debt depends on the real holdings of the foreign debts i n the entire home economy. T h i s means that domestic households take d as given when deciding on the op t imal holding of the foreign debt. T h e term ip2(eDt~d — 1) is a country-specific interest rate premium. A s borrowers, consumers w i l l be charged a premium over 10 the foreign interest rate; and as lenders, they w i l l receive a remuneration that is lower than the foreign interest rate. Another way to describe this cost is to assume the existence of intermediaries i n the foreign asset market (which are owned by foreign households) who can borrow from and lend to households of the rest of the world at the rate i*t , but who can borrow from and lend to households of the home country at the rate it. For characterizing the incomplete financial structure, we do not really need to introduce this addi t ional cost. Nevertheless, this, w i l l be useful for p inning down a well-defined steady-state for consumption and assets. T h e p r imary mot ivat ion for inc luding this term here is to remove the element of nonstat ionari ty in the model . Introducing the risk p remium term as a function of debts, ensures that wealth allocations i n the long-run converge to a unique steady state. In practice the parameter ip2 w i l l be set at a very low level in cal ibrat ing the m o d e l . 5 1.3 E q u i l i b r i u m The Flexible-Price Equilibrium A s a l l households and firms are identical , we can drop the subscript i. W e first consider the case when al l prices are perfectly flexible. Combine equations (1.2.3) and (1.2.7), we can get (1 - H t _ i ) A - i = A + P;[ATtH}t - (1 - a)(^)-"Ct] (1.3.12) T h i s is the market equi l ibr ium condit ion. The labor working i n either traded or non-traded goods sector w i l l get the same wage rate, -j^-jetPtATaH}? = aANtH%-lPm (1.3.13) T h e equi l ibr ium of this fully flexible price economy is a collection of eight sequences (Ai Pt-, Ct, et, Hxt, HNt, PNt, h) satisfying 8 equi l ibr ium conditions. These include (1.3.12), (1.3.13), the three household first-order equations, the debt-elastic interest rate equation (1.2.11), the price index (1-2.2) and the non-traded goods market clear condi t ion. The steady-state Stat ionar i ty fails for an open economy whenever the equi l ibr ium rate of aggregate per capi ta consumption growth is independent of the economy's 5 In our paper, this elasticity of the interest rate premium is set at 7 x 10~3. 11 aggregate per capi ta net foreign assets. T h e constant consumption i n the steady-state does not determine a unique steady-state for net foreign assets. In our model , the •debt elastic interest rate ensures the stat ionari ty of the model . T h e cost ip2(eDt~d— 1) captures that, when the economy-wide holdings of foreign-currency denominated debts are above (below) the steady-state level, d, ind iv idua l agents receive less (more) than the gross rate of return. W h e n the equi l ibr ium aggregate level of foreign debts i n the economy is equal to d, this cost w i l l disappear. We denote steady-state levels of variables wi thout a t ime subscript t. T h e steady-state level of domestic interest rate i is equal to the wor ld interest rate i* = i ^ p . G i v e n the in i t i a l steady-state level of net foreign currency debt D, we can solve for the steady state level of the real variables. The Sticky-Price Equilibrium For the st icky price equi l ibr ium, we also have eight equations and now we have equation (1.2.9) instead of (1.3.13). T h e current account w i t h s t icky prices may be computed as: e t ( A ~ A - i ) etit-iDt-i , Pn S v n x p— = p r -p-\YTt ~ CTt) 1 ' ^ v ' net exports The current account is determined by the interest payment for past debt and net exports. T h e real exchange rate is: ~ Pt 1.4 Calibration and Solution 1.4.1 Cal ibrat ion A s this is a perfect foresight, ra t ional expectation model , we can use a non-linear solut ion technique - mul t ip le shooting method. T h i s method is used to solve two-point boundary problems. L i p t o n et al. (1982) give a detailed discussion. T h i s solut ion technique allows us to avoid the approximat ion errors from log-linearization and gives us a trajectory for the current account after a shock. We follow the open economy macro literature i n picking parameter values for our basic experiments. T h e inverse of the inter temporal elasticity of subst i tut ion is set at 2, following Backus , Kehoe , and K y d l a n d (1995). T h e rate of t ime preference 12 is set at 0.01, so that the subjective discount factor is 0.99. T h e value of 77 is just a scale factor, so we set it a rbi t rar i ly to unity. T h e share of non-traded goods i n the consumer price index d is set at 0.5, following the evidence cited i n C o o k and Devereux (2001) for M a l a y s i a and Tha i l and . T h e inverse of the elasticity of labor supply is set to 0.5, so that ip = 0.5. T h i s is roughly following Rotemberg and Woodford (1998) 6 . T h e elasticity of subst i tut ion between non-traded and traded goods is set at 0.75 which follows direct ly from Os t ry and Reinhar t (1992). T h e elasticity of subst i tut ion between varieties of non-traded goods is A, and this governs the equi l ibr ium markup of price over cost in the non-traded goods sector. We follow Rotemberg and Woodford (1998), where they set A = 7.66 which implies an average mark-up of 15%. We assume that non-traded goods product ion is relatively labor intensive, w i t h a = 0.7, and traded goods is relatively non-labor intensive, w i t h 7 = 0.3, T h e consumption elasticity of money demand for household is equal to ^ in this model . Accord ing to M a n k i w and Summers (1986), this variable is very close to uni ty and hence e is set to 1. For the price stickiness, we follow Ireland (2001) and choose the price adjustment cost parameter S = 77. Table 1.1 reports the baseline cal ibrat ion assumptions. In this paper, we also investigate whether or not in i t i a l holdings of foreign debts/assets w i l l affect the current account responses to the various shocks. A s documented by Lane and Miles i -Fer re t t i (2001), net foreign assets over G D P vary across countries and are different from zero. T h e authors argue that the level of net foreign assets is a key state variable and a crucial determinant of the benefits of internat ional financial integration. T h e values of the steady-state net debt posi-t ion, relative to consumption, set between -0.5 and 0.5, appears to be reasonable for O E C D countries. 1.4.2 Impulse Response Analysis We calibrate the model to explore how the current account dynamics w i l l respond to a positive technology shock w i t h and without op t imal monetary policy. Before we can do. that, however, we investigate how the current account responds to a 6 In Rotemberg and Woodford (1998), the inverse of the elasticity of labor supply is set to 0.47. 13 positive technology shock or an expansionary monetary shock. Recent research shows that the current account can go into either deficit or surplus when facing a positive technology shock or an expansionary monetary shock. Lee and C h i n n (2006) show that the current account improves in response to temporary technology shocks for G 7 countries. In this section, we demonstrate the factors that affect the current account dynamics. Current Account Dynamics with Flexible Prices W i t h Flexib le prices, the monetary shocks w i l l only affect the prices, and not the real variables in this economy. W i t h real shocks, the prices w i l l adjust instantly. Money Supply Shocks W i t h perfectly flexible prices, money is indeed neutral i n our model . Under an unforeseen permanent money supply shock, the prices (domestic price and non-traded goods prices) instantly adjust and j u m p to the new equi l ibr ium levels by the same magnitude, while the current account remains at zero at a l l times. T h e fact that the smal l open economy holds a positive or negative in i t i a l debt does not affect the results i n this analysis. World Interest Rate Shocks T h e in i t i a l debt holding in the smal l open economy w i l l affect the interest payment and hence the real economy. W i t h zero in i t i a l debt holding, upon a temporary 1% increase in. the world interest rate, domestic currency depreciates, exports increase, and imports decrease! T h e current account w i l l go into surplus upon the positive temporary world interest rate shock i n period 1, and then go back to balance in the next period, since the shock is only temporary and prices adjust instantly. W i t h a positive debt holding, the current account goes into surplus i n period 1 upon the shock, then goes into a deficit i n per iod 2, and then returns again to zero in per iod 3. U p o n the shock, as the domestic interest rate increases, consumption decreases, households work more, and the traded good output increases. Households consume less and produce more of the traded good, so the current account improves right away. A s the economy is holding debt, however, higher interests must be paid , this increased interest payment causes the current account to dive into a deficit i n the next period. Since the shock is only temporary, everything returns to the in i t i a l equi l ibr ium level again. Similar ly , if the country holds foreign assets 14 in i t ia l ly , the current account improves right away in period 1, and w i l l not go to zero in period 2, because of the increased interest income from holding foreign assets, though it w i l l s t i l l go back to a balance, finally. See Figure 1.1 for the current account responses to a 1% temporary increase in the world interest rate. Technology Shocks We also calibrate the current account dynamics to technology shocks i n the traded good sector. The technology shock process takes the following form: where ext+i is the technology shock i n the traded good sector, p,i=0.95 (following the real business cycle li terature). U p o n a persistent technology shock, as the traded good's product ion is more efficient, the output of the traded good w i l l increase. F r o m the households' first-order conditions, we get: T h i s equation is the Euler Equa t ion , governing the dynamic movement of consump-t ion. T h e consumption growth depends on the sequence of relative prices - the so called "consumption-based real interest rate effect". If the aggregate price level rel-ative to the price of traded goods, is currently low compared to its future value, present consumption is encouraged over future consumption. Meanwhi le , the rel-atively higher future traded goods price also encourages consumption subst i tut ion from traded goods to non-traded goods. T h e former effect dominates if the in -ter temporal elasticity of subst i tut ion ^ is greater than the infratemporal elasticity of subst i tut ion p, i.e., ^ > p, as the spillover between non-traded and traded con-sumpt ion is positive. W h e n ^ < p, however, the spillover between non-traded and traded consumption is negative, as the intra temporal subst i tut ion effect dominates the inter temporal subst i tut ion effect. F i r s t , we show the current account response w i t h zero in i t i a l net foreign assets. W i t h an unexpected 1% temporary persistent technology shock i n the traded good sector, the output of the traded good sector w i l l increase as the product ion of the traded good is more efficient. A s the real exchange rate appreciates, domestic con-sumpt ion of the traded good also increases. The increase in the output of the traded lnATt+1 = HilnATt + eTt+1 et+i 15 good is greater than the increase i n the consumption of the traded good, so that the current account goes into surplus at the t ime of the shock. W i t h our basic cal ibrat ion, upon the 1% technology shock i n the traded good sector, the current account demon-strates a 0.27% surplus in the current per iod G D P ratio. Lee and C h i n n (2006) show that the current account improves in response to temporary technology shocks for G 7 countries. Our results are consistent w i t h this empir ical result. A s the technol-ogy shock is very persistent, the output of the traded good gradual ly decreases from the m a x i m u m level upon the impact . After 18 quarters, the consumpt ion becomes greater than the product ion of the traded good, and the current account sinks into deficit, reaching a 0.039% G D P ratio deficit after 37 quarters. W i t h the technology shock, i n our basic cal ibrat ion a = 2, i.e. ^ < p = 0.75, the spillover between non-traded and traded consumption is negative. If we change a = 0.5, so that ^ > p, the non-traded consumption moves together w i t h the traded consumption. In this case, the traded good consumption increases more than it would when a = 2. U p o n the shock, the current account s t i l l goes into surplus, but by a smaller amount i n the basic case, i.e., by only 0.24% of the current G D P . Figure 1.2 shows the current account, traded and non-traded consumpt ion responses to the traded sector technology shock w i t h \ < p and ^ > p. T h e impulses are percentage deviations from the in i t i a l steady-states except for the current account responses. W i t h a non-zero in i t i a l net foreign debt, the current account response to the technology shock is almost the same. Current Account Dynamics with Sticky Prices Lane (2001a) shows that monetary shocks empir ical ly are a significant source of var ia t ion i n the current account and can help explain the high vola t i l i ty of the current account. Hence, we must also include monetary shocks i n our model . T h e price stickiness is modeled by a price adjustment cost in the non-traded goods sector. T h e parameter 8 measures the degree of price stickiness i n the non-traded goods sector. In our basic cal ibrat ion 8 = 77, 78% of the non-traded goods prices fully adjust w i th in one year w i t h a 1% one-time money supply shock. In this section, we first s tudy the case where the country has zero in i t i a l net foreign assets. Then , 16 we study the case where the small open economy has positive/negative initial net foreign debt. Money Supply Shocks For simplicity, we assume that money supply is character-ized in terms of a money supply rule. The money supply rule is: lnMt = lnMt_x + [i{lnMt_x - lriMt-2) •+• eMt where /i=0.5, and eMt is the unexpected money supply shock. This section shows that there are a few factors affect the current account response to a monetary shock. First, the intratemporal effect and risk aversion will affect how the current account responds to a monetary shock. With a 1% unexpected temporary money growth shock, money supply increases permanently. The surprise monetary expansion stimulates extra demand, and hence, the production of non-traded goods. When ^  < p, the spillover between non-traded consumption and traded consumption is negative, as the intratemporal substitution effect dominates the intertemporal substitution effect. Increased consumption of non-traded goods means that the traded good consumption decreases. Increased production of non-traded goods also crowds out the production of the traded good. As a result, the traded production decreases upon the shock. Both the consumption and production of the traded good decrease. In our basic calibration, the inverse elasticity of labor supply is 0.5, the production of traded good decreases by a smaller amount, so the current account improves upon the impact of the shock. If we change the value of a so that 1 > p, the spillover between non-traded consumption and traded consumption is positive, increased consumption of non-traded goods also stimulates consumption of the traded good, since the elasticity of substitution between the consumption of non-traded and traded goods is low (relative to the intertemporal elasticity of substitution). At the same time, non-traded goods production crowds out the traded good production. So the current account goes into deficit. Figure 1.3 shows the current account responses to an unexpected 1% positive money growth shock with a = 2 and a = 0.5 when p — 0.75. The magnitude of the current account response is comparable to Lane (2001a)'s result. This analysis further confirms that the elasticity of substitution between consumption of non-traded and traded goods (the 17 infratemporal effect) and risk aversion (the inter temporal consumption smoothing effect) w i l l affect how the current account responds to shocks. Second, the elasticity of labor supply also affects the in i t i a l response of the current account to a monetary shock. W i t h the same monetary shock, we show how the inverse elasticity of labor supply ip affects the responses of the current account. W h e n the inverse elasticity of labor supply increases, and the elasticity of labor supply decreases, households are wi l l ing to provide less labor w i t h the shock. A s the demand for non-traded goods increases, the labor supply for non-traded goods has to increase. T h e traded good output decreases more w i t h the same shock. T h e current account w i l l show less surplus upon the shock. Figure 1.4 gives the current account responses to the monetary shock w i t h different elasticities of labor supply. We can see that w i t h different elasticities of labor supply, the current account responds differently to the same monetary shock. T h i r d , Equa t ion (1.2.10) shows that changes i n monetary supply generate changes i n p"t1}^ dynamics, which w i l l affect the markup and the current account. W i t h monopolis t ic compet i t ion, the output of non-traded goods falls below the competi t ive level. A t steady-state, as A increases, the markup of the non-traded goods' firms decreases. W i t h less monopolis t ic compet i t ion, the non-traded goods' firms are more competi t ive and have a higher output of non-traded goods. T h e surprise monetary expansion shock w i l l increase the output in the non-traded goods sector and tend to correct the monopolist ic dis tort ion. W i t h a higher A, as the equi l ibr ium of outputs for the non-traded goods is already higher, the monetary expansion w i l l cause the non-traded goods' outputs to increase less. So the traded good output w i l l decrease less w i t h a higher A 7 . In our basic cal ibrat ion, when £ < p, the increased the consumption of non-traded goods is accompanied by decreased consumption of the traded good. A s non-traded consumption increases less w i t h a higher A, the traded consumpt ion also decreases less. Now, we have both consumption and product ion of the traded good decreasing less, w i t h lower markup. Figure 1.5 demonstrates the current account responses to an unexpected 1% positive money growth shock w i t h different levels of markup - 15%, 7% and 2%. W i t h the lower markup, net exports 7 The increased production of non-traded goods will crowd out the traded good production. 18 fall when ^ < p. T h i s is the same result as was found by Lombardo (2002). We have shown, i n equation (1.2.10), that the markup is necessary for the monetary shock to have an effect. W i t h less markup distort ion, the real economy w i l l respond less. T h e markup matters quant i ta t ively for the current account response to a monetary shock. T h e markup also works w i t h the infratemporal subst i tut ion effect and the inter temporal subst i tut ion to affect the current account response. Figure 1.6 shows that w i t h ^ > p, the current account improves w i t h the lower markup. T h e reason for this result is that, as ^ > p, the traded consumption increases together w i t h the non-traded consumption, and w i l l increase less w i t h a higher A. Thus , w i t h the higher A and the lower markup, the traded output decreases less and traded consumption increases less. Therefore, the current account improves. In this section, we confirm that the inter temporal elasticity of subst i tut ion and the infratemporal elasticity of subst i tut ion, as well as the firm markup, affects the current account response to monetary shocks. We show that the elasticity of labor supply also affects the in i t i a l current account response. These factors not only affect the current account response independently,, but also work together to affect the in i t i a l response of the current account. We need to keep these factors i n m i n d when investigating whether or not these factors w i l l have any effect on our op t imal monetary pol icy analysis. Current Account Dynamics with Non-Zero Initial Net Foreign Asset In this section, we explore how the non-zero in i t i a l net foreign asset w i l l affect the current account dynamics w i t h technology and monetary shocks. In the smal l open economy w i t h flexible exchange rates, the domestic interest rate is decided by the wor ld interest rate. A n y feasible technology and monetary shock w i l l not change the domestic interest rate sufficiently to alter quant i ta t ively the dynamics of the current account response. Technology Shocks Before examining the op t imal monetary pol icy for the current account responses, we first must see what impact a technology shock w i l l have on the current account dynamics. We s t i l l use the technology shock Equa t ion (1.4.14) i n the traded good sector. Figure 1.7 shows the current account response to an unexpected 1% increase of the traded good technology. U p o n the technology shock, the traded 19 good sector w i l l increase its product ion since the product ion becomes more efficient. T h e increase i n output is greater than the increase i n the consumption of the traded good, and the current account goes into surplus upon the impact of the shock. W i t h our basic cal ibrat ion, the magnitude of the current account response is 0,27% of the current per iod G D P upon the shock. D u r i n g convergence, the current account has an over-adjustment deficit of up to 0.039% of the current G D P . ' 1.4.3 Current Account Dynamics with Optimal Monetary Policy Discretionary Monetary Policy In this section, we s tudy op t imal monetary pol icy w i t h current account move-ments and see how the op t imal monetary pol icy w i l l affect the current account dy-namics. We first s tudy the case when the net foreign asset is in i t i a l ly zero, and explore the op t imal monetary pol icy to a temporary positive technology shock i n the traded good sector. The op t imal monetary pol icy maximizes the welfare of the representative household in this smal l open economy. T h e technology shock process takes the form of (1.4.14). In our model , we assume that the monetary authori ty receives perfect information about the current state of the economy. So, the op t imal active monetary pol icy is that the money supply responds instantly to the technology shock. U p o n the unexpected positive technology shock in the traded good sector, the monetary authori ty immediate ly recognizes the temporary technology shock and sets the appropriate money supply to maximize the welfare of the household. T h e money supply rule is: lnMt = lnMt-i + p(lnMt_i - ZraMt_2) + <^Mt £ M I = a\^Tx and now, ^u=0.5 and a\ is the parameter that has to be determined op t imal ly by the monetary authority. In this economy, we have two kinds of distortions - nomina l r ig idi ty and monop-olistic compet i t ion i n the non-traded goods sector. W i t h a 1% unexpected positive technology shock i n the traded good sector, the sector increases its product ion since the product ion is more efficient now. T h e increased product ion of the traded good crowds out the non-traded goods product ion. A s the real economy is expanding 20 w i t h the positive technology improvement, intuit ively, welfare w i l l improve i f the smal l open economy increases the monetary supply at the same time. Due to the monopoly dis tor t ion in the non-traded goods sector, the output of the non-traded goods is already sub-opt imal ly low. The monetary authori ty takes pre-set. prices as given, and can increase the money supply and choose the op t imal money supply level to increase the welfare of the household. T h i s unanticipated monetary expansion, upon the technology shock, leads to increases i n real economic activities. A s the price adjustment is slow i n the non-traded goods sector, domestic output is demand-determined in the short-run. T h e surprise monetary expansion boosts the current demand for the non-traded goods and causes the outputs of the non-traded goods to increase, above the sub-opt imal level and even above the natural rate level of output. T h e increased amount of money also works w i t h the technology shock, boosting the to ta l employment above the steady-state level (natural rate level) i n the short-run. Equivalent ly, this nominal shock increases the employment above the natural rate level and eliminates part of the exist ing distortions in the economy. T h i s effect comes from two channels: 1) the increased money supply w i l l enable the firm to adjust less to decrease the welfare loss from st icky prices; and 2) the increased money supply w i l l min imize the monopolis t ic dis tor t ion from the non-traded goods sector. T h i s result corresponds to the benefits from surprise inflation i n Bar ro and G o r d o n (1983). T h e surprise monetary expansion gives rise to the discretionary monetary pol icy problem, as the monetary authori ty can exploit the firms' pr ic ing decision when it makes the monetary policy. T h i s gives us the no-commitment result. T h e benefit from this surprise monetary expansion, i n this case, is an expansion of real economic activ-ity. Because of the exist ing distortions in this economy, the economy starts off i n a sub-opt imal ly low equi l ibr ium, and the expansion of the non-traded goods sector improves the household's welfare. W i t h the basic cal ibrat ion, the op t imal value of the magnitude of the money injection parameter a\ is equal to 12.4. U p o n the unforeseen 1% technology shock i n the traded good sector, the monetary authori ty increases its money supply by 12.4%. T h i s op t imal monetary pol icy w i l l increase the household's welfare by 0.0095% than w i t h no monetary policy. If the monetary authori ty has just a passive 1% monetary 21 expansion to the 1% technology shock, the welfare of the household can still increase by a small amount 0.0017%, which is much lower than 0.0095%. The optimal monetary policy also affects the current.account responses and mag-nifies the initial current account/GDP responses upon the shocks, by 39.4%, accord-ing to our basic calibration. Figure 1.8 shows the impulse responses under sticky prices with the optimal monetary policy and a passive 1% money supply increase (basic calibration). As analyzed in the above section, the intratemporal effect and risk aversion, elasticity of labor supply, and the firms' markup affect the current account response to monetary shocks. We checked how these factors will affect the optimal monetary policy to the 1% technology shock, and found similar results, in that the expansionary monetary supply will increase the household's welfare. Nev-ertheless, the current account response to the optimal monetary policy does change with different factors. Figure 1.9 shows that with ^ > p, the current account will actually go into deficit with the optimal monetary policy. In general, the optimal monetary policy welfare analysis can still be carried out with different parameters and the optimal monetary policy is seen to have a big impact on the current account dynamics. The optimal monetary policy can either amplify or dampen the initial responses of the current account. In this case, the current account does not respond efficiently to the technology improvement alone. Optimal Monetary Policy with Commitment In this discretionary regime, the monetary authority can print more money in response to the technology improvement and expand the money supply in surprise to the public. Because people understand the monetary authority's incentives in our perfect foresight model, however, this type of surprise and the resulting benefits cannot arise systematically. The monetary authority cannot systematically move the small open economy away from the natural rate of output in the long-run. To have a time-consistent optimal monetary policy, the monetary authority must follow a commitment policy. For a time consistent monetary policy, the monetary authority should not be able to exploit the firms' pricing behavior. In our welfare analysis, we use firm subsidies to eliminate the monopolistic distortion in the non-traded 22 goods sector. T h e non-traded goods firms s t i l l set their prices over a markup, but get a lump-sum subsidy from the government, each period, to compensate for the monopolis t ic dis tor t ion. G i v e n the firm subsidy, the monetary authori ty 's incentive to exploit the firms' pr ic ing decisions is removed. In the smal l open economy, after we use the firm subsidy to eliminate the markup distort ion, the only dis tor t ion i n the model is the.sluggish price adjustment i n the non-traded goods sector. W i t h the technology improvement i n the traded goods sector, the real economy is expanding, and theoretically, the monetary authori ty can increase the monetary supply to match the real economy expansion. T h e increase of money supply in response to the expansion of the real economy w i l l be welfare improving, i n theory. In our smal l open economy, however, the technology improve-ment is temporary, and the price stickiness is only i n the non-traded goods sector, so the welfare gain from a expansionary monetary pol icy is very smal l i n our model . Now, no monopolis t ic compet i t ion exists in the non-traded goods sector, and the benefit for the monetary authori ty to use the surprise money expansion disappears. T h e exist ing dis tor t ion in the economy is then only the slow price adjustment i n the non-traded goods sector. T h e monetary authori ty w i l l not be able to exploit the non-traded goods firms' pr ic ing decisions, which amounts to a commitment monetary policy. T o see whether or not the monetary pol icy can increase the household's welfare w i t h only one dis tor t ion - price rigidity, we use firm subsidies to get r id of the markup dis tor t ion i n the non-traded goods sector. We assume that the government runs balanced budgets. T h e government taxes non-traded goods firm revenues at a rate that compensates for monopoly power i n the zero-inflation steady-state, and removes the markup over the marginal cost charged by firms i n a flexible-price world. T h e output of the non-traded good firm, Fjvt(*)^vt(*)) * s taxed at a tax rate of r . T h e tax rate is determined by 1 — r = ^ z ^ , which gives r = — p r f • Because A > 1 , the tax rate is negative, firms receive a subsidy on their revenues and pay lump-sum taxes Tft = TPNt(i)YNt(i). T h e profits of the non-traded good firm becomes TrNt(i) = ( 1 - T)Pm(i)Ym(i) + Tft -WtHNt(i) - 6-In this case, no monopolis t ic dis tor t ion exists in the non-traded goods sector. T h e 23 PmW Pm-i(i) Pm{i)Ym{i) non-traded outputs are already at the op t imal level, so the monetary authori ty does not have an incentive to generate a surprise monetary expansion. T h e opt imal mon-etary pol icy only increases the household's welfare by a negligible amount. W i t h only price r ig idi ty dis tor t ion i n this economy, the op t imal level of the money injec-t ion parameter a\ is equal to 0.01. U p o n the unforeseen 1% technology shock i n the traded good sector, the monetary authori ty can increase its money supply by 0.01%, T h i s op t imal monetary pol icy w i l l increase the household's welfare by a negligible amount. T h e op t imal monetary pol icy also affects the current account response, but the pol icy has a smal l impact on the in i t i a l current a c c o u n t / G D P responses at the t ime of the shock. Figure 1.10 shows the current account responses w i t h the op t imal monetary pol icy under the basic cal ibrat ion. T h e intratemporal effect and risk aver-sion, elasticity of labor supply, and the firm markup do not affect our welfare result. A s the op t imal monetary pol icy now is basically "do-nothing quant i ta t ively", the opt imal monetary pol icy has a m i n i m u m impact on the current account dynamics. T h e current account responds to the unexpected technology improvement quite efficiently, even wi thout the op t imal monetary policy.. Thus , when we remove the monopolis t ic markup dis tor t ion in our economy, we find that, theoretically, a role s t i l l exists for the op t imal monetary pol icy i n current account adjustment. Quant i -tatively, however, this role is very smal l . Thus , we conclude that, at least in relat ion to product iv i ty-dr iven movements in the current account, the current account ad-justment is quite efficient i n the.absence of compensating monetary policy. F r o m a pract ical point of view, this suggests that l i t t le reason exists to argue that central banks should t ry to adjust exchange rates or interest rates to affect net exports or the current account i n smal l open economies. We investigated whether or not the non-zero in i t i a l net foreign asset w i l l affect our results here. W i t h either in i t i a l debt or asset, we get the same result for our welfare analysis. We also explored the case when we have st icky prices i n the non-traded goods sector and local currency pr ic ing, and the same result was found. 24 1.4.4 Current Account Dynamics with Interest Rate Rule In this section, we investigate the current account dynamics w i t h a popular mon-etary pol icy - an interest rate rule. Here, we describe the current account responses and welfare comparison under several interest rate rules that stem from the work of Taylor (1993). In C h a r i , Kehoe, and M c G r a t t a n (2002), the authors show that any interest rate rule can be interpreted as a money growth rule and vice versa. In practice, however, some simple interest rate rule can be a very good approximat ion of the monetary policy. Thus , we consider the implicat ions of the interest rate rules as being similar to the rules studied by Taylor (1993) and C la r ida , G a l i , and Gert ler (2000). Fol lowing the recent l i terature (Woodford 2003 and C l a r i d a et al . 2000), i n abstracting from the details of the monetary mechanism, we s imply assume that the monetary authori ty is commit ted to a domestic targeting rule w i t h interest rate smoothing. In p a r t i c u l a r w e assume that nominal interest rates it are a function of lagged nomina l interest rates, inflation rates, and exchange rate changes, according to: •{1 + it) = {l + it-i)pr where -^-^ is the inflation from t — 1 to t; and i* is the steady-state world interest rate. T h e parameter js represents the degree of exchange rate f lexibi l i ty or the coefficient of exchange rate intervention. A s long as 7 S > 0, a determinate equi l ibr ium value is present for the nominal exchange rate. 7 S is exogenously given and measures the preference of pol icy makers. W h e n 7 S approaches zero, it represents a flexible exchange rate regime. pr measures the.degree of interest rate smoothing, and ert is a normal ly dis tr ibuted, mean-zero shock. We set pr = 0.79 and 7 P = 2.15 following C la r i da , G a l i , and Gert ler (2000). In this smal l open economy, no capi ta l control means that we cannot have autonomous monetary policies w i t h the fixed exchange rate. Nevertheless, we can s t i l l analyze the cases having different levels of f lexibi l i ty i n the exchange rate movements. W i t h the non-linear solut ion method, it is accurate to compare the welfare results under different interest rules. It is also possible to incorporate large interest rate 25 (I + 0(TA-) 7 P - 1 t-i ' shocks for our welfare analysis. Technology Shocks W h e n we use the Taylor rule i n our model , the positive tech-nology shock generates similar responses for the current account, consumption, and output as was the case w i t h the money supply rule. W i t h our basic cal ibrat ion, the magnitude of the current account response is 0.55% of the current per iod G D P upon the shock. Figure 1.11 shows the responses to an unexpected 1% increase of the traded good sector technology. Interest Rate Monetary Policy Shocks Figure 1.12 shows the current account dynamics to an unexpected 50% negative interest rate pol icy shock (The nominal interest rate decreases from 0.01 to 0.005). W i t h a lower interest rate, the returns to product ion decline and the output decreases. W i t h this looser monetary pol icy shock, consumption increases and the current account goes into deficit. For the welfare analysis, we find that w i t h lower js (more f lexibi l i ty i n the exchange rate) i n the interest rate rules, the household's welfare increases w i t h the expansionary monetary policy. T h e current account responds less w i t h the more flexible exchange rate since some of the shock is absorbed by the changes i n prices. A higher weight on inflation (higher 7p = 2.15) i n the interest rate rule w i l l also generate higher welfare w i t h the expansionary monetary pol icy shock. Figure 1.13 shows the current account responses to unexpected 50% and 200% positive interest rate pol icy shocks, respectively (The nominal interest rate increases from 0.01 to 0.015, and to 0:03). We also explore the cases w i t h firm subsidies and non-zero in i t i a l debt levels, and obtain similar results. 1.5 Conclusions In this paper, we show that the current account dynamics depend on the pa-rameter values of inter temporal and infratemporal elasticities of subst i tut ion, the elasticity of labor supply, the degree of monopolist ic compet i t ion, the in i t i a l net for-eign asset posit ion, and the types of shocks. W i t h flexible traded good prices and only slow price adjustment i n the non-traded goods sector, the current account re-acts quite efficiently to technological shocks i n the smal l open economy. T h e welfare gain for households from adopting op t imal monetary policy, in contrast to having 26 a constant money growth rule, is quant i ta t ively negligible for zero/non-zero in i t i a l debt/asset positions. For a smal l open economy, the op t imal monetary pol icy has an insignificant effect on the welfare of the economy and current account dynamics. 27 Table 1.1: Baseline Ca l ib ra t ion Parameters for Chapter 1 Variable Value Economic Meaning a 0.5 Share of non-traded goods in the consumer price index a 0.7 Labor share in non-traded goods production 7 0.3 Labor share in traded goods production P 0.75 Elasticity of substitution between non-traded and traded goods a 2 Inverse of the intertemporal elasticity of substitution . 0.99 Subjective discount factor A 7.66 Elasticity of substitution between varieties of non-traded goods 0.5 Inverse of the elasticity of labor supply € 1 Inverse elasticity of money demand for household Figure 1.1: A 1% Temporary W o r l d Interest Rate Increase w i t h Different Ini t ia l Net Debt Levels (Flexible Prices) 0 2 4 6 . 8 10 12 0 2 4 6 8 10 12 -el 6 • • • • - 1- - i l • u ^ . , • I 0 2 . 4 6 8 10 12 0 2 4 6 8 10 12 28 Figure 1.2: A Posi t ive 1% Technology Shock in the Traded G o o d Sector w i t h Differ-ent Intertemporal Elast ici t ies of Subst i tu t ion (Flexible Prices) Figure 1.3: An Unexpected 1% Money Supply Increase with Different Intertemporal Elasticities of Substitution (Sticky Prices) CA/GDP % o -0.1 -0.2 -0.3 -0.4 -0.5 " — — — M M o=2 , p=0.75 • - a=0.5, p=0.75 30 40 0.2 0 0.5 40 30 Figure 1.4; A n Unexpected 1% M o n e y Supply Increase w i t h Different Elast ici t ies of Labor Supply [a = 2, p = 0.75] (St icky Prices) 31 Figure 1.5: A n Unexpected 1% Money Supply Increase wi th Different F i r m Markups [a = 2, p = 0.75] (St icky Prices) 3 2 Figure 1.6: A n Unexpected 1% M o n e y Supply Increase w i t h Different F i r m Markups [a = 0.5, p = 0.75] (St icky Prices) 33 • F igure 1,7: A 1% Posi t ive Technology Shock in the Traded G o o d Sector w i t h Basic Ca l ib ra t i on (St icky Prices) 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 34 Figure 1.8: O p t i m a l Mone ta ry Po l i cy W i t h N o F i r m Subsidy [cr 35 Figure 1.9: O p t i m a l Mone ta ry Po l i cy W i t h N o F i r m Subsidy [a = 0.5, p = 0.75] 0 20 40 60 80 .100 0 • 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 36 Figure 1.10: Optimal Monetary Policy With Firm Subsidies (Basic Calibration) 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 37 Figure 1.11: A Posit ive 1% Technology Shock i n the Traded G o o d Sector w i t h In-terest Rate Rule (St icky Prices) . CA/GDP% 0.6 | • : - . • . 1.5 0 10 20 30 40 0 10 20 30 40 38 Figure 1.12: An Unexpected 50% Interest Rate Decrease (Sticky Prices) CA/GDP% 10 20 30 40 10 20 30 40 39 40 Chapter 2 Current Account Dynamics and Opt imal Monetary Pol icy in a Two-Count ry Economy 2.1 I n t r o d u c t i o n T h e U S current account deficit to G D P ratio reached an unprecedented level of 5.7% i n 2005. W i t h the U S being a large country in the global economy, this huge current account deficit is a t t ract ing new attentions i n studies of the current account. T h e sustainabil i ty of the U S current account has been discussed substantial ly i n the l i terature . 1 To study the dynamics of a large country such as the U S , however, we need a two-country economy model . Whether monetary policies can have any impact on the current account move-ments is also an important research topic. Lane (2001a) shows that monetary shocks are a significant source of variat ion in the external account, and a surprise monetary expansion can generate a persistent external account surplus. A s monetary shocks can have a great impact on current account dynamics, a role may exist for the monetary authori ty in regards to the current account dynamics. Extensive research has been done on the open macroeconomics li terature on op t imal monetary policies w i t h country-specific p roduc t iv i ty shocks. A notable feature in this l i terature is the lack of study on how the op t imal monetary policies w i l l affect the current account movements. If some part of the current account movements can be a t t r ibuted to Please refer to Clarida (2006) for an overview, and Engel and Rogers (2006) and Obstfeld and Rogoff.(2005) for different views. 41 the op t imal monetary policies by the monetary authority, we would not need to be concerned about the current account balances/movements. To address the question of how the op t imal monetary pol icy w i l l affect the current account movements, we introduce the current account into a s tandard two-country general equi l ibr ium model w i t h st icky prices. A s the current account plays a crucial role i n the transmission of shocks, according to Obstfeld and Rogoff (1995a), it is important to take the current account dynamics into consideration when s tudying monetary policies. Thus , in this research, we use sluggish price-adjustment and monopolis t ic compet i t ion to introduce money into the model and study the impact of monetary pol icy on current account movements. A s this is a two-country model , we can focus on international spillovers and macroeconomic interdependence in the world economy. Moreover, we use a non-linear solution method, which allows for a rigorous welfare analysis, by providing an index of social welfare - the expected u t i l i ty of the representative agent i n Home and Foreign countries. In the literature, two lines of research are related to this paper. The first line of research uses new open economy macroeconomic models, following Obstfeld and Rogoff (1995a), to study the factors affecting the current account movements w i t h monetary shocks. These research studies show that the in i t i a l current account re-sponse to an expansionary monetary shock depends on the values of the elasticity of subst i tut ion between the consumption of traded and non-traded goods (the " in-fratemporal effect") and the risk aversion measure (the "intertemporal consumption smoothing effect"), the degree of monopolis t ic compet i t ion, and the degree of lo-cal currency pr ic ing ( L C P ) . These current account responses are different from the smal l open economy model of the previous chapter. In a smal l open economy model , Lane (2001a) shows that the spillover between traded and non-traded consump-t ion is always negatively correlated when the "infratemporal effect" dominates. In our two-country model , however, the spillover between traded and non-traded con-sumpt ion is negatively correlated when the "infratemporal effect" dominates w i t h a technology shock, but is most ly posi t ively correlated w i t h an expansionary mone-tary shock. T h e difference w i t h the monetary shock from the smal l open economy model is main ly from the relative future traded goods price changes i n the economy. 42 A s i n Lombardo (2002), we also show that i n the two-country economy, changes i n the monetary supply w i l l generate changes in the goods price dynamics. T h i s w i l l affect the markup, and the current account w i l l respond to the monetary shock i n different ways when the markups are different. Bet ts and Devereux (2000a) show that, in a two-country world , monetary pol icy can have different impacts on the current account w i t h pricing-to-market. We show how the price setting in home or foreign currencies w i l l affect the current account dynamics. A s found by Devereux (2000), when a l l export prices are set in the producers ' currency, the current account response is. dominated by the "expenditure-switching" effects of the exchange rate change. W h e n al l export goods are priced under pricing-to-market, the response of the current account depends on the "intertemporal consumption smoothing" effect. The second line of research explores the op t imal monetary pol icy response to domestic technology shocks i n an open economy. Obstfeld and Rogoff (2002) argue that a monetary policy, i n which the monetary authorities respond solely to their domestic shocks, delivers the best possible outcome. Corset t i and Pesenti (2004) show that, w i th local currency pr ic ing, a common monetary pol icy is the op t imal choice for a l l countries. Devereux and Enge l (2003) find that op t imal monetary pol icy requires a fixed exchange rate under L C P pricing. Thus , home and foreign monetary authorities respond identical ly to country-specific p roduc t iv i ty shocks. If prices are set i n the currency of producers, and the pass-through from exchange rate to consumer prices is complete, then the flexible exchange rate is a central part of the op t imal monetary policy. Nonetheless, none of the above studies i n the op t imal monetary pol icy li terature analyze the impact of the op t imal monetary pol icy on the current account responses. Our ma in contr ibut ion is the welfare analysis for the op t imal monetary policies i n this two-country economy, and the investigation of how the op t imal monetary policies w i l l affect the current account movements i n the home country. Engel and Rogers (2006) show that the U S may have had more favorable technology improvements than the rest of the world in recent years, which may contribute to the current account deficit. In our two-country model , therefore, we allow the op t imal monetary pol icy to respond to an unexpected technology improvement shock in the home traded goods 43 sector. In our welfare analysis, we consider the case where most of the firms use local currency pr ic ing. T h e exchange rate pass-through is low. T h i s has been shown empir-ical ly by Enge l (1993), Engel and Rogers (1996), Enge l (1999), and many others. We find that, w i t h st icky prices and monopolis t ic compet i t ion, given a monetary supply rule, the no-commitment home ind iv idua l op t imal monetary pol icy i n response to an unexpected positive home technology shock in the home country traded goods sector, is to increase the money supply for the home country when foreign monetary pol icy is absent. T h i s op t imal monetary pol icy w i l l increase the home household's welfare at the expense of the foreign household's welfare w i t h the terms of trade improving for the home country. The foreign ind iv idua l op t imal monetary pol icy is to increase the foreign money supply when the home monetary pol icy is absent. These two in-d iv idua l op t imal monetary policies can improve the implementing country 's welfare through three channels - by matching the expansion of the real economy; exploi t ing the firms' pr ic ing decisions w i t h markup distortions; and w i t h favorable terms of trade changes. We also find that the op t imal monetary policies of a supranat ional monetary authori ty is to increase the money supply i n both home and foreign coun-tries symmetr ica l ly and by a large amount. W i t h low exchange rate pass-through, the best course of act ion for the supranational monetary authori ty is to avoid any asymmetric pol icy response to the asymmetric unexpected positive home technol-ogy shock. T h e real exchange rate is stable, the terms of trade between the home and foreign country do not change, and welfare improvement is symmetr ic for both countries w i t h the op t imal supranat ional monetary policy. These op t imal monetary policies have a large impact on the current account in i t i a l responses to the home technology improvement. We also show that, in the case of no monopolist ic compet i t ion, w i t h only st icky prices, a supranational authori ty 's op t imal monetary pol icy is to increase the home and foreign money supply by a quanti tat ively smal l amount. A home pol icy maker, however, w i l l s t i l l choose to increase home money supply to improve home welfare at the expense of the foreign country. W i t h the home monetary injection, the home price level w i l l increase, which w i l l cause the home country to have more favorable 44 terms of trade. A s for the foreign consumer, w i t h a higher home price level, the foreign consumer w i l l consume less home produced goods and work more to satisfy the increased demand from the home consumer. Thus , the foreign consumer w i l l consume less and work more, and end up w i t h lower welfare. The ind iv idua l op t imal monetary policies can improve the implementing country's welfare through two channels - by matching the expansion of the real economy, and through favorable terms of trade changes. T h e current account s t i l l responds to the ind iv idua l op t imal monetary policies and the response may be substantial , in contrast to what was shown i n the previous chapter. T h e supranational op t imal monetary policy, however, only requires smal l monetary injections to the economy. A s no markup distort ion is present, the supranational op t imal monetary pol icy is time-consistent. In this environment, the current account response to the op t imal monetary pol icy of the supranat ional monetary authori ty w i l l be quanti tat ively small . T h e detailed structure of the work is as follows. T h e two-country model is pre-sented in Section 2.2. Section 2.3 discusses the cal ibrat ion, and Section 2.4 presents the current account dynamics to shocks. Section 2.5 studies the current account dy-namics in response to the op t imal monetary policies. F ina l ly , Section 2.6 concludes the chapter. 2.2 Structure of the Model 2.2.1 General Features T h i s section reviews the ma in bui ld ing blocks of the model . To capture the dynamics of the current account, we set up a perfect foresight two-country general equi l ibr ium model w i t h incomplete financial markets. T h e countries are denoted as home and foreign. E a c h country has a representative household, firms, a monetary authori ty and a domestic government. There are two sectors i n each country - non-traded goods sector and traded goods sector. Foreign variables are marked by an "*", and by an / subscript when necessary. 45 2 . 2 . 2 Market Structure There are two types of goods in the economy: non-traded and traded goods. F i n a l traded goods i n this economy (Yrt) are produced by aggregating over a cont inuum of home goods indexed by i 6 [0,1] along w i t h aggregating over impor ted foreign goods indexed by i € [0,1]. T h e aggregation technology for producing final traded goods is: Yrt — [b*YTZt-+(l-b)*YT;t where 9- > 0 is the constant elasticity of subst i tut ion between home and foreign produced traded goods and b is the share of home traded goods i n the traded goods price index. YTM is an index of the home produced traded goods, Yr/t is the home imported of the foreign produced traded goods. T h e non-traded good, home and foreign produced traded goods are in tu rn defined as: i r-l Y_Nt — Y, ~;*di Uo Yn Tht Uo Yn Tft VTht(i)1 'xdi yrft(i)1 *di T h e cont inuum of home produced non-traded goods are indexed by i £ [0,1]. A l l goods are perishable. Y^t(i) (yrht(i), VTft(i)) denotes date t non-traded (home pro-duced, foreign produced traded) good i and A > 1 denotes the elasticity of substi tu-t ion among the differentiated goods. F i n a l traded goods producers behave competit ively, max imiz ing profit each pe-r iod: Max UTt = PrtYrt-PrhtYvht-PrftYrfu where Prt is the overall price index of the final traded good, Prnt is the price index of home goods and Prft is the price index of foreign goods, a l l 'denominated in the home currency. T h e price indices are defined as PTt=[bPT-hi+(l-b)PT^ . 46 Pi Nt Pm(i)l-Xdi i l - A It is assumed that fraction s of firms exhibit local currency pr ic ing (pricing-to-market P T M ) , i.e., they set the price of goods in the currency of the buyer. A n d the remaining fraction 1 — s of firms exhibi t producer currency pricing, pxt is the home currency prices, while p*Tt is the foreign currency price. R, Tht PTht{if~Xdi+ I p T h t { i ) l - x d i Uo Prht(i) represents home currency price of the home produced good. Prft -Uo PTftii)1 xdi + f {etp*Tft(i)) J s l - A di where Prft(i) is home currency price of a foreign P T M good i, while pTjt(i) is foreign currency price of a foreign n o n - P T M good. T h e exchange rate ( home unit cost of foreign currency) is given by et. A n d the price index of home exports may be expressed by: rTht — P*Tht(iy-xdi+ / .(-PTht(i)y-*di 0 Js et where pf / i tW 1 S f ° r e i g n currency price of a home P T M good i, while PTht{i) is home currency price of a home n o n - P T M good. G i v e n the aggregation functions above, demand w i l l be allocated between home and foreign goods according to: ,P, YTft = (l-b)(-^)-eYTt *Tt Yn Tht b(^reYTt iTt w i t h ind iv idua l demands for non-traded and traded goods are given as: YNt(i) VThtii) = PNt PThtii) Y, Nt Tht Yn Tht. VTft(i) PTfM Tft Yn Tft for i = 0,...,s 2/T/fl ^  = e*Pr /t(*)' Tft Yn Tft for i = s , 1 Analogous conditions apply to the foreign country. 47 2.2.3 F i r m Behavior T h e firms hire labor Ht at the nominal wage rate Wt, and labor is immobi le internationally. Reset t ing prices is assumed to be costly because of quadratic menu costs. T h e per iod t product ion functions of the firms producing non-traded good and traded good r a r e concave and given by: Ym(i) = AmHm[i)a YTt(i) = ATtHTt(iy where Ym and Yrt are the firms' outputs and a < 1,7 < 1, while Am and Axt are period t labor productivi t ies . ANt and Art are exogenous random variables and subject to shocks. 2 In the non-traded goods sector, w i t h st icky prices, firms choose PNtji) to |2 • Pm(i)YNt(i) s Max nNt(i) =-pm{i)Ym(i) - WtHNt(i) - -where price adjustment cost is f / • \ n 2 PNtW _ J Ym{i)- T h e parameter 5 measures _ p j v t _ i ( i ) the degree of price stickiness. T h e higher is 5, the more sluggish is the adjustment of nomina l prices. If 5 = 0, then prices are flexible. T h e producer of non-traded good i maximizes 00 ftm(i) = y^^t,t+jKNt(i) 3=0 where C~° P *h,t+j = P — ' is the pr ic ing kernel used to value date t + j pay-offs. A s firms are owned by the representative household, it is assumed that firms value future payoffs according to the household's inter temporal marginal rate of subst i tut ion i n consumption. T h e firms' pr ic ing decision is given by: n , \ v , . y , XWtHm(i) \ pNt{i) (1-A) Pm(i) [l-X)YNtW + 777 0< 777 T,—( — T - r - 1 ) a PNtW [pm-iW 2 pm-iW ( -pr - l)YNt{i) + oS2*,*+i( 7-7 1) 2 Ym+iji) = 0 Pm-iW Pmji) PmW 2In this perfect foresight model, ANI and Art are assumed to be unity at steady-state. 48 In the domestical ly produced traded goods sector, the pricing-to-market firms choose PTht(i) and PxkSf) to Max 7TTt(i) = prht(i)yTht(i) + etpTht{i)yTht{i)* - WtHTt(i) 2pTht-i{t) 2 P T W _ J W N o n - P T M firms choose prht{i) and prhti}) to Max ,7rrt(i) = PTht{i)yTht{i) + PTht{i)yTht{i)* - WtHTt(i) * PTht-l[l) 2 PTht-l{l) The op t imal price setting rule for domestic sales of a l l home firms (i = 0 , 1 ) is: (1 - AJVThtW + -WtHTt(i) ' - 77r-<H F \ - U T^VThtW . 7 ATtPThM PTht-iW PThi-iW r> (- - 1) VrhtW + oflt,t+i( T T T - - 1) 2 . . . yrht+iW = 0 2 PTht-iW PThtW • Prhti1) The opt imal price setting rule for exports for P T M - f i r m s (i = 0 , s ) is: The op t imal price setting rule for exports for non P T M - f i r m s (i = s , 1 ) is: ( i - A ) ^ ( i ) + _ ^ (JWI _ i) « ^ L ^ ( 0 7 ATtPThtW . Prht-iW PTht-\Vi) - — ^ 5 — ( - 77T- - i ) yThM + oiit,t+i{——TTT - i —2—rr-vrht+iW = o ^ PTht-lW PThtW PThA1) If prices are perfectly flexible, the price-setting is a markup over the marginal cost. However, in the presence of price adjustment costs, price-setting deviates from the simple markup w i t h addi t ional forward-looking and backward-looking terms. Pr ice stickiness generated through adjustment costs allows al l firms to reset prices i f the costs of price stickiness become large. 2.2.4 Household Behavior T h e two countries are symmetric . T h e "Home" country is modeled directly, and the "Foreign" country can be derived using the same method. The home economy 49 is populated by a cont inuum of consumers/households of measure unity. T h e rep-resentative household is endowed w i t h a certain amount of t ime, which is d iv ided between leisure and work. She consumes two types of goods - traded and non-traded goods. T h e household can hold three nominal financial assets: non-interest bearing home money M, and two types of non-contingent bonds, one denominated i n home currency, B^, paying returns i, and the other denominated in foreign currency, Bf, paying returns i*. She gets income from labor income W, profits from domestic firms and lump-sum government transfers, and pays back interest on the one-year bonds. T h e household derives u t i l i ty from consumption C ( , and supplying labor Ht lowers ut i l i ty. For simplici ty, real money balances ^ are also introduced i n the u t i l i ty function. Preferences are addi t ively separable i n these three arguments. T h e lifetime u t i l i ty of the home representative household is: max < I t=o oo I - a l - e \ P t J l + ip (2.2.1) where 3 € (0,1) is the discount rate, a is the inverse of intertemporal elasticity of aggregate consumption, e is the inverse of consumption elasticity of money demand for households, ip is the inverse of the elasticity of labor supply. Supply ing labor lowers uti l i ty, x and rj are scale factors. T h e consumption index Ct is a C E S aggregate of traded and non-traded goods: Ct where p > 0 is the constant elasticity of subst i tut ion between traded and non-traded goods and a is the share of non-traded goods i n the consumer price index. Cm is an index of consumption of the non-traded goods, Crt is the consumption of the traded goods. T h e household can provide different types of labor services. There exists a con-t inuum of labor types, indexed by i G [0,1]. Let Ht(i) denote the number of hours of type i labor. T h e variable Ht that appears i n the u t i l i ty function is defined as a sum of labor services i n non-traded and traded goods sectors: H t = [ (HNt(h) + HTt(h))dh Jo 50 Analogous conditions apply to the foreign country. The household's budget constraint in period t is: Bht + etBft + ^et(Bft-Bf)2 + Mt + PtCt = (1 + i t - O f i h t - i + et(l + iU)Bft-i + M t _ ! +Tt + WtHt + Ut (2.2.2) where %g-et(Bft — Bf)2 is the household's convex costs of adjusting foreign currency asset holding. Bht and Bft are stock of home and foreign currency bond holdings that become due i n per iod t. Bf is the long-run level of foreign currency bond, holding, and fa is a constant parameter defining the portfolio adjustment cost function. it-\ and are the nominal interest rate on the home and foreign bonds. T h e household owns the firms and n t is the profit from the firms and Tt is the government lump-sum transfer. T h e domestic households take Bf as given when deciding on the opt imal holding of the foreign assets. T h e term ^et(Bft — Bf)2 can also represent a country-specific interest rate premium. Introducing the risk premium term as a function of debts forces wealth allocations in the long run to return to their in i t i a l d is t r ibut ion. Such a term is common device i n macro models . 3 In practice, the parameter (fid w i l l be set at a very low level i n cal ibra t ion of the model . T h e household problem implies the following opt imal i ty conditions. F i r s t , house-holds w i l l smooth consumption across t ime periods according to: 1 = /?(! + * t ) § ^ ~ (2-2-3) Households prefer marginal uti l i t ies to be constant across t ime periods, unless a rate of return on saving exceeding their t ime preference induces them to lower con-sumpt ion today relative to the future. Second, household's op t imal money demand schedule is: xxi) = c r [ 1 ~ p K ^ c ^ ] ( 2 - 2 - 4 ) T h i r d , op t imal portfolio choices i m p l y the interest rate par i ty condit ion: (2.2.5) e t+i 1 + i*t et (l + tt)(i + (fid(Bft-Bf)) 3Please see Kollmann (2001), Kollmann (2002), McCallum and Nelson (2000) and McCallum and Nelson (1999) for details. 51 Fourth, households supply labor to the point that the marginal disutility of labor equals its marginal product: ^ = VC?H? ' (2,2.6) The home government issues the local currency, has no expenditures, and runs a balanced budget every period. The nominal lump-sum transfers from seigniorage revenues are then given by: T, = M,-We consolidate the public and private sectors to determine the resource constraint for the home economy. The current account may be computed in this model as: CAt - - —— it-xBht-\+ <h&-\Bft^-\ + etPThtY^ht - PT1tYTft 2 Pt Pt ' * ' net exports The current account is determined by the interest payment for past debt and net exports. The real exchange rate is: qt = ^j^-- Hence, an increase (decrease) in the real exchange rate represents a real depreciation (appreciation). The home goods market clearing conditions are: domestically produced non-traded goods are all consumed domestically, domestically produced traded goods are consumed by both domestic and foreign consumers: YTt = yTht + yTht. -(2.2.8) The home bonds market clearing conditions are: Bht + B*ht = 0 (2.2.9) Equilibrium is a set of 33 equations determining 33 sequences: Bht, Bjt, B*hVl Bjt, et, it, i*t, Wt, Wtl Ct, CI HNU H*Nt, HTU HTt, YThu YTft, Y*hv Y*ft, Pu P* and all the prices for the individual traded and non-traded goods. The 33 equilibrium condi-tions are: the definition of demand conditions for non-traded goods and home/foreign produced traded goods, the overall price index, the home consumed traded goods 52 price index (2.2.1), the price index for imports , the four price setting rules for do-mestic firms, the money demand condi t ion (2.2.4), labor supply condi t ion (2.2.6), the Euler equation (2.2.3), the interest pari ty condi t ion (2.2.5), market clearing con-ditions for goods (2.2.8) and bonds (2.2.9), along w i t h foreign counterparts for each of the above condi t ion and the balance of payments constraint (2.2.7). The equi l ibr ium conditions are a l l used i n the form of non-linear equations. We use a l l the equations to solve for trajectories of the 33 sequences. In our two-country model , the foreign bond holding adjustment cost ensures the stat ionari ty of the model . T h e cost ^et(Bjt — Bf)2 captures the s i tuat ion when the economy-wide holdings of foreign-currency denominated assets are different from the steady-state level, Bf, ind iv idua l agents have to pay extra cost. W h e n the equi l ibr ium aggregate level of foreign debts in the economy is equal to Bf, this cost w i l l disappear. T h i s yields unique steady-state levels of real variables. 2.3 Baseline Calibration We use a non-linear solut ion technique to evaluate the welfare of the representa-tive household. The solved model is also used to extract impulse responses subject to technology and monetary shocks. L i p t o n et al. (1982) give a detailed discussion. Our ca l ibra t ion serves to i l lustrate the properties of the model and is not intended to match any part icular pair of economies. We assume that Home and Foreign are two countries equal i n size. T h e elasticity of subst i tut ion between non-traded and traded goods is set at 0.75, which follows direct ly from Os t ry and Reinhar t (1992). T h e elasticity of subst i tut ion between home and foreign traded goods 0 is set to 2, following Obstfeld and Rogoff (2005). T h e share of local pr ic ing firms s is set to 0.9, according to estimates of Berg in (2006). To see how the local currency pr ic ing w i l l affect our welfare analysis, we also set the share of local pr ic ing firms s to 0.1 for high exchange rate pass-through. We set the rate of t ime preference at 0.01, so that the subjective discount factor is 0.99. We follow Rotemberg and Woodford (1998) i n setting the inverse of the elasticity of labor supply to 0.5, so that Tp = 0.5. T h e degree of monopolist ic compet i t ion is also taken from Rotemberg and Woodford (1998), where they set A = 7.66, which implies an average markup of 53 15%. We follow the open economy macro literature i n choosing parameter values for our basic experiments. T h e inverse of the inter temporal elasticity of subst i tut ion a is set at 2, following Backus , Kehoe, and K y d l a n d (1995). T h e value of 77 is a scale factor for labor supply, so we set it to 2.5. The share of non-traded goods i n the consumer price index a is set at 0.5. T h e share of home goods i n the traded goods aggregator, b, is set at 0.7, reflecting the 30% share of imports in G D P for the home country. We assume that non-traded goods product ion is relatively labor intensive, w i t h a = 0.7, and traded goods is relatively non-labor intensive, w i t h 7 = 0.3. T h e consumption elasticity of money demand for the household is equal to ^ in this model . Accord ing to M a n k i w and Summers (1986), this variable is very close to uni ty and hence, e is set to 1. For the price stickiness, the price adjustment cost is set at 5 = 20, which implies that 87% of the price has adjusted 4 periods after a monetary shock. B o n d adjustment cost, ip^ = 0.007, following the estimate of Schmit t -Grohe and Ur ibe (2003), and is necessary i n to negate the unit root associated w i t h the incompleteness of the asset markets. For the in i t i a l debt/asset posi t ion, we set it to zero first and then change it to non-zero to see whether the non-zero net foreign debt/asset w i l l affect the current account dynamics. Table 2.1 reports the baseline cal ibrat ion assumptions. We calibrated two types of shocks: shocks to the traded goods sector and shocks to the monetary supply. The technology shock process takes the following form: lnATt+i = /J-ilnArt + e T t + 1 (2.3.10) where ert+i is the technology shock in the traded good sector, p i=0 .42 measures the persistence of the technology shock (following Schmit t -Grohe and Ur ibe (2003)). For simplici ty, we assume that money supply is characterized i n terms of a money supply rule. T h e money supply rule is: lnMt = lnMt-i + p{lnMt_x - lnMt_2) + (-Mt (2.3.11) where p=0.5 measures the persistence of the monetary shock, and CM* is the unex-pected money supply shock. 54 2.4 Impulse Response Analysis for the Current Account In this two-country model, we explore the factors which may affect the qualitative responses of the current account to shocks. We have: 1) the intertemporal elasticity of substitution (1), which governs whether or not the household wants to consume more today or tomorrow; 2) the elasticity of substitution between traded and non-traded goods (p), which shows whether or not the consumer prefers more traded or non-traded goods; 3) the elasticity of substitution between home and foreign traded goods (0), which demonstrates whether or not the consumer prefers home or foreign goods; 4) the elasticity of substitution between varieties of differentiated goods (A), an elasticity that reveals the equilibrium markup for the firms; and 5) the share of firms that set the price of goods in the currency of the buyer s. The effects of intertemporal elasticity of substitution and the elasticity of sub-stitution between traded and non-traded goods are different from the small open economy case. The elasticity of substitution between home and foreign traded goods (ff), and the local currency pricing cannot be well discussed in a small open economy framework. In our welfare analysis, we also investigate whether or not these factors have any welfare implication. 2.4.1 Intertemporal and Intratemporal Elasticity of Substi-tution Driskill (2001) illustrates that, in a small endowment economy with non-traded goods, the key effects for transmission of shocks to the current account depends on the relative price of traded and non-traded goods. He also shows that whether or not the trade balance effect is dampened depends on the sign of the cross-partial derivatives with respect to traded and non-traded goods of the utility function. Lane (2001a) also shows that, in a small open economy, the intertemporal elasticity of substitution (1) and the elasticity of substitution between traded and non-traded goods (p) affect the current account responses qualitatively. In the small open economy, the price of traded goods is the nominal exchange rate, while in our two-country economy, the price of traded goods is a CES aggregation of the price of home produced and 55 impor ted foreign produced traded goods. In the two-country world , the effect of the existence of non-traded goods on current account is not the same as i n the smal l open economy. F r o m the household's first-order conditions, we can get: G Tt+l C-Tt = [0(1+ **)]' Prt a ' Pt ' Prt .F*Tt+l. Pt+i . Prt+i -(2.4.12) T h e consumption growth depends on the sequence of relative prices, (p^+1) ° and the 2nd term In Lane (2001a) and D r i s k i l l (2001), the consumption growth only depends on the second term pTt+l . For this term, this is the price ratio of current and future aggregate price level relative to the price of traded goods. If is currently low compared to its future value, this encourages present consumption over future consumption, i.e. Crt w i l l tend to increase. Meanwhi le , the relatively higher future traded goods price also encourages consumption subst i tut ion from traded goods to non-traded goods, i.e. Crt w i l l decrease. T h e former effect dominates if the inter temporal elasticity of subst i tut ion ~ is greater than the infratemporal elasticity of subst i tut ion p, i.e. ^ > p. Consider ing the above effects, we can see that non-traded and traded goods consumption is highly posit ively correlated when ^ > p. We also have the extra term {jf^ )° also affecting'the consumption growth. A s a increases, the intertemporal elasticity of subst i tut ion declines, agents are less wi l l ing to smooth consumption across states of nature and time. A s a result, relative consumption moves by less than the real exchange rate, so that the current account improves more. Figure 2.1 shows the result when ^ < p and ^ > p w i t h a 1% positive technology shock in the home traded goods sector. T h e technology shock follows equation (2.3.10). W i t h higher a, the current account responds less to the shock due to the agents' unwillingness to smooth consumption across t ime. W i t h the positive technology shock, home traded goods' product ion is more efficient, the real prices of the home produced traded goods has to decrease to clear the market. T h e relative price changes lead to a negative correlation between traded and non-traded goods consumption when - < p and a positive correlation when - > p. T h e impulses 56 are percentage deviations from the in i t i a l steady-states except for the current account responses. In the case of an expansionary monetary shock, the household w i l l use more money to buy consumption goods, and both traded and non-traded goods consump-t ion w i l l increase. The monetary shock follows equation (2.3.11). Now, the correla-tions between the traded and non-traded goods consumption are both positive. A s the domestic currency is depreciating, the traded goods consumption increase w i l l be less when a is high, as the household is less wi l l ing to smooth consumption across t ime. Figure 2.2 shows that when ^ < p, the current account improves following the monetary shock. W h e n a is low, the household is wi l l ing to smooth consumption, w i t h higher money holding, and w i l l increase its traded goods consumption, so the current account deteriorates. In this two-country model , the non-traded goods and traded goods consumption are not always negatively correlated when ^ < p. In the smal l open economy, the non-traded goods and traded goods consumption are always negatively correlated when \ < P- T h i s is explained by the relative future traded goods prices p laying one more role i n the traded goods consumption growth in the two-country wor ld . 2.4.2 Home and Foreign Goods Elasticity of Substitution In a model without non-traded goods, the cross-country subst i tu tabi l i ty 6 cap-tures the sensit ivity of the consumption al locat ion between home and foreign, w i t h respect to the terms of t rade. 4 T h i s shows whether goods produced i n different coun-tries are poor or close substitutes. W h e n 6 > 1, home and foreign produced goods are close substitutes i n consumption, and a positive correlation exists between the terms of trade and the current account. W h e n the relative price of impor ted goods increases, households substitute away from impor ted towards home produced goods, and w i t h other things remaining constant,, the current account improves. W h i l e 6 < 1, home and foreign produced goods are poor substitutes, the increase of the impor ted goods' relative price w i l l cause the current account to worsen. E m p i r i c a l evidence suggests that values of 9 are above unity. Hence, in the following analysis, 4The terms of trade is defined as the relative price of export and import prices fi^Tt P r i c e -57 we focus on the case where 9 > 1. 2.4.3 Local Currency Pr ic ing We have assumed that a fraction s of firms use local currency pricing when pricing their exports. Devereux (2000) shows that, when all export prices are set in the producers' currency, the current account responses are dominated by the "expenditure-switching" effects of exchange rate change. When export prices are all set by pricing-to-market, the response of the current account depends on the intertemporal "consumption smoothing" effect. In the Mundell-Fleming model, the trade balance will improve if the Marshall-Lerner conditions hold with a currency devaluation. This is an "atemporal condi-tion" that hinges on the elasticities of demand for home and foreign traded goods. As we have shown, the intertemporal elasticity of substitution (1) and the cross-country substitutability 9 affect the current account responses through the intertem-poral "consumption smoothing" considerations and the "atemporal elasticity" ef-fect. When all export prices are set in the producers' currencies, the "expenditure-switching" effects of the exchange rate change will dominate the response of the current account. On the other hand, when all export prices are set in the buy-ers' currencies, the "expenditure-switching" effect is irrelevant, since exchange rate movements will not affect the consumption prices. As the consumption prices do not change, the Marshall-Lerner conditions are also irrelevant. Now, the impact of shocks on the current account movements depends on the intertemporal consumption smoothing in consumer preferences. To demonstrate this point, we change our basic calibration to 9 = 0.75. In our calibration, we use more realistic pricing fractions. We have high local currency pricing, with 90% of the firms pricing their exports in the foreign currency s = 0.9; and low local currency pricing, with 10% of the firms pricing their exports in the foreign currency s = 0.1. Now, the impact of a monetary expansionary shock on the current account is determined by the relative strength of the "atemporal elasticity" of substitution between traded goods and the "intertemporal elasticity" of substitution between future and current consumption. The monetary shock follows 58 equation (2.3.11). W h e n both the foreign and the home country are symmetr ic (b — 0.5), and the elasticity of subst i tut ion between home and foreign produced traded goods is low, 6 = 0.75, the higher fraction s = 0.9 of firms using local currency pr ic ing w i l l cause the current account to respond more (0.1% of G D P ) , compared to the s i tuat ion w i t h a low degree of P T M (s = 0.1), which w i l l cause a smaller (0.005% of G D P ) current account improvement. W i t h higher s, the consumption smoothing effect dominates, and the expansionary monetary shock w i l l cause the current account to improve more. Figure 2.3 shows the current account responses w i t h different levels of local currency pricing. Accord ing to another cal ibrat ion, b = 0.7, the home country has a larger share in the traded goods consumption. A g a i n , lower s means a lower consumption smoothing effect. W i t h the home country 's higher share in the consumption, lower s means the "consumption-switching" effect dominates, and the current account improves more w i t h s — 0.1. F igure 2.4 shows the current account responses wi th different levels of local currency pricing. 2.4.4 M a r k u p Changes i n monetary supply generate changes i n j ip^-y dynamics, and this effect exists for a l l prices. These changes w i l l affect the markup and the current account. W i t h monopolis t ic compet i t ion, the output falls below the competi t ive level. A t steady-state, as A increases, the markup decreases. W i t h less monopolis t ic competi-t ion , the firms are more competi t ive and have higher output . The surprise monetary expansion shock w i l l increase the output and tend to correct the monopolis t ic distor-t ion . In the case of our basic cal ibrat ion, w i t h ^ < p, the current account improves upon the monetary shock. W i t h a higher A, the monopolist ic dis tor t ion w i l l be less, and the output level at the steady-state w i l l be less distorted, to indicate the con-vergence to high output level without the monopolist ic compet i t ion. T h e output expanding effect of the monetary shock w i l l be less effective w i t h less monopolis t ic dis tor t ion. T h e home traded goods consumption w i l l increase less w i t h less monopo-list ic distort ion, which w i l l be reflected by either more exports or less imports . Under 59 the specification ^ < p, w i t h higher A, the positive current account response to the monetary shock w i l l become less significant. Figure 2.5 demonstrates the current account responses to an unexpected 1% positive money growth shock w i t h different levels of markup. W i t h the lower markup, the net exports i n the current account (2.2.7) deteriorates when £ < p. T h i s is the same result as found by Lombardo (2002). Nevertheless, Figure 2.6 shows that .wi th ~ > p, the current account improves w i t h the lower markup. T h i s can be explained by ^ > p, as the home traded consumption increases, it w i l l increase less w i t h a higher A. Now, w i t h the higher A and lower markup, the home country's imports increase less, and therefore, the current account improves. F rom Figure 2.5 and Figure 2.6, we can see that the current account responds by less to the monetary shock wi th less monopolist ic compet i t ion. 2.4.5 Non-Zero Initial Debt/Asset Positions A s documented by Lane and Miles i -Ferre t t i (2001), net foreign assets over G D P vary across countries and are different from zero. In this section, we investigate whether or not the in i t i a l net debt/asset posi t ion w i l l affect our current account analysis. W e find that, w i t h either a net debt or an asset posi t ion, the result impulse responses are very much the same as in the case of no in i t i a l debt/asset. T h i s is because neither the temporary technology shock i n the traded goods sector nor the temporary money growth shock w i l l affect the interest rate enough for the interest payment to matter in this two-country economy. 2.5 Optimal Monetary Policies In this section, we use the two-country model to study the op t imal monetary pol-icy for each country and the op t imal monetary pol icy for a supranational monetary authority. We s t i l l use the technology shock as in Equa t ion (2.3.10) in the home traded goods sector. Enge l and Rogers (2006) show that the U S may have had more favorable technology improvements than the rest of the world , i n recent years, which may contribute to the current account deficit. So, in our two-country model , we have the op t imal monetary pol icy to respond to the home unexpected technology improve-60 merit shock i n the traded goods sector. We study the case where the net foreign asset is in i t i a l ly zero, then we explore whether or not the net debt/asset holdings w i l l af-fect our welfare results. We assume that the monetary authori ty receives perfect information about the current state of the economy. T h e op t imal active monetary pol icy is thus money supply responding instant ly to the technology shock. U p o n the unexpected technology shock in the home traded goods sector, the monetary authori ty immediate ly realizes what the temporary technology shock is, and sets an appropriate money supply to maximize the welfare of the household. T h e money supply rule is the same as i n Equa t ion (2.3.11). Now, wi th bo th home and foreign monetary authorities, different objectives can exist for the op t imal monetary policy. 1) T h e home monetary authori ty can choose its op t imal monetary pol icy to maximize the home household's welfare, taking the foreign monetary authori ty 's current pol icy as given. 2) T h e foreign monetary authori ty can choose its op t imal monetary pol icy to maximize the foreign household's welfare, taking the home monetary authori ty 's current pol icy as given. 3) We assume that a supranational monetary authori ty is in place. 5 It can choose the op t imal monetary pol icy for both home and foreign mon-etary supply, to maximize the home and foreign household's welfare. In our model , no interaction occurs between the home and foreign monetary authorities, and they each take the other's current pol icy as given. 2.5.1 Opt imal Monetary Policies W i t h M a r k u p Distort ion In this economy, the shock is an unexpected technology improvement i n the home traded goods sector, which is an asymmetric technology shock. W h e n markup distor-t ion is present i n the economy, the monetary authorities have perfect information on the way i n which firms set their prices. So, w i t h the technology shock, the monetary authori ty can take advantage of the price setting behavior of the firms. Due to the monopolis t ic compet i t ion i n both the traded and non-traded goods sectors, output is sub-opt imal ly low. Domest ic output is demand-determined in the short-run because of the slow price adjustment. W i t h slow price adjustments, the monetary expansion can boost the current demand for the non-traded goods consumption above the nat-5 We will study the case of the Nash game in future research. 61 ura l level of output in the short-run. T h i s is the no-commitment monetary policy, as discussed in detai l for the smal l open economy model . Next , we focus on the welfare implicat ions of the three different op t imal monetary policies. 1) The home monetary authori ty chooses its op t imal monetary pol icy to maximize the home household's welfare, taking the foreign monetary authori ty 's current pol icy as given. In our model , no interaction occurs between home and foreign monetary authorities. The home monetary authori ty can use expansionary monetary pol icy to match the real expansion of the economy. W i t h our basic cal ibrat ion, 90% of firms price their exports i n the buyers' currency, and when the foreign monetary authori ty does not respond to the home technology shock, the op t imal monetary pol icy for the home country is to increase the home money supply by 15%, which w i l l improve the home household's welfare by nearly 0.05%. W i t h this home monetary expansion, the home currency w i l l depreciate and the home country's terms of trade w i l l improve. W i t h most firms using L C P , the exports are mostly i n foreign currency, and the exchange rate depreciation w i l l cause the export prices i n the domestic currency to increase; but since the imports are mostly priced i n home currency, the depreciation w i l l not change the home impor t prices. Thus , the home country 's terms of trade w i l l improve. Par t of the home country's welfare improvement comes from its terms of trade improvement. T h i s op t imal monetary pol icy w i l l decrease the foreign household's welfare by 0.04%. W e can see that the to ta l welfare of bo th home and foreign countries is s t i l l improving w i t h this policy, and the to ta l welfare of bo th home and foreign increases by 0.004%. T h e to ta l welfare improvement comes from two channels. F i r s t , the home real economy is expanding w i t h the technology improvement, and the matching w i t h home money expansion is efficient. Second, w i t h the markup, the monetary authori ty can expand the economy by pushing the output above the natural rate level. T h i s w i l l improve the household's welfare. 2) T h e foreign monetary authori ty chooses its op t imal monetary pol icy to maximize the foreign household's welfare, taking the home monetary authori ty 's current pol icy as given. W h e n the home monetary authori ty is not responding to the technology shock, the foreign monetary authori ty can increase it 's monetary supply by 15% to 62 maximize the foreign household's welfare by nearly 0.05%. A s i n case 1), the home welfare is decreased by 0,04%, while the to ta l welfare improves by 0.004%. T h e home welfare is lower because of the unfavorable terms of trade. W i t h the asymmetric home technology shock, we can see that the home and foreign monetary authorities can have the same response. 3) A supranational monetary authori ty chooses the op t imal monetary pol icy for both home and foreign monetary supplies to maximize the home and foreign household's welfare. T h e home money supply should increase by 14% and, at the same t ime, the foreign monetary authori ty increases its money supply by the same amount, 14%. Then , the welfare of home and foreign countries each increases by 0.007% and the to ta l welfare of both countries improves by 0.007%. In our basic cal ibrat ion, the exchange rate pass-through is low: 90% of the firms set their exports in the buyers' currency. In this case, a supranational monetary authori ty has a symmetr ic monetary pol icy for both home and foreign counties. Thus , the real exchange rate is very stable. Corset t i and Pesenti (2004) show that the use of exchange rate for s tabi l izat ion purposes would entail excessive welfare costs, i n the form of higher impor t prices and lower purchasing power across countries. T h e best course of action for the supranational monetary authori ty is to use symmetr ic pol icy to respond to the asymmetric home technology shock. Figure 2.7 shows the current account responses to an unexpected 1% increase of the home traded goods technology w i t h different op t imal monetary policies. U p o n the technology shock, the traded goods sector w i l l increase its product ion since it w i l l be more efficient. W i t h our basic cal ibrat ion, the magnitude of the current account response is 0.08% of the current period G D P , at the t ime of the technology shock. W i t h home or foreign op t imal monetary policy, the current account has big movements. T h e ind iv idua l home or foreign opt imal monetary pol icy tends to inject a lot of money into the economy. A s Bet ts and Devereux (2000b) show w i t h L C P , an unanticipated home monetary expansion w i l l increase the home welfare and decrease the foreign welfare. T h e welfare improvement for the ind iv idua l country main ly comes from the terms of trade improvement following the injection of money. W i t h the injection of money, the current account has excessive movement as the result 63 of the ind iv idua l country 's op t imal monetary policy. T h i s suggests a coordinat ion gain from having a social planner. T h e social planner's op t imal monetary pol icy eliminates the terms of trade effect caused by the welfare improvement for a single country, and the supranational monetary authori ty w i l l maximize the welfare of bo th countries. T h e symmetr ic op t imal monetary pol icy from the social planner also has a big impact on the current account movements. T h e welfare improvement for the social planner's op t imal monetary pol icy comes from matching the expanding real economy and the markup distort ion. T h i s supranational op t imal monetary pol icy also has a certain impact on the movements of the current account. 2.5.2 Optimal Monetary Policies With No Markup Distor-tion A s the firm markup enables the monetary authorities to exploit the firms' pr ic ing decision, the monetary authorities can expand the economy above its long-run output level. T h i s pol icy is not time-consistent. In this section, we use firm subsidies to eliminate the markup distort ion. After e l iminat ing the markup distort ion, the monetary authorities use time-consistent monetary pol icies . 6 T h e three different op t imal monetary policies are as follows: 1) T h e home monetary authori ty can use expansionary monetary pol icy to match the real expansion of the economy. T h e opt imal monetary pol icy for home is to increase the home money supply by 15%, which w i l l improve the home household's welfare by nearly 0.03%. W i t h this home monetary expansion, the home currency w i l l depreci-ate and the home country's terms of trade w i l l improve w i t h most of the firms using L C P . Th i s op t imal monetary pol icy w i l l decrease the foreign household's welfare by 0.04%. Moreover, the to ta l welfare of bo th the home and the foreign countries de-creases by 0.0002%. T h e to ta l welfare deteriorates. Now without markup distort ion, the monetary authori ty cannot exploit the firms' pr ic ing decisions, and the home opt imal monetary pol icy causes an excess movement i n the exchange rate, which is a welfare cost i n the economy. 2) The foreign monetary authori ty chooses to increase i t 's monetary supply by 15% 6Please see Chapter 1 for a detailed discussion about firm subsidy. 64 to maximize the foreign household's welfare. T h i s w i l l increase the foreign house-hold's welfare by nearly 0.03%, and the home welfare is decreased by 0.04%, while to ta l welfare deteriorates by 0.0002%. T h e home welfare is lower because of the un-favorable terms of trade. W i t h the asymmetric home technology shock, we can see that the home and foreign monetary authorities can s t i l l have the same response. . 3) A supranational monetary authori ty chooses the following op t imal monetary pol -icy: the home money supply should increase by 0.3% and, at the same t ime, the foreign monetary authori ty increases its money supply by 0.2%. Then , the welfare of the home country increases by 0.0004% and the welfare of the foreign country de-creases by 0.0004%, while the. total welfare of both countries improves by 0.000003%. T h e welfare gain from this op t imal monetary pol icy comes from the matching of the real economy expansion. W i t h no markup distort ion, the ind iv idua l country's op t imal monetary pol icy is actual ly total welfare deteriorating. T h i s confirms the results of Corset t i and Pesenti (2004) in that the use of exchange rate for s tabi l izat ion purposes would entail excessive welfare costs. T h e opt imal monetary pol icy of the ind iv idua l home country and the foreign country is s t i l l welfare improving for the single country, as it main ly comes from the favorable terms of trade change because of the L C P . Figure 2.8 shows the in i t i a l response of the current account to a 1% home traded goods sector technology improvement, w i th and without the op t imal monetary policies. A s the home monetary expansion is welfare improving for a single country, the op t imal monetary pol icy for the home country is to inject lots of money into the home country to get a favorable welfare improvement. T h e ind iv idua l op t imal monetary pol icy s t i l l has a great impact on the current account dynamics. W i t h no markup distort ion, however, the supranational authori ty 's op t imal monetary pol icy is t ime consistent. It only requires a smal l monetary injection i n both countries to match the real economy expansion. Now, the current account responses to the home technology shock w i t h no monetary pol icy is almost the same as w i t h an op t imal monetary pol icy chosen by a supranat ional monetary authority. T h e current account response to the home technology shock is fairly efficient even wi thout any monetary pol icy from the social planner's view. 65 2.5.3 Discussions We also explore the case where the technology shock is not l imi ted to the traded goods sector. W i t h technology shocks i n bo th the traded and non-traded goods sectors, we get s imilar result for our welfare analysis. A s documented by Lane and Miles i -Fer re t t i (2001), net foreign assets over G D P vary across countries and are different from zero. In this section, we investigate whether or not the in i t i a l net debt/asset posi t ion w i l l affect our welfare analysis. The result is the same as i n the case of no in i t i a l debt/asset. We find that, w i t h either a net debt or an asset posi t ion, the welfare improvement is s t i l l quanti tat ively small from the supranational op t imal monetary pol icy w i t h no markup distortions to a positive home traded good sector technology shock. 2.6 Conclusions In this paper, we show that the current account dynamics depends on the degree of in t ra temporal and intertemporal elasticities of subst i tut ion, the degree of monop-olist ic compet i t ion, and the degree of local currency pricing. W i t h monopolis t ic compet i t ion, surprise monetary expansion responding to an unexpected technology shock, w i l l increase the home household's welfare at the expense of the foreign coun-try 's welfare, w i th most of the firms using L C P . We also study the case where a time-consistent op t imal monetary pol icy is used. W i t h only one dis tor t ion - s t icky prices, for a supranational monetary authority, the current account response to the op t imal monetary pol icy is quanti tat ively small . For a home monetary authority, however, a home monetary injection at the t ime of the technology shock can s t i l l improve the home household's welfare w i t h most firms pr ic ing their exports in the buyers' currency. 66 Table 2.1: Baseline Ca l ib ra t ion for Chapter 2 Var iable Value Variable Value Var iable Value Var iable Value a 2 a 0.5 P 0.75 S 20 7 0.3 b 0.7 e 2 A 7.66 a 0.7 0.5 e 1 X 0.005 0.007 V 2.5. s 0.9 ' 0.99 67 Figure 2.1: A n Unexpected Positive 1% Technology Shock in the Traded G o o d Sector w i t h Different Consumpt ion Smoothing CA/GDP% Reale Y* Y 0 10 20 0 10 20 0 10 20 Figure 2.2: A n Unexpected Posi t ive 1% Mone ta ry Shock w i t h Different Consumpt ion Smoothing cr. Figure 2.3: A n Unexpected Positive 1% Mone ta ry Shock w i t h Different Exchange Ra te Pass-Through [b = 0.5; 9 = 0.75] CA/GDP% Real e Figure 2.5: A n Unexpected Posit ive 1% Moneta ry Shock W i t h Different Markups [a = 2] Figure 2.8: Home 1% Technology Shock W i t h and W i t h o u t the O p t i m a l Mone ta ry Po l i cy W i t h F i r m Subsidies Chapter 3 Software Upgrade, Consumer Behavior and Software Vendor's Choice 3.1 Introduction In the last two decades, the software industry has grown rapidly. P rud 'homme and Y u (2002) show that to ta l software investments increased by almost 1400% from 1981 to 2001. W i t h the rapid growth of the software industry, decisions about software upgrade have become increasingly, important to software vendors. The ma in reasons for upgrading software, from the viewpoint of the software vendor, are as follows: 1) Software is a durable good. W i t h o u t any new functionality, exist ing consumers w i l l not buy the software again. So, "planned obsolescence", where the current software is made deliberately out-of-date due to the release of a new version, can help the software vendor to extract more surplus from exist ing consumers; 2) A s the market demand evolves, consumers need addi t ional software functions. T h r o u g h upgrades, N g (2001) shows that the software vendor can not only better serve the exist ing market but. also stimulate new demand;. 3) If operat ing system software is upgraded, Mar inoso (2001) shows that appl i -cat ion software may need to be upgraded to ensure compat ibi l i ty ; and, 4) T h e rapid growth of information technology results in faster C P U , and larger R A M and hard disks, a l l o f .wh ich supply more f lexibi l i ty for new generations of software. 76 Some constraints affect the software vendor, i n relation to software upgrades. F i rs t , upgrading software requires an extra investment and a great effort. Second, i f the old version is not wi thdrawn from the market after the new version is introduced, the old software competes w i t h the new, and can draw potential consumers away from the new version. We cal l this a "cannibal izat ion problem". If the old version is retired, the software vendor may lose those consumers who do not care about the enhanced features of the new version and would prefer using the less expensive older version. T h i r d , while such upgrades usually include many worthwhile features, software users are beginning fo resist the steady stream of upgrades. For example, Q i u (2002) estimates that there had been s t i l l mil l ions consumers were s t i l l using Windows 3 . X and Windows 95, even after Windows X P was released. Software users compla in that upgrading software is expensive as they need to spend t ime and money to learn how to instal l and use the new versions. Specifically, Bulkeley (1990) finds that the compat ib i l i ty problem of new versions of software forces users to upgrade their hardware, which can be even more expensive than the upgraded software. Fudenberg and Ti ro le (1998) argue that some consumers may choose to leapfrog the new version. Consequently, the "disappearance market" tends to grow (i.e., a group of people choose to wi thdraw from the market) as software users become less sensitive to technological progress. Obviously, for the software vendor, the problem of max imiz ing profit or market share under the above conditions or environment is a big issue. To achieve a maxi -miza t ion goal, the software vendor needs to know how consumers respond when the vendor releases a new version of software onto the market. Accordingly , the vendor needs to make the right decision on several issues involving software upgrades: 1) W h i l e the new version is released, should the old version s t i l l be kept in the marketplace? If so, for how long? 2) Should price d iscr iminat ion be undertaken, and how? Several earlier papers have investigated software upgrades from the technical viewpoint or from the view of social welfare. 1 Another stream of software research Please see Xie and Hong (1998), Xie and Hong (1999), Shinohara et al. (1997) , Ellison and Fudenberg (2000) and Fudenberg and Tirole (1998). 77 focused on software editions (static versioning), that is, delivering several versions w i t h different quali ty levels (or feature sets) such as a professional edit ion, a student edi t ion and a home edit ion at the same t ime . 2 To the best of our knowledge, to date no research has been conducted to fully and systematically investigate the relation-ship between consumer behavior and software upgrading decisions for the software vendor. T h i s s tudy contributes to the li terature by exploring this untapped area. We model the consumer behavior of software users facing software upgrading. Based on the behavior, we investigate the software vendors' choices on quali ty levels, pr ic ing strategies, customer services, and market structures. We study the behavior of software users under two market structures: Structure 1 - where both the old and new versions coexist in the market; and Structure 2 -where only the new version is available i n the market. We find that keeping the old version i n the market is an effective tool to reduce the "disappearance" of exist ing users. T h e "cannibal izat ion" effect between the old version and new version, for the exist ing users, is different from that for new users. T h i s finding is non-intui t ive since we believed that the new version could be substi tuted by the old version for bo th exist ing and new users. We also find that, when the price of the old version is low enough, the to ta l market share under market structure. 1 is greater than the market share under market structure 2. Nevertheless, even if the market share under market structure 1 is greater than that under market structure 2, the total profit under market structure 1 is not necessarily greater than that under market structure 2. We show the conditions under which the software vendor should choose market structure 1 to maximize the total profit. In addi t ion, one important finding in this s tudy is that, as software users become more heterogeneous, the difference between the two market structures diminishes, i n terms of market share and profit to the software vendor. G i v e n the lack of exist ing research on the choice of market structure for the software vendor, our work is directed towards filling the gap and provid ing some insights for the software vendors' op t imal upgrade choices. We argue that, for a new version, the software vendor can charge a lower price to existing users and a higher price to new users, under either of two market structures, providing that the 2Please see Varian (1997), Raghunathan (2000) and Haruvy and Prasad (1998). 78 software consumers are sufficiently heterogeneous. Furthermore, if the price of the new version is quite high, the software vendor can charge a lower price to existing users and a higher price to new users for the old version. T h i s finding is different from previous studies inc luding Y a n g (1997b), where the condi t ion to implement discr iminated prices to exist ing users and new users is based on the relative qual i ty of the new version compared to the old version. • The software vendor has some advantages in conduct ing upgrades, compared to physical durable goods (such as hardware) producers for two key reasons. F i rs t , software, is an information good and its marginal cost is almost zero. Second, unlike the secondhand markets for other durable goods, Fudenberg and Ti ro le (1998) show that the secondhand market for software is restricted by difficulty i n preventing dupl icat ion, and thus, copyright infringement. Legit imately, only the software vendor has the right to sell the old version i n the market. The prohibi t ion of a secondhand market for software is motivated by the difficulty in preventing dupl ica t ion and thus copyright infringement in the presence of such a market. Or equivalently, the software vendor is able to completely control the secondhand market of his products. O n the contrary, as hardware users can also sell their used items, the hardware vendor is not the only supplier of the old-model hardware i n the market. In the next section we provide a review of relevant research. In Section 3.3, we present our formal models of both exist ing users' and new users' behaviors under two different market structures. T h e op t imal choices of the software vendor are analyzed i n Section 3.4, which is followed by our discussion of results in Section 3.5. T h e concluding remarks and summary are arranged in Section 3.6. 3.2 Literature Review M a n y durable goods, such as textbooks, automobiles, and electronics have well-established secondhand markets that can dra in revenue from the vendor who is t ry ing to sell new products. M u c h prior research has been devoted to the impact of used-good markets on durable product sales of the monopol is t . 3 These studies show, for example, that a monopolist can be effective in reducing compet i t ion from such used-3Please see Miller (1974), Liebowitz (1992), Berkovec (1985) and Rust (1986). 79 good markets through "planned obsolescence" (like revised textbook editions). A n alternative to "planned obsolescence" for the monopolist is to eliminate the compe-t i t ion from product durabi l i ty by refusing to sell the product and only renting or leasing it as shown i n B u l o w (1982). Coase (1972) points out that a durable goods monopoly that sells products has less market power than a monopoly that rents the goods (known as Coase Conjecture). Shy (1995).states that "a monopoly selling a durable good earns a lower profit than a renting monopoly" . Extensive li terature can be found that debates the relationship between the de-gree of a firm's monopoly power and the qual i ty or durabi l i ty it chooses to release each t ime . 4 K l e i m a n and Oph i r (1966) and Levhar i and Srinivasan (1969) show that a monopolist has an incentive to release lower quali ty products, in comparison to what can be seen i n a competi t ive market structure. Nevertheless, Swan (1970a) and Swan (1970b) argue that such "planned obsolescence" is never actually opt imal . Swan demonstrates that the monopolist can maximize profits by setting a durabi l -i ty level equal to the competi t ive level and raising prices. T h i s result is known as "Swan's independence result" and has proven to be the impetus for several subse-quent debates. 5 A recent series of papers by W i l h e l m and various co-authors focus on the upgrading of contents and the t iming of high-tech products (hardware such as desktops and laptops) . 6 Software is a durable good that can potential ly be used infinitely wi thout physical erosion. Typ ica l ly , durabi l i ty of a product greatly affects the frequency of repeated purchase by consumers. Software users have no incentive to re-purchase the software. Upgrades or revisions, however, may be effective ways to make exist ing software "ob-solete" and provide an incentive for current users to purchase the updated version. Ex i s t i ng li terature about software upgrade can be d iv ided into three categories. The first considers software upgrades and release decisions from the technical perspec-t ive . 7 Papers i n this category main ly discuss the tradeoff between new software 4Please see a survey by Schmalensee (1979). 5See, for example, Miller (1974), Kihlstorm and Levhari (1977), Spence (1975) and Liebowitz (1992). 6Please see Wilhelm and Xu (2002), Wilhelm et al. (2003) , Damodaran and Wilhelm (2003b) and Damodaran and Wilhelm (2003a). 7Please see Xie and Hong (1998), Xie and Hong (1999) and Shinohara et al. (1997) . 80 re l iabi l i ty (time required to check software error and failure) and the software's re-lease t ime i n the market. These papers do not consider the demand side or market factors. . T h e second category of research papers considers software upgrades from the perspective of welfare economics. E l l i son and Fudenberg (2000). examine two reasons explaining why a monopolist has an incentive to introduce more upgrades than is social ly opt imal , when upgraded software is backwardly compatible. Fuden-berg and Ti ro le (1998) s tudy conditions under which a monopolist can choose to offer upgrade discounts to exist ing consumers. In Fudenberg and Ti ro le (1998), only one new update is present, and customers w i l l not "leadfrog" the new update if the marginal cost of product ion is zero. T h e th i rd category of research papers is centered on software editions or static versioning. Var i an (1997), Raghunathan (2000) and H a r u v y and Prasad (1998) conduct studies on delivering several versions w i t h differ-ent qual i ty levels or feature sets, such as a professional edit ion, a student edit ion, and a home edit ion, simultaneously. T h e purpose of static software versioning is main ly to segment the market and to take advantage of the sub-markets by price and qual i ty d iscr iminat ion. In addi t ion, H u i and Tarn (2002) investigate the op t imal levels of software functionali ty i n a duopoly market and Y a n g (1997a) discusses the op t imal upgrade t ime i n software provision w i t h network effects. In Sahin and Zahedi (2001), by using the consumer satisfaction index, the authors bu i ld a M a r k o v Decis ion P ro -cess ( M D P ) model to identify op t imal and near-optimal policies among the choices: warranty, maintenance, and upgrade of software systems to software vendors. A s far as we know, no existing research offers insights into the software vendor's decision on software upgrade, based on the behavior of software users. Our work is related to W i l h e l m and X u (2002)'s work, but is different i n several aspects. F i r s t , W i l h e l m and X u focus on the upgrade of computer hardware, which involves product design engineering, process design engineering, product planning and supply chain, and outsourcing. O u r work focuses on software upgrading, and the product ion pro-cesses and operations for software differ from those of hardware products. Second, W i l h e l m and X u assume no cannibal izat ion problem i n the upgrading process, while we investigate the cannibal izat ion problem i n the software market. 81 3.3 Modeling Software Users' Behavior for Soft-ware Upgrade When the software vendor releases a new software version, the hope is that both existing users and new users will buy the product. Thus, we analyze both existing users and new users, under the following two market structures: Structure 1 - both the old and the new versions coexist in the market, and Structure 2 - only the new version is available in the market. We perform our analysis for two sequential software upgrades. We call it the 2x2x2 model, where the first 2 refers to two categories of software users (existing and new), the second 2 refers to two market structures, and the third 2 refers to two sequential software upgrades. 3.3.1 Existing Users' Upgrade Demand Under Structure 1 We assume that the software user has the following utility function: U = U(Q, C , M) subject to the budget constraint Y — M + P, where Q > 0 is the software qual-ity which is measured by software features or functionalities; C > 0 are the other gains from using the software such as customer services (e.g., technical supports), and are functions of aggregate sales (e.g., positive network externalities); M > 0 is the consumption cost (value of the consumption) of all other commodities except the software; Y > 0 is the income; and P > 0 is the software price. We set up three time periods 0, 1, and 2 and the software vendor will release the new versions at period 1 and period 2, with strictly increasing quality, respectively. We assume that for any software version, during the period when it is just released C = C'\ in the next period C = wC, where 0 < w < 1; and after the next period, as the old version has compatibility problems, the software vendor reduces or stops customer services offered for the old version. We take it; as a decision variable to the software vendor in our model (see the summary description of notations used in the Appendix). We assume that the software vendor only keeps the previous version as an old version, when he releases a new version under Structure 1. So, in periods 1 and 2, one new version and one old version are co-existing in the market. To simplify our analysis, we only investigate the existing users who are using the initial version.at period 0. We assume that the information for the upgrade plan is perfectly known to the existing 82 software users, or that the existing users can perfectly forecast the software vendor's upgrade plan. Software users t ry to make an op t imal decision about whether or not they want to upgrade the software and at which period(s). Since the fact that the user has the same or different income i n each per iod does not affect the analyt ical results, we s imply assume that the user has the same income Y i n each of the three periods and the t ime discount rate is 0. See Figure 3.1 for the software releasing sequence under Structure 1. A t per iod 0, the exist ing user uses the in i t i a l software, of qual i ty Qo, which is greater than 0. A t period 1, the software vendor releases the new version of software, which is of qual i ty Qj = aQo, where a > 1. A t period 2, the software vendor releases another new version of qual i ty Q2 — BQ\ = B(aQo), where B > 1. T o simplify our analysis and solve the inequalities afterwards, we assume B > 2 — ^ , i.e. Q2 — Q\ > Q\ — Qo-8 T h e exist ing software user, at period 1, has five options when responding to the software upgrade plan: 1) no upgrade at a l l ; 2) only upgrade to Q\ at period 1; 3) only upgrade to Q2 at per iod 2; 4) upgrade to Qi at per iod 1 and upgrade to Q2 at period2; or, 5) only upgrade to Q\ at per iod 2. Definitions: For exist ing software users, "Disappearance" demand refers to those consumers who finally choose O p t i o n 1, denoted as D00. "Oppor tunis t" demand refers to the consumers who finally choose Op t ion 2, denoted as D10. "Leapfrog" demand refers to the consumers who finally choose O p t i o n 3, denoted as D02. "Loya l " demand refers to the consumers who finally choose Op t ion 4, denoted as D12. "Lagged" demand refers to the consumers who finally choose O p t i o n 5, denoted as Dm. N o w let us look at the software user's u t i l i ty at different periods. Fol lowing the u t i l i ty function i n W i l s o n (1980) and Aker lof (1970), 9 we can write one software user's general u t i l i ty i n any period as U = M + ntQ + nC subject to the budget 8Under the condition 6 < 2 — we can derive similar results as those under 6 > 2 — -—. For simplicity, we only demonstrate the results under the condition P > 2 — ^ . 9 We choose the utility function from Wilson and Akerlof because the software user has similar choice options as described in them. 83 constraint Y = M + nP, where n is an indicator to show whether to buy the software (n = 1) or not (n = 0); t is the u t i l i ty index to measure the differences among the exist ing users. We assume t € [^o^n] w i t h to > 0 and t has the density function of h(t). Therefore H(t) = f* h(x)dx is the number of buyers w i t h u t i l i ty index less than t. If we substitute M into the objective function, we get the new objective function without a constraint: U = Y — nP + ntQ + nC. Suppose the buyer chooses to buy the software Q, i.e., n = 1, his u t i l i ty is U = Y — P + tQ + C; suppose the buyer chooses not to buy, i.e., n = 0, his u t i l i ty is U — Y. Therefore, the buyer chooses to buy the software Q i f and only if his u t i l i ty under n = 1 is greater than his u t i l i ty under n = 0, i.e., t > ^r"- So, the tota l demand for the software is D = f*n h(x)dx.10 In similar manner, we can write the existing user's u t i l i ty in per iod 1 as Ui = M + ntQ + (1 - n)tQ0 + (1 - n)wC + nC = M + nt(aQ0) + (1 - n)tQ0 + (1 - n)wC + nC subject to Y = M + nP\ , where n is an indicator variable to show whether the user upgrades his software (n = 1) or not (n = 0) at period 1, P\ is the price for the new version Qi at period 1 (Later on, we use the notat ion P2 for the price for version Q2 at per iod 2, and P 3 for the price of the old version Q\ at period 2). Subs t i tu t ing the constraint into the objective function, we get U\ = Y + nt(aQo) - nPx + (1 - n)tQ0 + (1 - n)wC + nC From this u t i l i ty function, if the user decides to upgrade to Q\, i.e., n = 1, his u t i l i ty i n per iod 1 is U\ = Y + atQo — P\ + C. If he decides not to upgrade, i.e., n = 0, his u t i l i ty i n per iod 1 is U\ = Y + tQ0 + wC. Note that the exist ing user has no incentive to buy the old version of Qo i n per iod 1 because he has already had it . In the same fashion, we can derive the user's u t i l i ty in per iod 2, U2, depending on whether he decides to upgrade and which version he chooses to upgrade in period 2. T h e to ta l u t i l i ty in the three periods 0, 1 and 2 under the five options is defined as Ul = UQ + Ui + U2, i = 1 ,2 ,3 ,4 ,5 . W i t h o u t considering a discount rate, the tota l uti l i t ies dur ing periods 0, 1 and 2 under the five options are described i n Table 3.1. 1 0 I n this study, we only study the case where upgrades prices are not endogenous. 84 Now, let us look at various demands for software upgrade. F rom the u t i l i ty formulas i n Table 3.1, the exist ing user finally chooses not to upgrade at either of period 1 or per iod 2 under the condit ion: Ul >.U2,Ul > U3,!!1 > Ui and U1 > U5. F rom this condit ion, we get the following constraints for t to be satisfied simultaneously: P i - C " P2 - (1 + w)C -2C + P, + P2 P 3 - C'w < 2{a - 1 ) Q 0 ' • Q0(ap - 1) ' < Q0(a + aB - 2) ' < Q0(a - 1)" Define . / P 2 - (1 + w)C -2C + Pi + P 2 P 3 - C'w t = 7 7 2 2 7 1 I • • \ Q o ( ^ ^ - 0 , ccB-l)' Qo(a + aB - 2 ) ' Q0(a - 1) So, if tu > td, the to ta l number of existing users who do not upgrade at either of ctu period 1 or period 2 is D = J d h(x)dx. In the same fashion, we get the various other demands for existing users. To simplify our analysis, we assume h(t) has a uniform density on the support [0,tn] and normalize C and Qo into l ' s . Table 3.2 shows the various demands, and the associated tu,s and td's. Note that how the various demands, D00, Dw, D02, D12 and D 0 1 , par t i t ion the whole market depends on the values of a, B, w, P i , P2 and P 3 . In the real world, these five demands w i l l , i n a l l l ikel ihood, co-exist i n the market. W e assume that the five demands co-exist to conform to the reality i n our analyt ical f ramework. 1 1 P r o p o s i t i o n 1 If all of the five demands of existing users: "disappearance", "oppor-tunist", "leapfrog", "loyal" and "lagged" demands, i.e.,D00, Dw, D02, D12 and Dm, co-exist in the market, then the five demands partition the support of t as Figure 3.2. Proof : D00, Dw, D02, D12 and P> 0 1 co-exist in the market implies that D00, D10, D02, D12 and D01 par t i t ion the whole support of the random variable t. So, four pairs of demands share four boundaries of t values. O n checking the t values i n Table 3.2, we can see D 0 0 and P> 0 1 share In the same fashion, Dm and P> 1 0 ' a—1 " share ; Dw and D02 share ^ " f 1 ; D02, D12 share h±m=l. Therefore, a—1 ' 1+ap—2a' ' a—1 ' for P> 0 0 , td = 0 and tu = ; for D°\ td - ^ and tu = w ~ l - p ^ • for D 1 0 , ' a—1 ' ' a—1 a—1 ' . ' 1 1 We tried numerical calibrations that under a set of parameter values, the five demands do co-exist. and t 85 td = v-i-Pt+Pi d t u = p-w-Pff D02 td = E ^ p f i a n d tu = Ei+vLzl. for £>i2 • •a—1 l+a/3—2a'. ' l+a/3—2a a—1 ' ' ^ = and t" = *„. Q E D The importance of proposition 1 is that, if the five demands co-exist, we can draw the demand partition pattern as in Figure 3.2, and can see how each demand varies as some parameters are changed. Note that the order of the different demands on the support of t from 0 to tn are D00, D01, D10, D02 and D12. In order to investigate the role of the old version, we then assume the old version Qi is withdrawn from the market. Given the other conditions are unchanged, the distribution of various demands without the old version Qi at period 2 is described as Figure 3.3. Since p*~™ in Figure 3.2 is less than 21{a-i) m F i g u r e 3-3 as . ( Px-l ' P 2 - l - w -2 + P, + P2 P 3 - . u A Pz-w t = min — r , -x , 7,—-—, : = \ 2 ( a - l ) aB-l a + aQ - 2 . a - 1 ) a - 1 for in Figure 3.2, obviously the "disappearance" market D 0 0 is smaller under Struc-ture 1 than under Structure 2. On the other hand, 2 f a l j ) m Figure 3.3 is less than w~1a-l+Pl m F i g u r e 3.2, implies that the "opportunist" demand under Structure 1 becomes smaller due to the appearance of the old version. Figure 3.2 shows that some existing users with low t will choose to upgrade his version in a "lagged" man-ner from Qo to Qi at period 2 instead of in an "opportunist" manner, and some existing users with the lower t, who would disappear from the market, now upgrade their software from Qo to Qi at period 2. The "lagged" demand is determined by a, P1/P3 and w. This finding indicates that keeping the old version with the new ver-sion in the market can attract some of the consumers who initially had no intention to upgrade at all, and thus salvage some of the "disappearance" demand. Usually, the software vendor would like to salvage the "disappearance" demand as much as possible. From Figure 3.2, we get the following proposition. Proposition 2 If P 3 < w, the software vendor can salvage the whole "disappear-ance" demand. Proof: In order to eliminate L>00, we need td > tu for D00, that is, 0 > which ' — ' ' — a—1 ' implies P 3 < w. Q E D This proposition tells us that if the software vendor wants all the "disappearance" demand to upgrade their software, he can adjust P 3 and w separately or jointly to 86 satisfy the condi t ion P3 < w. For example, one strategy would be to decrease P3 at least to some level below w. O r the software vendor can increase w greater than P3. In either of these ways, some of exist ing users, who in i t i a l ly had no intention of buying any upgraded versions, now choose to buy version Q\ at period 2. In addi t ion, the co-existence of "lagged" demand implies tu > td for D°\ that is W ~ 1 ~ ^ + P l > ^Ef i n F igure 3.2. Re-arranging this inequality, we get P3 < pi+2™~~1. T h i s tells us that if the price of version Q\ i n period 1 is decreased, that is, lower P i , then the software vendor has to further decrease the price of such a version i n per iod 2, that is, lowering P3 in order to salvage the "disappearance" market. T h e reason for this is that = \ > 0 if we set P 3 = In most cases, the software vendor not only wants to salvage the "disappearance" demand, but also wants more "loyal" consumers among existing users, that is, he wants more existing users to upgrade in bo th periods 1 and 2. T h e following proposi-t ion provides some insights into how to increase the proport ion of "loyal" consumers among existing users. Proposition 3 . Decreasing the price of version Qi in period 1, decreasing the cus-tomer services in the second period, or increasing the quality of version Q\ in period 1, will help to increase the proportion of "loyal" consumers. Proof: In order to increase the interval for D12, we need to decrease EI+HLZI w h i c h 1 Q _ i > suggests to either lower P i or w, or increasing a, or doing them jointly. QED Lowering P i has mult iple effects on the demands of exist ing users. T h e overall effects of P i is such that decreasing P i w i l l increase the demands D10 and D12, but decrease the demand Dm and D02. Lowering P i leads to more "loyal" consumers is reasonable since some exist ing users, who are wi l l ing to upgrade to Q2 at per iod 2, are now also wi l l ing to upgrade to Q\ at period 1 as the version Qi is cheaper. A t the same time, some exist ing users who would have been wi l l ing to upgrade to Q2 at period 2, now choose to upgrade to Qi at period 1 instead. T h i s causes the shrinkage of D02, and implies that there is a positive cross price elasticity between version Qx and version Q2. Increasing the quali ty of version Qi at per iod 1 has similar effects as lowering P i . T h e reason that decreasing w can increase the "loyal" 87 demand is that, as the software users cannot get more customer services, this forces some existing users who in i t i a l ly only buy Q2 in per iod 2 to buy Q\ in period 1 as wel l . However, from Figure 3.2 we can see that while decreasing w increases "loyal" demand, it w i l l increase the disappearance market as a side-effect. Therefore, i f the software vendor cares about both "loyal" demands and "disappearance" demands, decreasing P\ should be a better choice than decreasing w as decreasing P i w i l l lead to a larger "loyal" demand, but w i l l not lead to a larger "disappearance" demand. Comparative Static Analysis It is helpful to understand how a change i n the value of one parameter affects various upgrade demands. Our comparative statics results are listed in Table 3.3. T h e impact of lowering the price of version Q\ at per iod 1 can be summarized as follows: It w i l l increase the "opportunist" and "loyal" demands, and decrease the "leapfrog" and "lagged" demands, which is consistent w i t h Propos i t ion 3. B u t , changing P i does not affect the "disappearance" demand. Lowering the price of version Q2 at per iod 2 has no impact on the "disappearance" demand, the "loyal" demand or the "lagged" demand. It w i l l decrease the "opportunist" demand, increase the "leapfrog" demand. If we combine the impact of P i and P2 together, we can find that decreasing the relative price of version Q2 at per iod 2 w i t h respect to the price bf version Qi at period 1, that is, R = g , w i l l lead to decreasing the "opportunist" demand and increasing the "leapfrog" demand. T h e proof is out l ined as follows: dDw _ OD^ _ dD^_ dD^ _ 'dry* _ OD^_ ~d~R ~ "dPT ~ ~dK a n ~cW ~ ~dP2~ ~ ~~dF\' We know QjjW d D W dD02 QD02 a p 7 > 0 ' 5 P 7 < 0 ' - d p t < 0 a n d - d p T > 0 t therefore, dDio gD02 ^ — > 0 a n d . w < 0 . T h i s result shows that there is a subst i tut ion effect between versions Q\ at period 1 and Q2 at period 2, which is referred to as the "horizontal" cannibal izat ion in the exist ing users market. 88 T h e price of version Qi as an o ld version i n per iod 2 has some impacts on the "dis-appearance" demand, the "lagged" demand and "opportunist" demand. Lowering the price w i l l decrease the "disappearance" demand and "opportunist" demand, and increase the "lagged" demand. T h i s is consistent w i t h Propos i t ion 1. In addi t ion, we have the following proposit ion. Proposition 4 Keeping the old version of software with the new version will only lead to the "cannibalization" effect on the "opportunist" demand of the existing users. Proof: F r o m Figure 3.2 and Figure 3.3, D02 and D12 do not change w i t h / w i t h o u t the old version. F rom Table 3.3, we can see only Dw is the function of P 3 . Neither D02 nor D 1 2 is the function of P 3 . These suggest that the appearance of the old version Q i has no influence on the demands D02 nor D12. QED T h i s proposi t ion shows that keeping the old version and changing its price w i l l only affect the purchase choices of the "opportunist" demand. T h i s finding is non-intui t ive as we believed al l the exist ing users inc luding D02 and D12 should be affected by the appearance of the old version Q\ i n per iod 2. T h e impl ica t ion of this propo-si t ion is that if the software vendor does not care that some "opportunist" users become "lagged" users, keeping the old version w i t h the new version is a better choice because the old version has no negative impact on the new version, but can salvage some "disappearance" demand. T h e reason that the appearance of version Qi as old version i n period 2 has no effect on the demands D02 is that i f the exist ing user decided not to buy Q\ at per iod 1, he w i l l not buy it at per iod 2 when it has become a l i t t le out-of-date. For D12, i f the exist ing user decided to buy the version Q\ at per iod 1, it is senseless for h i m to buy the old version at per iod 2 again as he w i l l buy the new version Q2. T h e impact of lowering the qual i ty level of version Qi can be briefly described as follow: the "disappearance" and "lagged" markets become larger and the "loyal" market becomes smaller, the "opportunist" demand becomes smaller, and the "leapfrog" demand becomes larger. T h e impact of lowering the qual i ty level of version Q2 is that the "opportunist" demand becomes larger, and the "leapfrog" demand becomes smaller. We can see that lowering the software qual i ty has the equivalent 89 effects as increasing the software price has. In addi t ion, the impact of the qual i ty level of version Qi is relatively comprehensive, and the qual i ty levels of the two versions Qi and Q2 have some subst i tut ion effects between the "opportunist" demand and the "leapfrog" demand. 3.3.2 Existing Users' Upgrade Demand Under Structure 2 If the software vendor chooses not to keep any old version along w i t h the new version in the market, the software releasing sequence can be described as Figure 3.4. Generally, the existing user has four upgrade choices, which are described as the first four options in Table 3.1. F rom the u t i l i ty formula i n Table 3.1, the exist ing user chooses not to upgrade at either of per iod 1 or per iod 2 under the condit ion: U1 > U2, U1 > U3 and U1 > U4. F rom this condi t ion, we get the following constraints for t that need to be satisfied simultaneously: t Pi-c' P 2 - (1 + w)C -2C + P i + P 2 < 2 ( a - l ) g 0 ' < Q0(a(3-\) ' < Q0(a + af3 - 2 ) ' " . Define . , P i - C P 2 - (1 + w)C -2C + 1\+P2. I — fYitfii - ) V 2 ( a - 1 ) Q 0 ' Qo(ap-l) ' Q0(a + a(3 - 2) and td = trj. So, if tu > td, the tota l number of existing users who do not upgrade at period 1 or per iod 2 is D = J ( d h(x)dx. In the same fashion, various demands for exi t ing users can be figured out. Because the existing users do not have the opt ion of upgrading their software to Qi in per iod 2, we only have the following demands: D00, D10, D02 and D12 in Structure 2. If we assume that a l l these demands co-exist i n the market, and t is uniformly distr ibuted i n [0,t n] and normalize C and Qo into l ' s , we get the par t i t ion of these demands as shown in Figure 3.3. So, we can see. that most of the analysis about the behavior of exist ing users under the Structure 1 is s t i l l va l id under the Structure 2, except that the "lagged" demand disappears, which means that the software vendor has lost one tool for reducing the "disappearance" market. 90 3 . 3 . 3 New Consumer's Demand of Software Versions We need to analyze the new user under the two market structures. There are two streams of new consumers i n our 2x2x2 model , one in period 1 and another one i n period 2. To simplify our analysis, let us only analyze the stream i n per iod 2 as bo th streams share the same behavioral characteristics. Case One: New User's Demand under Structure 1 In this case, at per iod 2 the qual i ty of old version is Q\ and the qual i ty of new version is Q2. T h e price of the old version is P 3 and the price of new version is P2 as previously. T h e u t i l i ty function of the new user can be described as U = M + tn(2 - n)Ql+ n(2 - n)wC + '*~ L "2—-Qi + n{n - 1)C subject to Y = M + n(2 - n)P3 + ^ 1 ) P 2 . Note that n is an indicator variable w i t h values of 0, 1 and 2. T h e new consumer has three options i n the software market: buy the old version, buy the new version and not buy any versions at a l l . For the u t i l i ty function, n = 0 means that the new consumer does not buy any version, n = 1 means he buys the old version and n = 2 means he buys the new version. Table 3.4 shows the three options to the new user. T h e new consumer buys the old version if and only if U\ > Uo and U\ > U2, which suggests + P3-wC ' P2-P3 + wC -c t > ana t < Define aQo Qoa(B-l) P3-wC ,+u- P2-P3 + wC'-C and t = ocQo Qoa{8-\) Therefore the to ta l demand for. the old version is Dlld = ffd h(x)dx.12 T h e new consumer chooses to buy the new version i f and only if U2 > Uo and U2 > U\, which 1 2In this section, we use D%ld to refer to the new users' demand for the old version Qi, D\ew to refer to the new users' demand for the new version Q2, and D2dis to refer to the "disappearance" demand. 91 suggests P2-a R - ^ + W C - C ' t > — T T T C — and t > — — . aBQo Q0a(B-l) Define ,d _ (P2-C' P2-P3 + wC'-C\ ~ m a X [ a8Q0 ' Qoa(B-l) h and get D2iew = f*dn h(x)dx. By the same token, the new consumer does not buy any version if and only if f/0 > U\ and Uo > U2, which suggests Pz-wC , !'•> - C" t < and t < ———. aQo apQo Define . fPs-wC P2-C t = mm — , ——— V aQo apQo and get D2dis = f*d h(x)dx. Note that for the above max and- min functions, if Qo(PP3 - P2) + C - BwC > 0, tu = td P2-C aPQo If Qo(pP} - P2) + C - BwC < 0, r•= and ta = —-——— . aQo Qoa((3 - 1) . Proposition 5 If Qo(BP3 — P2) + C — BwC > 0, there are no new consumers willing to buy the old version even if both the old and the new versions co-exist in the market. In this case, the market with two versions co-existing (i.e., Structure 1) is identical to the market with only the new version (i.e., Structure 2) to the new user. Proof: If Qo(BPo, — P2) + C — BwC > 0, the t value of the consumer who will buy the new version is between ( P ^ Q q , tn), and the t value of the consumer who will not buy any versions is between (to, P ^ Q 0 )• It is easy to observe that these two intervals of t partition the entire support of t. Therefore, no new consumers will buy the old •version. QED The condition Q0(BP3 - P2) + C -BwC > 0, that is, (BP3 - P2) > ^wCQ-c', suggests that P 3 is not significantly below P2. If this happens, the consumer has few 92 incentives to buy the old version. If we draw the dis t r ibut ion of new consumers, we w i l l get a figure identical to Figure 3.6 i n Case T w o in the next section. Under the condi t ion Qo.(PP$ — P2) + C — 3wC > 0 and the assumptions.that t has a uniform dis t r ibut ion on .[0, tn], we normalize C and Qo into l ' s and figure out D2ew and D\is as follows: tnap tnap which are the same as the demands i n Case T w o (see the next section). T h i s propo-si t ion states that the new version release gives the vendor pressures to decrease the price of the old version. In other words, i n order to induce some new consumer to buy the old version, the software vendor has to largely decrease the price of the old version. Under the condi t ion Qo{3P3 — P2) + C — BwC < 0, which suggests that the price of the old version is significantly lower than the price of the new version, we have a different story: some new consumers choose to buy the old version; some choose to buy the new version; and some choose not to buy either of them. See Figure 3.5 for the dis t r ibut ion of the new consumers. Us ing the same assumptions that t has a uniform dis t r ibut ion on [0, t n ] , and normal iz ing C and Qo into l ' s , we can figure o u t Dlw D l e w a n d Ddis a s follows: 2 _ P2 ~ m - 1 + PW n 2 P2 - P 3 + W - 1 n 2 _P2~W tna(p-l) tna(p-l) atn Case Two: New User's Demand under Structure 2 In this case, the new consumer's u t i l i ty function can be described as the following: U = M + tnQ2 + nC subject to the budget constraint Y = M + nP2, where n is an indicator w i t h the values of 0 and 1. T h e new consumer has two options: buy the version Q2 or not, which are the options 1 and 3 in Table 3.4. The new consumer w i l l choose to buy the new version if and only if U\ > Uo, which suggests t > ^=g. Define td = and g e f / J ^ = . H { x ) d x . T h e new consumers w i l l not choose to buy the new version if and only i f U\ < Uo, which suggests t < P^QA • Define tu = P^QQ and get D\is = f*dn H{x)dx. We can demonstrate the dis t r ibut ion of the new consumers i n Figure 3.6. If we assume that 93 t has a uniform dis t r ibut ion on [0, tn] and normalize C and Qo into l ' s , we can figure out D2new and D%, as follows: £ n a / ? tna(3 T h e interesting observation for new users is that once some new users buy the old version in Structure 1, the total demand i n Structure 1 must be greater than that i n Structure 2. T h e in tu i t ion for this observation is that the old version at a low price w i l l attract some new users who would in i t i a l ly have no intention to buy the new version at a h igh price. T h e proof is as follows. G i v e n the condi t ion Qoi-il', - 1>>) -C- 3,cC>{). which implies that some new users w i l l buy the old version i n Propos i t ion 5, we can see P3~Q^ in Figure 3.5 is less than P^pQ0 in Figure 3.6, which proves the observation. Comparative Static Analysis Compara t ive static analysis can shed light on how demands change along w i t h the changes of some relevant parameters. Table 3.5 shows us the ma in comparative static results for the new user. In Case One, as the table shows that under Structure 1, the demand for the o ld version is negatively related to the qual i ty of the new version and, its own price, but is posi t ively related to the price of the new version. T h i s demonstrates that there is a cross subst i tut ion effect between the old version and the new version, called the "vertical" cannibal izat ion. Conversely, the demand for the new version is posi t ively related to its qual i ty level and the price of the o ld version, but negatively related to its own price. T h e number of people who. w i l l not buy either of the versions is only posi t ively related to the price of the o ld version since it is either the o ld version or no purchase. Figure 3.5 shows that the cr i t ica l point between not buying at a l l and buy ing o ld version is Pi~Q^ • In Case T w o , the demand for the new version is posit ively related to the qual i ty level upgraded. T h e negative relation between the demand for the new version and the price of the new version, P2, is straightforward. It follows that the number of new consumers who decide not to buy the new version, AL>: is negatively related to the higher upgraded quality, /?, and posi t ively related to the price of the new version, P2. 94 Another interesting question is that how the change of relative price of the o ld version w i t h respect to the new version, that is R — | | affects the behavior of the new consumer. A s the relative price of the old version to the new version becomes higher, the market for the old version become smaller, and the market for the new version become larger. T h e proof is straightforward: d D l u _ dL*M dDlld dDlew _8Dlew 8D2new We know therefore, OR ap, dP2 OR ap, ap2 ^ < 0, ^ > 0, ^  > o and < 0, ap, ap2 ap, ap2 d V ° l d < 0 and > 0. OR OR T h i s result clearly shows there is a subst i tut ion effect between the old version and the new version, that is, the "vertical" cannibal izat ion, when both versions co-exist i n the market. T h e impact of the parameter w is interesting. Increasing w w i l l increase the old version sale and salvage some disappearance demand, but decrease the new version sale. So, w can be used as a potential tool to balance the sales from different categories of new users. 3.4 Choice of Software Vendor Now, let us look at the choice of the software vendor for software upgrade. Qua l i t y levels, prices of different versions, customer services and market structures are the choice variables for the software vendor. Table 3.6 summarizes various demands under the two market structures. It is obvious that there are associated upgrading costs to the software vendor. For the software vendor, H u et al. (1998) show that this k ind of costs is determined by the upgrading efforts. More specifically, i n our analyt ical framework, among the parameters we analyzed, adjusting P\, P 2 and P3 w i l l not incur any cost, but affect the market share or profit. However, while changing a or B w i l l affect bo th market share and profit, increasing a or f3 implies investing more in software design and 95 development. T h e similar case is applied to w. T h a t is, changing w w i l l affect bo th market ing share and profit, but increasing w w i l l incur some costs. T h e software vendor might t ry to maximize either profit or market share (more discussion in the next section). We can see the opt imiza t ion problem to the software vendor is a max-max problem. He firstly tries to maximize the profit or market share under each of the two market structures, then compares the two maximums to choose the higher one. W e can write the software vendor's two basic objective functions as: Max: [(Max: T h e total market profit under Structure 1), (Max: The tota l market profit under Structure 2)] or Max: [(Max: T h e tota l market share under Structure 1), (Max: T h e tota l market share under Structure 2)] O f course, the software vendor might have an objective to maximize to ta l profit subject to the market share being not less than some level, or to maximize to ta l market share subject to the market profit being not less than some level. T h e to ta l profit is composed of the profit from the existing users and the profit from the new users. T h e tota l market share is composed of the share from the exist ing users and the share from the new users. Note that the profit is equal to the revenue (the market share times the price) minus the associated upgrade cost (we assume that the upgrade cost is linear in the increased quali ty) and the associated customer service cost (we assume that the customer service cost is linear i n the customer service level C). In our analyt ica l framework, instead of implementing the two-step max-max optimizat ions, we compare the two market structures more straightforwardly. We derive the following proposi t ion about the market share. Proposition 6 The total market share under market structure 1 is greater than the market share under market structure 2 if P% < max ^ p i + ^ - i ; P^+ffw-i^ Proof: If we compare the tota l demand under Structure 1 w i t h the tota l demand under Structure 2, we can see bo th structures share some same components. T h e different components for each of total demands are listed in Table 3.7. For the exist ing users, the demand under Structure 1 is greater than the demand under Structure 2 if and. only if ^ t ^ a - i ) ^ > 0 which means P 3 < P l + 2 2 w ~ 1 . O n 96 the other hand, if P 3 > P l + 2 2 ^ " ~ 1 , bo th market structures have the identical existing users because the "lagged" demand disappears as we analyzed after Propos i t ion 2. Reca l l Propos i t ion 5, to make Structure 2 different from Structure 1 to new users, we need (BP3 - P i ) + 1 - Bw < 0, i.e., P 3 < P 2 + ^ " ~ 1 , which suggests the demand under Structure 1 dominates the demand under Structure 2 for new users. O n the other hand, i f P 3 > P 2 + ^ ~ 1 , bo th market structures are identical to the new users. Therefore, i f P 3 < max , P2+PW-I^ t h e t o t a l d e m a n d u n d e r Structure 1 is greater than the to ta l demand under Structure 2.. QED Proposition 7 If the software vendor wants the maximum profit, he needs to choose market structure 1 if { P 3 _ ( * a > ( P 2 _ ^ / f t - m - 1 + 2 ; \ a - I J v ' " ° y V 0 - 1 Otherwise, he needs to choose.market structure 2. Proof: Let us figure out the different profits for exist ing users between the two structures. Compared w i t h Structure 2, Structure 1 has addi t ional amount 'w - 1 + P i - P 3 P 3 - w\ 1 a — 1 a — 1 / tn of "lagged" demand. O n the other hand, Structure 2 has addi t ional amount 'w-l + Pi-P3 P i - 1 \ 1 a - 1 2(a-l)J in of "opportunist" demand. Whether the total profit from exist ing users under Struc-ture 1 is greater than that under Structure 2 depends on whether W - I + P 1 - P 3 P 3 - w \ l fw^l + IY-Ps J \ - l \ 1 p T - P 5 > " : 777: T-Pl-a - I a - 1 J tn \ a - I 2(a - 1) J tn Figure 3.7 shows the different demand for exist ing users under two market struc-tures. N o w let us look at the different profits for new users under the two market structures. In Figure 3.8, for new users the to ta l support of t is par t i t ioned into three segments under Structure 1 and into two segments under Structure 2. Whether the total profit from new users under Structure 1 is greater than the to ta l profit from new users under Structure 1 depends on whether P 2 - P 3 - w - l P 3 - w \ l D f P 2 - P 3 - w - l P 2 - w \ l J D P? > -TT, 7\ : T --'2-a{B - 1) a :) tn 0 \ a(B - 1) a J tn 97 Therefore, the total profit Structure 2 i f and only i f from Structure 1 is greater than the total profit from ™ - l + P i - P 3 Po~w\ D , /P2-P3 + W-I P 2 - w \ 0 "3.+ -775—TT~— 7— F o > a - 1 a- 1 J V aW~ 1) a ^ - l + P r - P g P i - l \ n , / ^P 2 - P 3 + w - 1 P 2 . , „ . ^1 + ^ — T r - ^ 2-a - 1 2(a 1 ) ; V a ( / 3 - l ) . ad After some algebra, we get 2 y V a - 1 J V QED T h i s proposi t ion shows that even i f the market share under Structure 1 is greater than under Structure 2, the to ta l profit under Structure 1 is not necessarily greater than that under Structure 2. ( P 3 — ^-) (Pl~2^12w~l) measures the profit gain for the old version from the existing market as the o ld version salvages some "disappearance" demands. However, since the old version has such a cannibal izat ion effect to the new users, ( P 2 — /3P 3 ) ^p2-/3P3-i+/3^ j stands for the relative lost or oppor tuni ty cost when the o ld version attract some new users from using the new version to using the old version. O n l y under the condi t ion that the profit gained from the o ld version is greater than the profit foregone due to the appearance of the old version, it is op t imal for the software vendor to choose Structure 1. The ideal s i tuat ion for the software vendor is that Structure 1 can br ing both m a x i m u m tota l market share and m a x i m u m tota l profit. If not, he has to consider the tradeoff between the m a x i m u m tota l market share and the m a x i m u m total profit from each market structure. Let us wri te the difference between the left side and right side of the condi t ion in Propos i t ion 7 as: 2 ' \ a - 1 J v ' " ° ' \ 6 - 1 We can see that if w is reduced when the price of Q2 is relative higher than the price of Qi in period 2, such the difference is becoming larger. T h a t is, ^^p- < 0 i f ( 2 P 3 - PX)(B - 1) < ( P 2 - P3)B(a - 1). 98 T h i s observation tells us if the software vendor sets the price for the new version is higher enough to satisfy the condi t ion (2P3-P1)($-l)<(P2-P3)B(a-l), reducing the customer services, i.e., w, w i l l help h i m benefit from the market struc-ture 1. Another question is whether and how the software vendor can use price discr im-ina t ion on existing users and new buyers. T h e significant difference between the exist ing users and new buyers is that the existing users can s t i l l use their software even when the new version shows up. T h e in tu i t ion is that, i n order to induce the exist ing users to buy the new version, the software vendor should give them some price discounts. Proposition 8 The software vendor is able to use price discrimination on the ex-isting user and the new user to reap higher profit from the new version and/or the old version. For the new version, the software vendor should charge a lower price to the existing user and a higher price to the new user.under either of the two market structures if tn is sufficiently large. For the old version, the software vendor should charge a lower price to the existing user and a higher price to the new user if the price of new version P2 is greater than / 3 P l ~ / 3 + 2 . (See the proof in Appendix.) T h i s proposi t ion tells us that the software vendor should charge a lower price for the new version to the existing user and a higher price to the new user if consumers are highly heterogeneous, i.e., tn is sufficiently large. T h e proposi t ion also tells us for the o ld version under Structure 1, the software vendor can charge a lower price to the exist ing user and a higher price to the new user i f the price of new version is greater than some value. The previous s tudy by V a n Ackere and Reyniers (1995) states that by using trade-ins (some discounts) to exist ing buyers and int roductory offers to new users, the monopolist can use third-degree price d iscr iminat ion among consumers to reap higher profit. T h i s s tudy shows the conditions under which the software vendor should offer price discounts to exist ing users and charge higher prices to new users. If the conditions are not satisfied, such discr iminated price pol icy is not op t imal at 99 a l l . For example, suppose the software vendor decides to adopt discr iminated prices of the new version to exist ing and new users. Suppose that tn is not sufficiently large, say, tn < 2 P l + ^ ~ P 3 - 1 under market structure 1, which implies that the. condi t ion i n Propos i t ion 8 is not satisfied. Theoretically, the op t ima l price d iscr iminat ion pol icy for the new version is that offering price discounts to new users and charge higher prices to exist ing users. If the software vendor s t i l l offer price discounts to exist ing users and charge higher prices to new users, this price pol icy is not op t imal at a l l , and even worse than the uniform price policy. Our finding is also different from previous studies including Y a n g (1997b), where the condi t ion to implement discr iminated prices to exist ing users and new users is based on the relative new version qual i ty to the old version. The advantage that the software vendor can use for price discr iminat ion is that it might be possible for h i m to dist inguish the exist ing users from the new users by using some relevant information technology. One practice is that the software vendor supplies only addi t ional software upgrade components for the exist ing users at a low price. In this way, such addi t ional upgrade components are useless to the new users if they have no chance to firstly own the old version. Under this arrangement, as the software users have to self-report or self-select them as the exist ing users or the new users, the third-degree price d iscr iminat ion is feasible and might be widely encouraged i n the software industry. 3.5 Discussions and Limitations 1. Market Share vs. Market Profit. In our analyt ical framework, we layout two objectives for the software vendor: market share or market profit. T h e software vendor's objective choice would depend on many issues inc luding the following: a) F rom product life cycle ( P L C ) theory, dur ing the in t roduct ion and growth stages, as the market share is the base for market profit, the software vendor should expand market share. A s long as the products are more widely used and the mar-ket share becomes larger, the software vendor might switch to market p ro f i t . 1 3 T h e higher switching costs for software users due to sunk costs in learning to use current 13See Greenstein and Wade (1998) for PLC in the computer market. 100 software will help the software vendor to implement the switching of his objective from market share to profit. Taking profit as an ultimate objective is very impor-tant for the software vendor to achieve consistent successes, since more profit can guarantee more investment into new system development and software upgrade; b) Another advantage of developing market share first, and reaping profit next, is that high market penetration is important for establishing a technology standard and for providing a platform for other application software. Once the software vendor's product is a standard or a platform for other software in the market, the initial soft-ware upgrade will generate associated sequential upgrades of other software. Thus, the software vendor might be able to achieve the first-move advantage; and, c) Regarding the competition in the software markets, even though we model the software vendor as a monopolist, in fact, there often are several software vendors with similar products following similar marketing strategies, which is especially common for non-leaders in the software market. For example, if a software vendor's competitor is trying to deploy market expansion strategies, the vendor may have tb follow similar strategies to maximize market share. To understand the detailed investigation of one software vendor's response to other vendors with similar products, duopoly modeling is required, which is one of the potential research opportunities related to this study. 2. Strategic Customer Services. Our research shows that customer services are an effective strategic tool to help the software vendor achieve his objective. Never-theless, when using w, the software vendor should bear in mind some of the tradeoffs involved. For example, for existing users,, decreasing w can increase the "loyal" de-mand, but will increase the disappearance market. In other case, we can see that decreasing Px should be a better choice as this will lead to larger "loyal" demands and will not lead to larger "disappearance" demands. For new users, increasing w will increase the old version sales, and salvage some disappearance demand, but will decrease the new version sales. In our model, for simplicity, we assume that the software vendor adopts the stationary customer services policy for any software ver-sion. In reality, however, the software vendor can use a more flexible non-stationary customer services policy. For example, the software vendor can adopt differentiated customer services for different versions in different periods, or implement service 101 discr iminat ion for different software users. 3. Perfect Information. In our analyt ical framework, to simplify the analysis, we assume that the information for the upgrade plan is perfectly known to the exist-ing software user, or that the existing user can perfectly forecast the upgrade plan of the software vendor. T h i s assumption is based on the fact that, in reality, the pre-announcement or "vaporware" market ing strategy is widely used in the software industry. Hoxmeier (2000) finds that software vendors can use software preannounce-ments for a competi t ive purpose w i t h few concerns about any negative impact on users, if the vendors deliver the promised features and functionality. Raghuna than (2000) shows that the software vendor can reap a higher profit when it pre-announces the future strategy. 1 4 Even i f the software vendors do not announce their upgrade plan i n the next period, given the common knowledge that any software needs to be upgraded i n the future, the software users should consider mult i-periods software upgrades when they make a purchase decision. Instead of knowing the exact up-grade plan, they can figure out the expected values of various parameters involved i n upgrading plans. Under this s i tuat ion, the sophisticated Bayesian analysis might be required. However, as the upgrade parameters are controlled by the software vendor, our simplified two-step backward analysis (i.e., analyzing consumer behavior first, then investigating the vendor's choices) s t i l l shows some insights into the software vendor's op t imal software upgrade choice. 4. Hardware Upgrading vs Software Upgrading. A s a software upgrade might need an associated hardware upgrade, software users incur some associated indirect costs of hardware for their software upgrade plan. Or , the to ta l costs to implement a software upgrade might be more than the cost of the new software. Except for the possibi l i ty that hardware users can sell used hardware, the consumer behavior for hardware upgrade is s imilar to the behavior for software upgrade. In our analyt ica l framework, one computer user has s imilar upgrade options for-hardware as described in Table 3.1. For simplici ty, we do not include the associated hardware upgrade cost for a software upgrade i n our framework as we believe that the consumer software 1 4 For more discussion.about product preannouncement and vaporware, see Cho and Kreps (1987), Eliashberg and Robertson (1988), Herbig (1996) and Jenkins (1988). 102 upgrade behavior represents the consumer's hardware upgrade behavior. In any case, .we should bear in m i n d that some differences exist between the hardware upgrade and software upgrade. F i r s t , software is an information good and its marginal cost is almost zero. Hardware does not share this property. Second, unlike the secondhand markets for hardware, the secondhand market for software is prohibi ted, due to the difficulty in preventing dupl icat ion, that could lead to copyright infringement as shown in Fudenberg and Ti ro le (1998). Legit imately, only the software vendor has the right to sell the old version i n the market. O r equivalently, the software vendor is able to completely control the secondhand market for the products. For hardware, used hardware and older models of hardware are widely available i n the markets. T h i s implies that the hardware vendor losses some power to use used-goods or older models as strategic tools to achieve his objectives. O n the other hand, we observe that the software upgrade has some barriers that hardware upgrade does not have. One important issue is that software vendor might incur losses from software piracy, which affect his upgrade plan. B u t , for hardware upgrade, no hardware piracy problem exists. 3.6 Conclusions Software upgrade is an important issue to software vendors. T h i s paper shows that several types of software upgrade demands exist in the software upgrade mar-kets: "disappearance" demand, "opportunist" demand, "leapfrog" demand, " loyal" demand and "lagged" demand. Keep ing an old version w i t h a lower price is an ef-. fective way to salvage the "disappearance" demand. Keep ing the old version w i t h the new version w i l l lead to different "cannibal izat ion" effects for existing and new users. Decreasing the price of the new version in period 1, increasing the qual i ty of the new version i n per iod 1 or decreasing customer services w i l l help to increase the proport ion of " loyal" demand. If the price of the old version is not low enough, Structure 1 is identical to Structure 2 to new users. However, if the price of the old version is low enough, the to ta l market share under Structure 1 is greater than the market share under Structure 2. So, if the software vendor wants to maximize his market share, lowering the price of the old version and choosing Structure 1 is 103 the right choice. However, even i f the market share under Structure 1 is greater than that under Structure 2, the to ta l profit under Structure 1 is not necessarily greater than under Structure 2. Propos i t ion 7 tells us the condi t ion under which the software vendor should choose Structure 1 to maximize the to ta l profit. T h e ideal s i tuat ion for the software vendor occurs when Structure 1 can br ing bo th the max-i m u m tota l market share and the m a x i m u m tota l profit. We find that as software users become heterogeneous, the difference i n market share or profit between the two market structures are d iminish ing to the software vendor. T h e software vendor is able to use price d iscr iminat ion on the exist ing user and the new user to reap a higher profit. We find that, for the new version, the software vendor should charge a lower price to the existing user and a higher price to the new user under either of the two market structures if consumers are sufficiently heterogeneous; for the old version, the software vendor should charge a lower price to the exist ing user and a higher price to the new user i f the price of the new version is quite high. T h e soft-ware vendor gains an advantage by implementing price d iscr iminat ion since, by using some information technology, the software vendor might dis t inguish the exist ing user from the new user, on the basis of the user's self-selection. Therefore, third-degree price d iscr iminat ion should be encouraged i n the software industry. 104 Table 3.1: Existing Users' Options for Software Upgrade Behavior Total Utility Option 1 No upgrade in two periods at all. U] = 3Y + 3tQ0 + (1 + w)C 2 Only upgrade to Qi at period 1. U2 = 3Y + (1 + 2a)tQ0 + (2 + w)C - /> . 3 Only upgrade to Q2 at period 2. u3 = 3y + 2/<20 + (2 + 2w)C + BatQ0 - P2 4 Upgrade to Qi in period 1 and u4 = 3Y + (1 + a + aB)tQ0 + (3 + w)C -Px-P2 upgrade to Q2 in period 2. 5 Only upgrade to Q\ in period 2. u5 = 3Y + 2tQ0 + (1 + 2w)C + taQ0 - P3 105 o Demand Conditions Type t" and td £)00 No upgrade in two periods at all. £,10 Only upgrade toQi at period 1. £,02 Only upgrade to Ch at period 2. .Ul >U2,U\U4,US P,-\ P2-l-w -2 + Px+P2 P3-w t =min( td =0 2(a-\)' aB-\ '(a + aB-2) ' a-\ u2 >u',u\u\us u} >u[,u2,u\u5 . ,P2-w-Pt P2-\ , t = min(— - , — ) l + aB-2a a(B-l) d • />-l w-l-p+P. t =max(———, L) 2(a-\) f = P+W-l IP >ul,u2,u\u* t = max( a-\ P2-\-w P2-w-P] />-/>-! aj5-\ '\ + aB-2a a(B-l) £,12 Upgrade (J4 > Ul,U2,U\U5 t"=t, toQi in period 1 and upgrade to Q2 in period 2. £,01 Only upgrade toQi in period 2. P.+w-\ R - l -2+P.+P, -2 + W-P.+P+P,, t" = max(—• , — , • - , : ~ a-\ a(B-l) a + aB-2 aB-l • min( w-l-P^ + P, P2-Pi-\ -2 + w-P^ + Px+P2 (or-1) a(B-l) aB-l t = P3-w Demand Function If t" > td ," t" D1 =jh(x)dx; ,d otherwise D' — 0 (('=00,10,02,12 and 01). cr 'oT co to O a B CL & 1-1 co o CD d xs CR 1-1 P ' CL CD ci 0 CL CD CD CO o 3 Table 3.3: Compara t ive Stat ic Analys i s for Upgrade Demands of Ex i s t i ng Users N.A. <*-N.A. N.A. 55 — or ~* — N.A. N.A. a r | a r N.A.' N.A. • N.A. N.A. ~* N.A. N.A. N.A. «- — Demand It II o o Q % i <C 55 <N 1 cc + . a." + 1 1 Ol 7 <§• + ar + is + aT i is II o cC 53 1 + cc 53 ^ a r ~ ^ 53 i t + *-53 + 1 7 II o Q a r i X 7 + 53 i II 7 <N + 1 «cS (N ^= 1 CC II o • Q 107 Table 3.4: New User 's Choice Under Structure 1 Option Behavior Total Utility / No Purchase of Software U0=Y (n=0) 2 Buy the Old Version (n=l) £/, =Y-P3 +taQQ + wC 3 Buy the New Version (n=2) U2=Y-P2+ taBQ, + C Table 3.5: Compara t ive Stat ic Analys i s for Software Demands of New Consumers Demand * t w t Case One: Both Old and New Versions Market , P2-j3P,-\ + fiw 1 1 t 1 t HCM' , r, ,. ?„«(/?-!) t t 1 t i t t N.A. N.A. 1 Case Two: Single New Version Market D2 = l-P^ new n tnaB t N.A. i N.A. N.A. 1 N.A. t N.A. N.A. Table 3.6: Types of Software Users in Simplif ied 2x2x2 model Market Structure Type of Software Users Types of Demands 1 :Both Old and New Versions Coexist Existing Users New Users D 0 0 , D I 0 , D 0 2 , D I 2 , D 0 ' D2 D2 D2 U-<" ' U"U ' Unew 2: Only New Version Exists Existing Users New Users D00, D10, D02, D12 Dl,D2m. 108 Table 3.7: Different Demands Under T w o Marke t Structures For existing users For new users The Difference P, + 2w-l-2P 3 P2 + Bw-BP^-l between Structure 1 2tn(a-\) tnaB and Structure 2 , 109 Figure 3.1: Upgrade Software Releasing Sequence Under Structure 1 Qo G, = <XQ, G„ Q2 = BQ,=P(aQ{)) G, Figure 3.2: D i s t r ibu t ion of Various Demands w i t h O l d Vers i sion D 0 0 D 0 1 ow D02 D 12 -A w A A A A V Y Y 0 P 3 ~ W W-\-P,+Pl J\-W-Pt Px+W-\ t a-\ ( « - ' ) i + o/?-2« Figure 3.3: D i s t r ibu t ion of Various Demands wi thout O l d Version t A Y ' Y — ^ y n 0 P,~l P:-w-Pt P l + w - \ 2(a-\) \ + a/3-2a a _ l tn 110 Figure 3.4: Upgrade Software Releasing Sequence Under Structure 2 —' 1 1 • 0 1 ? Figure 3.5: Di s t r ibu t ion of the New Consumers Under Structure 1 i A , / Y Y >i I Ps - wC' P,-P, + wC'-C ° aQ„ Q„a(P-\) Figure 3.6: Di s t r ibu t ion of the New Consumers Under Structure 2 2 new r DL D, A K t0 P2-C i n Figure 3.7: Demand of Existing Users 1 /* - w w-1 + P - P a - l a -I -I 2(flr-l) Structure 1 Structure 2 Figure 3.8: Demand of New Users P^-w fi-P,+ 4-1 a Structure Structure 2 fi - l a/? 112 B i b l i o g r a p h y A K E R L O F , G . (1970): "The Marke t for Lemons: Qua l i t y Uncer ta in ty and the Marke t Mechanism," Quarterly Journal of Economics, 84, 488-500. B A C K U S , D . , P . K E H O E , AND F . K Y D L A N D (1995): "International Rea l Business Cycles: Theory and Evidence," in T.F. Cooley ed. Frontiers of Business Cycle Rsearch, Pr ince ton Univers i ty Press. B A R R O , R . J . , AND D . B . G O R D O N (1983): "Rules, Discret ion and Reputa t ion ina M o d e l of Moneta ry Pol icy," Journal of Monetary Economics, 12(1), 101-121. B E R G I N , P . (2006): "How W e l l C a n the New Open Economy Macroeconomics E x -pla in the Exchange Rate and Current Account? , " Journal of International Money and Finance, 25(5), 675-701. B E R K O V E C , J . (1985): "New C a r Sales and Used C a r Stocks: A M o d e l of the Automobi l e Marke t , " The Rand Journal of Economics, 16(2), 195-214. B E T T S , C , AND M . B . D E V E R E U X (2000a): "Exchange Rate Dynamics i n a M o d e l of Pr ic ing- to-Market , " Journal of International Economics, 50, 215-44. ; (2000b): "International Moneta ry Po l i cy Coord ina t ion and Compet i t ive Depreciat ion: A Reevaluation," Journal of Money, Credit, and Banking, 32, 722-45. B U L K E L E Y , W . M . (1990): "Software Users A r e Beginning to Rebel Agains t the Steady Stream of Upgrades," The Well Street Journal. B U L O W , J . L (1982): "Durable-Goods Monopol is ts ," Journal of Political Economy, 90, 314-332. C H A R I , V . , P . K E H O E , AND E . M C G R A T T A N (2002): " C a n St icky Pr ice Models Generate Vola t i le and Persistent Rea l Exchange Rates?," Review of Economic Studies, 69(5), 533-63. C H O , I . , AND D . K R E P S (1987): "Signaling Games and Stable Equ i l i b r i a , " Quarterly Journal of Economics, 102(2), 179-221. C L A R I D A , R . , J . G A L I , AND M . G E R T L E R (2000): "Monetary Po l i cy Rules and Macroeconomic Stabi l i ty : Evidence and Some Theory," Quarterly Journal of Eco-nomics, 115, 147-180. 113 C L A R I D A , R . E . (2006): " G 7 Current Account Imbalances: Susta inabi l i ty and A d -justment," University of Chicago Press, Chicago, forthcoming. C O A S E , R . H . (1972): "Durab i l i t y and Monopoly ," Journal of Law and Economics, 15, 143-149. C O O K , D . , AND M . B . D E V E R E U X (2001): "The Macroeconomics of International F inanc ia l Panics," mimeo UBC. * C O R S E T T I , G . , AND P . P E S E N T I (2004): "Endogenous Pass-through and O p t i m a l Mone ta ry Pol icy: A M o d e l of Self-Validat ing Exchange Rate Regimes," CEPR Working Paper, (8737). D A M O D A R A N , P . , AND W . W I L H E L M (2003a): "Branch and Pr ice Methods for Prescr ibing Profitable Upgrades of High-technology Products w i t h Stochastic De-mands," Working Paper, Department of Industr ial Engineering, (Texas A & M U n i -versity). (2003b): "Prescr ibing the Content and T i m i n g of Produc t Upgrades: Par t II," P h . D . thesis, Texas A & M Universi ty, Department of Industr ial Engineering. D E V E R E U X , M . B . (2000): "How does a Devaluat ion Affect the Current Account? , " Journal of International Money and Finance, 19, 833-851. D E V E R E U X , M . B . , AND C . E N G E L (2003): "Monetary Po l i cy i n the Open Economy Revis i ted: P r i c i n g Sett ing and Exchange Rate F lex ib i l i ty , " Review of Economic Studies, 70, 765-83. D R I S K I L L , R . (2001): "Nontraded Goods and the Current Account ," Review of International Economics, 9(1), 16-23. E L I A S H B E R G , J . , AND T . R O B E R T S O N (1988): "New Produc t Preannouncing Be-havior: a Marke t Signal ing Study," Journal of Marketing Research, 25(3), 282-292. E L L I S O N , G . , AND D . F U D E N B E R G (2000): "The Neo-luddite 's Lament : Excessive Upgrades in the Software Industry," The Rand Journal of Economics, 31(2), 253 -272. E N G E L , C . (1993): "Real Exchange Rates and Relat ive Prices: an E m p i r i c a l Inves-t igat ion," Journal of Monetary Economics, 32, 35-50. (1999): "Account ing for U . S . Rea l Exchange Rate Changes," Journal of Political Economy, 107, 507-538. E N G E L , C , AND J . R O G E R S (1996): "How W i d e is the Border?," American Eco-nomic Review, 86, 1112-25. (2006): "The U . S . Current Account Deficit and the Expec ted Share of W o r l d Output ," NBER Working Paper no. 11921. F U D E N B E R G , D . , AND J . T I R O L E (1998): "Upgrades, Trade-ins, and Buybacks ," The Rand Journal of Economics, 29(2), .235-258. 114 G R E E N S T E I N , S. M . , AND J . B, W A D E (1998): "The Produc t Life Cyc le i n the Commerc ia l Mainframe Computer Marke t , 1968-1982," The Rand Journal of Eco-nomics, 29(4), 772-790. H A R U V Y , E . , AND A . P R A S A D (1998): " O p t i m a l Produc t Strategies i n the Presence of Network External i t ies ," Information Economics and Policy, 10, 489-499. H E R B I G , P . (1996): "Market Signaling: A Review," Management Decision, 34(1), 35-45. H O X M E I E R , J . (2000): "Software Preannouncements and The i r Impacts on C o n -sumers' Perceptions and Vendor Reputa t ion ," Journal of Management Informa-tion Systems, 17(1), 115-139. H u , Q . , R . T . P A N T , AND D . B . H E R T Z (1998): "Software Cost Es t ima t ion Us-ing Economic P roduc t ion Models ," Journal of Management Information Systems, 15(1), 143-164. H u i , K . L . , AND K . Y . T A M (2002): "Software Funct ional i ty : A Game Theoret ic Analys i s , " Journal of Management Information Systems, 19(1), 151-184. I R E L A N D , P . (2001): "Sticky-price Models of the Business Cyc le : Specification and Stabil i ty," Journal of Monetary Economics, 47, 3-18. J E N K I N S , A . (1988): "Long Overdue: the Reasons Beh ind Vaporware," Computer-world, 22 (44a), 11-13. K l H L S T O R M , R . , AND D . L E V H A R I (1977): "Quali ty, Regulat ion, Efficiency," KYK-LOS, 30, 214-234. K L E I M A N , E . , AND T . O P H I R (1966): "The Durab i l i t y of Durable Goods ," Review of Economic Studies, 33, 165-178. K O L L M A N N , R . (2001): "Macroeconomic Effects of N o r m i n a l Exchange Rates Regimes: New Insights into the Role of Pr ice Dynamics , " University of Bonn Memeo: (2002): "Monetary Po l i cy Rules i n the Open Economy: Effects on Welfare and Business Cycles ," Journal of Monetary Economics, 49, 989-1015. L A N E , P . R . (2001a): "Money Shocks and the Current Account ," i n G.Calvo, R. Donbusch and M. Obstfeld (eds.) Money, Factor Mobility and trade: Essays in Honor of Robert Mundell, M I T Press: Cambridge, M A . (2001b): "The New Open Economy Macroeconomics: a Survey," Journal of International Economics, 54(2), 235-66. L A N E , P . R . , AND G . M . M I L E S I - F E R R E T T I (2001): "The Ex te rna l Wea l th of Nations: Measures of Foreign Assets and Liabi l i t ies for Industr ial and Developing Countries," Journal of International Economics, 55, 263-94. 115 L E E , J . , AND M . C H I N N (2006): "Current Account and Rea l Exchange Ra te D y -namics i n the G-7 Countries," Journal of International Money and Finance, (25), 275-274. L E V H A R I , D . , AND T . SRINIVASAN (1969): "Durab i l i ty of Consumpt ion Goods: Compe t i t i on versus Monopoly ," American Economic Review, 59, 102-107. L I E B O W I T Z , S. (1992): "Durabi l i ty , Marke t Structure, and New-used Goods M o d -els," American Economic Review, 72(4), 816-824. L I P T O N , D . , J . P O T E R B A , J . S A C H S , AND L . S U M M E R S (1982): " M u l t i p l e Shooting i n Ra t iona l Expecta t ions Models ," Econometrica, 50(5), 1329-33. L O M B A R D O , G . (2002): "Price Rigidi ty , the M a r k - u p and the Dynamics of the Current Account ," Canadian Journal of Economics, 25(3), 531-555. M A N K I W , N . , AND L . S U M M E R S (1986): "Money Demand and the Effects of F i sca l Policies," Journal of Money, Credit, and Banking, 18 ,415-29 . M A R I N O S O , B . G . (2001): "Marke t ing an Upgrade to a System: Compa t ib i l i t y Choice as a Pr ice Disc r imina t ion Device," Information Economics and Policy, 13, 377-392. M C C A L L U M , B . , AND E . N E L S O N (1999): "Nomina l Income Target ing i n an Open-economy O p t i m i z i n g M o d e l , " Journal of Monetary Economics, 43, 553-78. (2000): "Monetary Po l i cy for an Open Economy: A n Al te rna t ive Frame-work w i t h O p t i m i z i n g Agents and St icky Prices," Oxford Review of Economic Policy, 16, 74-91. M I L L E R , H . L . J . (1974): " O n K i l l i n g off the Marke t for Used Textbooks and the Relat ionship Between Markets for New and Secondhand Goods," The Journal of Political Economy, 82(3), 612-619. N G , C . S. P . (2001): " A Decision Framework for Enterprise Resource P l ann ing Maintenance and Upgrade: a Cl ient Perspective," Journal of Software Mainte-nance and Evolution: Research and Practice, 13, 431-468. O B S T F E L D , M . , AND K . R O G O F F (1995a): "Exchange Rate Dynamics Redux," Journal of Political Economy, 103, 624-60. (1995b): "The Intertemporal Approach to the Current Account , " in Gene Grossman and Kenneth Rogoff (eds.) Handbook of International Economics, vol 3, Elsevior. Press: Ams te rdam, Nor th -Hol l and . (2002): "Globa l Impl icat ion of Self-Oriented Na t iona l Mone ta ry Rules," Quarterly Journal of Economics, 117, 503-35. (2005): "The Unsustainable U S Current Account Pos i t ion Revis i ted ," NBER working paper 10869. 116 O S T R Y , J . , A N D C . R E I N H A R T (1992): "Private Savings and Terms of Trade Shocks: Evidence from Developing Contries," IMF Staff Papers, 39, 495-517. P R U D ' H O M M E , M . , AND K . Y U (2002): " A Pr ice Index for Computer Software Us ing Scanner Da ta , " Working Paper, Department of Economics , (Lakehead University, Canada) . Q i u , C . (2002): "The Role of Upgrading O p t i o n for New Software : the M o n o p o l y Case," M A i n Economics Project , (Economics Department , S imon Fraser Univer-sity, Canada) . R A G H U N A T H A N , S. (2000): "Software Edi t ions : A n A p p l i c a t i o n of Segmentation Theory to the Packaged Software Marke t , " Journal of Management Information Systems, 17(1), 87-113. R O T E M B E R G , J . J . (1983): "Aggregate Consequences of F i x e d Costs of Pr ice A d -justment," American Economic Review, 73, 433-436. R O T E M B E R G , J . J . , A N D M . W O O D F O R D (1998): " A n Opt imizat ion-based Econo-metric Framework for the Eva lua t ion of Moneta ry Pol icy," In B.S. Bernanke and Rotemberg (eds.) NBER Macroeconomic Annual 1997, pp. 297-346, Cambridge, M A : M I T Press. R U S T , J . (1986): " W h e n is It O p t i m a l to K i l l off the Marke t for Used Durable Goods? ," Econometrica, 54(1), 65-86. S A H I N , I . , AND F . M . Z A H E D I (2001): "Cont ro l L i m i t Policies for Warranty, M a i n -tenance and Upgrade of Software Systems," HE Transactions, 33, 729-745. S C H M A L E N S E E , R . (1979): "Market Structure, Durabi l i ty , and Qual i ty : A Selective Survey," Economic Inquiry, 17, 177-196. S C H M I T T - G R O H E , S. , AND M . U R I B E (2001): "Stabil izat ion. Po l i cy and the Costs of Dol la r iza t ion ," Journal of Money, Credit, and Banking, 33, 482-509. (2003): "Clos ing Smal l Open Economy Models ," Journal of International Economics, 61, 163-185. S H E S H I N S K I , E . , AND Y . W E I S S (1977): "Inflation and Costs of Pr ice Adjustment ," Review of Economic Studies, X L I V ( 1 3 7 ) , 287-304. S H I N O H A R A , Y . , T . D O H I , AND S. O S A K I (1997): "Comparisons of O p t i m a l Release Policies for Software Systems," Computers Industrial Engineering, 33(3-4), 813— 816. S H Y , O . (1995): Industrial Organization, Theory and Applications. T h e M I T Press, Cambridge, M A . S P E N C E , A . M . (1975): "Monopoly , Qual i ty , and Regulat ion," Bell Journal of Eco-nomics, 6, 417-429. 117 S W A N , P . L . (1970a): "Durab i l i ty of Consumer Goods," American Economic Re-. view, 60(5), 884-894. . . (1970b): "Market Structure and Technological Progresses: the Influence of M o n o p o l y on Produc t Innovation," Quarterly Journal of Economics, 84, 627-638. T A Y L O R , J . B . (1993): "Discret ion Versus Po l i cy Rules i n Pract ice," Carnegie-Rochester Conference Series on Public Policy, 39, 195214. T H O E N I S S E N , C . (2003): "Current Account and Exchange Rate Dynamics and the Role of Net Foreign Assets," In preparation for Driver, Sinclair and Thoenissen edited "Exchange Rates and Capital Flows". V A N A C K E R E , A . , A N D D . R E Y N I E R S (1995): "Trade-Ins and Introductory Offers in a Monopoly ," Rand Journal of Economics, 26, 58-74. V A R I A N , H . R . (1997): "Versioning Information Goods," Working paper, School of Information Systems, (Universi ty of Cal i fornia , Berkeley)! W I L H E L M , W . E . , P . D A M O D A R A N , A N D J . L i (2003): "Prescribing the Content and T i m i n g of Produc t Upgrades," HE Transactions, 35(7), 647-664. W I L H E L M , W . E . , A N D K . X u (2002): "Prescribing Produc t Upgrades, Prices, and Produc t ion Levels Over T i m e i n a Stochastic Environment ," European Journal of Operational Research, 138(3)^ 601-621. W I L S O N , C . (1980): "The Nature of E q u i l i b r i u m in Markets w i t h Adverse Selec-t ion," Bell Journal of Economics, 11, 108-130. X l E , M . , A N D G . H O N G (1998): " A Study of the Sensi t ivi ty of Software Release T ime , " Journal of Systems and Software, 44, 163-168. (1999): "Software Release T i m e Determinat ion Based on Unbounded N H P P M o d e l , " Computers and Industrial Engineering, 37, 165-168. Y A N G , Y . (1997a): "Network Effects, P r i c i n g Strategies, and O p t i m a l Upgrade T i m e i n Software Provis ion ," PhD Dissertation, Chapter 2, Economics Depart-ment, (U tah State Univers i ty) . (1997b): "Upgrades, Commi tment , M o n o p o l y Profi t , and Network Effects," PhD Dissertation, Chapter 3, Economics Department, (U tah State Univers i ty) . 118 A p p e n d i x A A p p e n d i c e s to C h a p t e r 3 A . l Proof of Propositions Proof of Proposition 8: First Step Let us prove that the total profit under price d iscr iminat ion is greater than the to ta l profit under a uniform price. Approach One: W i t h price discr iminat ion, the software vendor maximizes the profit from existing users and maximizes the profit from new users by different prices respectively. Approach Two: W i t h a uniform price, the software vendor maximizes the sum of profits from both existing users and new users by the same price. We can see A p -proach T w o is one special case of Approach One by adding a constraint that the price for exist ing users is equal to the price for new users. F rom the op t imiza t ion theory, the op t imal value of objective function w i l l shrink if an addi t ional non-redundant constraint is applied. Therefore, the total m a x i m u m profit under Approach One should be greater than (at least equal to) the. total m a x i m u m profit under Approach T w o . Second Step For the same new version, let us show that the software vendor should charge a lower price for existing users and a higher price for new users if tn is suffi-ciently large, which means consumers are sufficiently heterogenous. N o w the problem is that how the software vendor sets up the different prices of version Q2 to exist-ing users and new users when he maximizes two objective functions. T h e demands from exist ing users related to the price of version Q2 is D 1 0 , D02 and D12. F r o m the first order condi t ion to maximize the profit from exist ing users, we can solve the 119 opt imal price level of version Q2, P2 = 2 P l + a / 3 t n - 2 a t n + w + t n under two market struc-tures respectively. In the s imilar fashion, we can figure out the op t imal price level of version Q2 for new users under market structures 1 and 2. Please see Table A.2 for details. Reca l l tn is the m a x i m u m value for the support of t and the variance of t is ^ . A high tn value suggests the consumers are highly heterogenous. It is easy to check out the op t imal price of version Q2 for exist ing users is lower than new users if tn > 2P1+2w-pd-i u n d e r S t r u c t u r e 1 ; a n d t n > Pi+w-i u n d e r s t ructure 2. Third Step Let us prove that for the old version, the software vendor should charge lower price to exist ing users and higher price to new users i f the price of new version P2 is greater than l 3 P l ~ l 3 + 2 , Investigating the objective function under Structure 1, we can figure out the op t imal price levels for exist ing users and new users l isted in Table A.3. It is easy to check out that the op t imal price of the old version to new users is higher than that to existing users (the left co lumn is greater than the right co lumn i n Table A.3) if and only if P2 > m ~ p + 2 . QED 120 Notation Table A . l : Notations Used in the Order of Appearance Economic Meaning U - Utility Q - Software quality c - Extra benefits from customer services etc M - Consumption of other products except software Y Income P Software price G„ - Initial software quality in period 0 c - The extra benefits of C available in the period when an upgraded software is released w Customer services level in the next period after an upgraded software is released G, - Software quality upgraded in period 1 a Magnitude of quality upgrade in period 1 G 2 - Software quality upgraded in period 2 P - Magnitude of quality upgrade in period 2 ^ - Software price for QX in period 1 Software price for Q2 in period 2 Software price for QX in period 2 n Indicator variable for consumer upgrade options t Utility index '<> " The lower boundary of r's support The upper boundary of r's support hit) - The density function off Utility in Period j,j=\,2 and 3 ul - Utility under consumer's option i, /=1,2,3,4 and 5 D - Software upgrade demand DON ' - No upgrade in periods 1 or period 2, i.e., "disappearance" demand D'° - A n upgrade to QX at period 1 and no upgrade in period 2, i.e., "opportunist" demand No upgrade in period 1 and an upgrade to Q2 in period 2, i.e., "leapfrog" demand D'2 - An upgrade to QX in period 1, and an upgrade to Q2 in period 2, i.e., "loyal" demand Dm - No upgrade in period 1, but an upgrade to QX in period 2., i.e., "lagged" demand R R = P / Pj, the relative price P; with respect to P. D2 -new New users demand to buy new version D;M - New users demand to buy old version i>l - New users choose not to buy any versions 121 Table A.2: O p t i m a l Prices of the New Version for Ex i s t i ng and New Users Structure 1 Structure 2 Existing Users P* 2Pl + aBtn - 2atn + w + tn P * 2PS + aBtn - 2atn + w + tn 2 2 I 2 New Users P2 aBtn-atn-w + \ + /> P2 _aBtn+\ 2 I 2 Table A.3: The O p t i m a l Prices of the O l d Vers ion to Different Users Existing Users New Users Structure 1 2w+P]-\ 3 " 4 P2-\ + 3w 3 " .26 122 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0100661/manifest

Comment

Related Items