.

International Outsourcing, Exchange Rates, and Monetary Policy

Wai-Ming Ho∗

Department of Economics

York University

May 2016

Abstract

Firms’ decisions to outsource the production of intermediate inputs abroad depend on the

macroeconomic environment set by governments’ monetary and foreign exchange policies, while

the relocations of production have important implications on the liquidity demands in the financial

markets, which in turn affect the effectiveness of the policies. By incorporating the microeconomic

foundations of firms’ make-or-buy decisions and highlighting the working capital needs of both of

the buyers and suppliers of intermediate inputs, this paper constructs a two-country, monetary

model with segmented financial markets to examine the interdependence of firms’ international

outsourcing decisions and governments’ conducts of monetary and foreign exchange policies. It

finds that under a fixed exchange rate regime, not only the presence of international outsourcing

activities but also the contractual payment arrangements between the domestic firms and their

foreign outsourcing suppliers matter for the international transmissions and welfare implications

of monetary and productivity shocks. It also shows how the endogenous adjustments of firms’

sourcing decisions alter qualitatively the impacts of a currency revaluation and result in a perverse

effect on the trade balance.

Keywords: International Outsourcing, Liquidity Constraints, Monetary Policy,

Currency Revaluation.

JEL Classification: E44, F41

∗ Department of Economics, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3

Tel: 416 736 2100 ext.22319. Fax: 416 736 5987. Email: waiming@yorku.ca.

1 Introduction

International outsourcing has become a key trend in globalization. Domestic firms outsource to

unaffiliated foreign suppliers to take advantage of lower costs of labor and intermediate inputs

abroad. The effects of these international transactions on the flows of goods and labor have been

studied extensively in the theoretical literatures on international trade and labor economics.1 How-

ever, the financial dimension of the transactions has been largely neglected; and the implications of

international outsourcing for the macroeconomy and for the conducts of monetary and foreign ex-

change policies have received relatively little attention. This paper incorporates the microeconomic

foundations of firms’ make-or-buy decisions into a two-country, monetary model with segmented

financial markets so as to analyze how the relocation of production of intermediate inputs affects

the liquidity allocations in the domestic and foreign loan markets. Monetary and foreign exchange

policies affect the availability of liquidity in financial markets, and the adjustments of prices, in-

terest rates, and exchange rates alter the competitiveness of domestic products relative to those

produced abroad. It is of interest to examine how these policies affect firms’ tradeoffs between

in-house production and sourcing abroad and to investigate how the effects of the policies depend

on the presence of international outsourcing. The understanding of this interdependence would be

important in the study of the macroeconomic consequences of international outsourcing.

The sourcing of intermediate inputs from foreign countries (offshoring) has been expanding

rapidly in the last two decades both through arm’s-length trade (international outsourcing) and

through intra-firm trade (foreign direct investment). Direct measurements of the economy-wide

offshoring in goods and services cannot be obtained due to data limitation. Hence, trade in certain

categories of goods and services which are more commonly used as intermediate inputs is often

used as a proxy measurement of offshoring. As reported in the World Bank’s International Trade

Statistics 2013, intermediate goods trade has become an increasingly important component of total

1Campa and Goldberg (1997), Feenstra (1998), Hummels, Ishii, and Yi (2001), and Yi (2003) document thegrowth of world trade due to the growing use of imported inputs in the production of domestic goods that areto be re-exported. Spencer (2005) and Helpman (2006) survey the theoretical literature that combines trade andthe organizational choices of firms to provide insights into the forces driving international outsourcing. A recentreview of the international trade literature on multinational firms has been presented in Antras and Yeaple (2013).Feenstra and Hanson (2001) provide a detailed discussion of the impacts of trade in intermediate inputs on wages andemployment. Some examples of recent theoretical studies are Antras, Garicano, and Rossi-Hansberg (2006), Baldwinand Robert-Nicoud (2007), and Holmes and Thornton Snider (2011) analyzing the wage effects, and Keuschnigg andRibi (2009) and Koskela and Stenbacka (2009, 2010) examining the employment effects of outsourcing.

1

world trade, the share of intermediate goods in world non-fuel exports has been around 50% since

2000, and it was equal to 55% in 2011.2 Some studies construct indices of vertical specialization to

measure the imported input content of exports. The World Trade Report 2013 presents preliminary

estimates of trade measured in valued-added terms and states that the share of total world trade

consisting of re-exports of intermediate inputs was about 30% in 2011, which had increased by

almost 10% since mid-1990s.3 Although the measures of international intermediate trade and

vertical specialization do not allow for a distinction between arm’s-length and intra-firm trade,

some firm-level data indicate that both types of trade have been substantial.4 In order to focus on

firms’ make-or-buy decisions and highlight the financial needs of both the buyers and suppliers of

intermediate inputs, this paper models only the arm’s length trade, avoiding the issues of vertical

integration and transfer pricing arising from intra-firm trade.5

The international trade of intermediates and vertical specialization have been studied in some

areas in macroeconomics. Kose and Yi (2001, 2006), Ambler, Cardia, and Zimmerman (2002),

Head (2002), Huang and Liu (2007), Burstein, Kurz, and Tesar (2008), and Arkolakis and Rama-

narayanan (2009) have examined the roles of intermediate input trade and vertical structure of

production in the propagating mechanism of international business cycles using dynamic stochastic

general equilibrium models.6 Kollmann (2002), Huang and Liu (2006), and Shi and Xu (2007)

examine the optimal monetary policy in the presence of intermediate input trade. Devereux and

Engel (2007) study the desirability of flexible exchange rate in a two-country model with interme-

2According to the OECD Economic Globalization Indicators 2010, the trade flows in intermediates among theOECD countries and with their major trading partners grew at an annual rate of 6.2% for goods and 7% for servicesduring the period 1995-2005, and the trade in intermediate inputs represented 56% of total goods trade and 73% oftotal services trade in 2006.

3Hummels, Ishii, and Yi (2001) construct an index of vertical specialization to measure the imported input contentof exports, capturing the fact that many imported intermediate goods are used in the production of other goods whichare then exported. They show that for 14 countries (10 OECD countries and Ireland, Korea, Taiwan, and Mexico),the growth in vertical specialization could account for about one-third of the growth in overall exports between1970 and 1990. According to the World Trade Report 2008, for the 29 countries for which the input-output datawere available from the OECD database, calculations of this index show that vertical specialization increased in 27countries between 1995 and 2000, and the growth in vertical specialization could account on average for more thanhalf of the growth in the export/output ratios.

4Lanz and Miroudot (2011) report that intra-firm trade statistics are available only for the United States, andthat in 2009, the share of arm’s length transactions in US imports of intermediate goods was 51.8%, and the shareof arm’s length transactions in US exports of intermediate goods was 71.3%.

5Some recent studies of intra-firm trade are presented by Bauer and Langenmayr (2013), Garetto (2013), Irar-razabal, Moxnes, and Opromolla (2013), and Ramondo and Rodrıguez-Clare (2013). It is noted that the effects ofmacroeconomic policies have not been examined in these studies.

6Bergin, Feenstra, and Hanson (2009) present the evidence from employment fluctuations in the offshoring in-dustries in Mexico to illustrate the cross-country comparison of volatility over the business cycles. di Giovanni andLevchenko (2010) find evidence that vertical production linkages across industries can help explain the impact ofinternational trade on business cycle comovements.

2

diate goods. Devereux and Genberg (2007) and Dong (2012) analyze the role of intermediate input

trade in the global imbalance adjustments and examine how large the exchange rate movements

are required for closing the US trade deficits.

This paper focuses on the trade in intermediates via international outsourcing and is different

from the previous studies in the macroeconomic literature in two important aspects. First, the

models in the literature make trade in intermediates necessary by assuming that each type of in-

termediate input is produced exclusively by the firms of one country, and that both the domestic

and imported intermediates must be used in the production of final goods.7 In contrast, this model

assumes that the intermediate inputs produced domestically and abroad are perfect substitutes so

that trade in intermediates is optional, but not necessary. By incurring a fixed cost of international

outsourcing, a domestic firm can import the intermediate input at a lower unit cost. Depending

on their productivity levels in producing the final good, some domestic firms choose to produce

their intermediate inputs in-house, while the other prefer to engage in international outsourcing.

Hence, firms’ reliance on the imported intermediate inputs is endogenously determined, the econ-

omy can adjust its use of the imported intermediate inputs not only at the intensive margin (the

changes in the quantities demanded for imports of the firms that have already been sourcing from

abroad) but also at the extensive margin (the changes in the number of firms entering in outsourc-

ing arrangements).8 By modeling explicitly firms’ make-or-buy decisions, we investigate how the

presence of international outsourcing affects the effects of monetary and foreign exchange policies.

In particular, the relationship between the level of a unilaterally fixed nominal exchange rate and

the trade balance is examined. Understanding the adjustment at the extensive margin of trade in

intermediates in response to changes in the market conditions would provide new insight into the

effectiveness of the policies.

Second, our study highlights the function of financial flows in facilitating the flows in goods

and labor related to international outsourcing activities and analyzes the effects of firms’ sourc-

ing decisions on the demands for liquidity in financial markets for financing the production and

trade of intermediate inputs, whereas the financial aspects of the transactions are often omitted

7For example, Kose and Yi (2001, 2006) assume that the elasticity of substitution between domestic and foreignintermediate goods is equal to 1.5. Ambler, Cardia, and Zimmerman (2002), Devereux and Engel (2007), Huang andLiu (2007) and Shi and Xu (2007 and 2010) assume Cobb-Douglas production functions so that the domestic andforeign intermediate inputs have a unitary elasticity of substitution. Devereux and Genberg (2007) assume a Leontieftechnology that imported intermediate inputs must be used in fixed proportion with domestic inputs.

8Bergin, Feenstra, and Hanson (2011) use a trade model to illustrate how the responses at the extensive marginof offshoring to demand shocks can amplify the employment volatility of the offshoring industries in the low-wagecountries.

3

in the macroeconomic literature studying trade in intermediates. Due to the time lags between

production and the receipts from sales, firms often face liquidity constraints and have to finance

their working capital externally. As surveyed by Manova (2010), the recent literature on trade and

finance has focused on the role of domestic financial market frictions and stressed the importance

of the negative impacts of credit constraints on a country’s international trade flows.9 This paper

demonstrates that firms’ make-or-buy decisions determine not only the locations of production of

the intermediate inputs but also the loan markets to which the producers seek external financing.

The domestic country’s exports of final good would therefore depend on the availability of liquidity

in both the home and foreign loan markets. The interdependence of liquidity flows across differ-

ent financial markets are examined by assuming segmented financial markets and cash-in-advance

constraints on all transactions. Market participants have asymmetric access to liquidity due to

financial market segmentation. The cash-in-advance assumption highlights the liquidity services

provided by money.10 Financial intermediaries channel funds collected from depositors to facil-

itate the production and international trade. Based on their individual optimization problems,

the households make their portfolio decisions between deposits for savings and cash holdings for

consumption purchases, while the firms and importers choose their working capital demands. Our

study of the interactions of the supplies and demands of loanable funds in a general equilibrium

framework shows that international outsourcing has important implications on the liquidity flows

and thus the real allocation in the world economy.

To keep the model simple, we assume that labor is the only primary factor of production in the

world economy, and that production fragmentation occurs in the final good sector of the domestic

country only. The outsourcing relationship involves the domestic country outsourcing some tasks

to the foreign country, referred to as the production of the intermediate good for convenience. The

value added by foreign labor to the domestic production is therefore captured by the value of the

domestic economy’s imports of intermediates. Hence, the trade in intermediates in this model is

defined in a broader sense to capture not only the trade in intermediate goods and services, but

also the trade in tasks introduced by Grossman and Rossi-Hansberg (2008) to describe the value

9The global trade collapse during 2008-2009 has raised attention to the negative impacts of domestic credit marketfrictions on the ability of firms to exports and on countries’ trade flows. Lanz and Miroudot (2011) find that duringthe period of 2008-2009, the United States experienced a larger decline in its imports of intermediate goods than inits imports of final goods, however, the decline in its intra-firm intermediate goods imports seemed to be less severe.

10In contrast, monetary models in the macroeconomic literature mentioned above introduce money by eitheradopting money-in-utility preferences or assuming the use of the nominal interest rate as a monetary policy instrumentwithout modeling the function of money explicitly.

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added contributed by the factors of production in different locations.

The results are summarized as follows. First, under a fixed nominal exchange rate regime,

the effects of monetary and productivity shocks depend not only on the presence of international

outsourcing activities but also on the contractual upfront payment arrangements between the do-

mestic (source) firms and their foreign suppliers. The domestic firms’ decisions to outsource shift

the financing of the working capital required for the production of the intermediate input from the

domestic loan market to the foreign loan market. The upfront payment arrangement determines

the foreign suppliers’ reliance on the foreign loan market in meeting their working capital needs

for the production of the intermediate input, affecting the responsiveness of liquidity demands to

adjustments in the market interest rates as well as the directions and magnitudes of the official in-

tervention required to accommodate the fixed nominal exchange rate. The model demonstrates the

importance of firms’ sourcing decisions and payment arrangements in determining the international

transmissions and welfare implications of shocks.11

Second, a foreign currency revaluation leads more domestic firms to outsource to foreign firms

and results in an improvement in the foreign country’s trade balance. The general equilibrium

adjustment mechanism is the reason behind these counter-intuitive results. When there is no inter-

national outsourcing, an increase in the value of foreign currency results in reductions (increases)

in the domestic (foreign) households’ demands for imported consumption goods, deteriorating the

foreign country’s trade balance. With international outsourcing, there are additional effects via

the adjustments of trade in intermediates. A foreign currency revaluation leads the domestic firms

that have been outsourcing abroad to adjust at the intensive margin by reducing their intermediate

imports and thus the production of the domestic consumption good. Hence, the domestic price of

the domestic consumption good will rise substantially, leading some domestic firms to switch from

producing their intermediate inputs in-house to outsourcing abroad. The adjustment of the imports

of intermediates at the extensive margin plays a dominant role in determining the trade flows, re-

sulting in a perverse effect on the trade balance. Many studies in the literature have explained why

a country’s trade surplus may not be responsive to its currency revaluations, but they could not

explain China’s accelerating increases in its trade surplus along with its currency revaluations.12

11For the responses to temporary shocks, the adjustments of imports of intermediates are at the intensive marginonly, given that the households’ saving decisions and firms’ outsourcing decisions are inflexible in the short run.

12Many recent empirical studies find that Chinese trade flows do not respond to exchange rate movements assuggested by conventional wisdom. Cheung, Chinn, and Qian (2012) re-examine Chinese trade flows and also providea detailed survey of the empirical literature.

5

Considering China as the foreign country in our model and taking into account the rise of China

as a major host country of international outsourcing, this paper offers a theoretical framework to

rationalize the puzzling positive correlation between China’s currency value and its trade surplus

over the recent period since 1994.13 It sheds light on the continuing improvements in China’s trade

balance in intermediate goods in spite of the substantial increases in its currency value.

Third, a reduction in the fixed cost associated with outsourcing makes the foreign country

better off and the domestic country worse off under both flexible and fixed exchange rate regimes.

Given that a revaluation of the foreign currency benefits the home country and makes the foreign

country worse off, it can be used as a policy tool to redistribute some of the welfare gain of the

foreign country to the domestic country so that both countries can benefit from a reduction in

the fixed cost of international outsourcing and attain a higher aggregate welfare level. This result

highlights the asymmetry in the welfare effects of international outsourcing and illustrates the

welfare-redistributive role of a fixed exchange rate regime when there is international outsourcing.

The remainder of the paper is organized as follows. The model is presented in Section 2. Section

3 analyzes the effects of changes in some exogenous variables on the world economy. Some welfare

analyses are presented in Section 4. Section 5 concludes the paper.

2 The Model

Consider a world economy consisting of two countries, home and foreign. There are two final (con-

sumer) goods, goods x and y, one intermediate input, referred to as good I, and one primary factor

of production, labor. In each country, there are a monetary authority and many infinitely-lived,

multi-member households. Each household consists of five members: a shopper, an entrepreneur,

a worker, a financial intermediary, and an importer. The households of each country are identical

ex ante, and the number of households in each country is normalized to one. All foreign variables

and parameters are indexed with asterisks (*).

Household members separate and conduct different tasks in different segmented markets during

a period, while pooling their income and consumption at the end of the period. All markets are

perfectly competitive so that every market participant acts as a price taker. To introduce money

13As will be illustrated in Figure 3, the correlation coefficient between the official USD/RMB rate and China’soverall trade balance to GDP ratio was positive and equal to 0.75 during the period of 1994-2008, and 0.17 for theperiod of 1994-2014. For its trade in intermediate goods, the correlation coefficient of the official USD/RMB rateand the intermediate goods trade balance to GDP ratio was equal to 0.91 for the period of 1994-2008 and 0.83 forthe period of 1994-2014.

6

to the world economy, all transactions are assumed to be subject to cash-in-advance constraints,

and payments must be made in terms of the sellers’ currency. Shoppers need cash to pay for

their consumption purchases. Entrepreneurs and importers need working capital to facilitate their

activities because of the timing frictions between the payments of operating costs and receipts from

sales. All loans are intermediated through financial intermediaries.

2.1 Preferences

The preferences of the representative home household has the following life-time utility function,

U =∞∑t=0

β t u(Cxt, Cyt, ht), 0 < β < 1, (1)

where β is the subjective discount factor, and u(Cxt, Cyt, ht) = a lnCxt + (1− a) lnCyt + v [ 1− ht ]

is the instantaneous utility function. In period t, the household consumes Cjt units of final good j,

j = x, y, and supplies ht units of work effort to the domestic labor market. The expenditure share

on good x is measured by the parameter a, 0 < a < 1.14 The time endowment of the worker is

normalized to one, and v is a positive parameter measuring the constant marginal utility of leisure.

Similarly, the representative foreign household has preferences given by

U∗ =∞∑t=0

β∗ t u∗(C∗xt, C∗yt, h

∗t ), 0 < β∗ < 1,

with u∗(C∗xt, C∗yt, h

∗t ) = (1 − a∗) lnC∗xt + a∗ lnC∗yt + v∗ [ 1 − h∗t ], 0 < a∗ < 1, and v∗ > 0. The

foreign household consumes C∗jt units of final good j, j = x, y, and supplies h∗t units of work effort

to the labor market of the foreign country. It has an expenditure share on good y, a∗, and a

constant marginal utility of leisure, v∗.

2.2 Technology and Production

In the home country, the home entrepreneurs operate the home firms and specialize in the produc-

tion of good x following the production function,

Qixt = Axt θit Iitα, (2)

where Axt > 0, θit > 0, and 0 < α ≤ 1. Given the country-wide production parameters, Axt and

α, the home firm with the firm-specific productivity parameter, θit, inputs Iit units of intermediate

good I to produce Qixt units of good x. In each period t, the parameter θit is uniformly distributed

on the unit interval [ θ, θ ], where 0 ≤ θ < θ and θ − θ = 1. Every home firm can produce its

14There is home bias in consumption in the case with 12< a < 1.

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intermediate input in-house following an identical, linear production technology, using one unit of

labor effort to produce one unit of good I.

Iit = lit, (3)

where lit is the labor input hired from the home labor market to produce intermediate good I

in-house.15 Alternatively, the home firm can import good I from the foreign country by engaging

in international outsourcing, purchasing it at the unit cost of q∗t units of foreign currency, plus a

real fixed cost of international outsourcing, κ units of good x.

Each home firm is subject to a cash-in-advance constraint, it has to borrow its working capital

from a home financial intermediary to finance its operation (hiring domestic workers or importing

good I) in order to maximize its profit. It is noted that the decreasing returns production technology

given by (2) can be interpreted as inputting Iit units of the intermediate good and one unit of the

entrepreneur’s effort, with the income shares denoted respectively by α and 1 − α, so that the

profit is the compensation to the entrepreneur. As the entrepreneur’s own effort is not subject to

a cash-in-advance constraint, the working capital need of the home firm is increasing in α.

Following the production functions (2) and (3), a home firm with productivity parameter θit

and producing good I in-house faces the decision problem of maximizing its nominal profit πt(θit),

πt(θit) ≡ maxbit, lit

PxtAxt θitlitα + (bit − wt lit)− bit(1 + it)

subject to the liquidity constraint,

wt lit ≤ bit. (4)

The home firm borrows bit units of home currency from the home financial intermediary to finance

its hiring of lit units of labor from the home labor market at the wage rate of wt units of home

currency. The output of good x will be sold at the home goods market at the price of Pxt units of

home currency. The first-order condition of lit is given by

αPxtAxt θitlitα−1 = (1 + it)wt. (5)

The home firm hires workers until the marginal revenue product of labor αPxtAxt θitlitα−1 is equal

to the effective marginal cost of labor (1 + it)wt. Solving this condition yields the optimal level of

lit. We can then derive the firm’s optimal profit, πt(θit).

lit =

(αPxtAxt θit(1 + it)wt

) 11−α

and πt(θit) = (1− α)PxtAxt θit

(αPxtAxt θit(1 + it)wt

) α1−α

.

15As all home firms face the same production technology of the intermediate input, there will be no domesticoutsourcing (trade in intermediates among the home firms).

8

The decision problem of a firm with productivity parameter θit and importing good I at the

price of q∗t units of foreign currency is to maximize its nominal profit, πot (θit). In order to import

good I, the firm must have an outsourcing contract that specifies the fraction ξ of the contract

payment that the home firm has to pay upfront to its foreign supplier before the beginning of the

production process, and the remaining fraction 1− ξ will be paid after the production and sale of

good x. The parameter value of ξ, 0 ≤ ξ ≤ 1, is assumed to be exogenously given.16 It allows us to

capture different types of outsourcing relationships between the home (source) firm and its foreign

supplier. The case of ξ = 0 represents a completely arm’s-length relation between the two firms in

which the foreign firm bears the liquidity burden of financing the production of the intermediate

good solely. When 0 < ξ < 1, the two firms shares the liquidity burden. The larger the value of the

fraction ξ, the lower the liquidity burden on the foreign supplier, and the less the working capital

that it needs to borrow from the foreign loan market. In the case of ξ = 1, the home firm provides

the foreign firm with the entire working capital required for the production of intermediate good

I.17 As will be illustrated in Section 3.1, the value of ξ plays a crucial role in determining the

effects of various exogenous changes on the world economy.

The home firm borrows boit units of home currency from the home financial intermediary to

finance the contractual upfront payment for the purchase of Iit units of good I from a foreign

supplier. Taking as given the home-currency price of good x, Pxt, the foreign-currency price of good

I, q∗t , and the nominal exchange rate (the price of foreign currency in terms of home currency), et,

and applying the technological constraint (2), the optimal profit πot (θit) is given by

πot (θit) ≡ maxboit, Iit

Pxt (Axt θitIitα − κ) + (boit − ξ etq∗t Iit)− boit(1 + it)− (1− ξ) etq∗t Iit

subject to the liquidity constraint,

ξ etq∗t Iit ≤ boit. (6)

The presence of the real fixed cost of international outsourcing κ implies that the net output of

good x produced by the home firm is Axt θitIitα − κ. The first-order condition of Iit is given by

αPxtAxt θitIitα−1 = (1 + ξ it) etq

∗t . (7)

16Some factors that determine the value of ξ are the existence of established trading relationship and the legalprotection of contract enforcement.

17This arrangement can be considered as the home firm’s direct investment in the intermediate-good producingbranch of the foreign firm. However, this interpretation is just a loose characterization. Foreign direct investmentinvolves more than the provision of working capital, and this paper does not intent to model the choice betweenoutsourcing and foreign direct investment. See Grossman and Helpman (2003) for an examination of the choicebetween outsourcing and foreign direct investment.

9

The home firm inputs I to the production of good x until the marginal benefit αPxtAxt θitIitα−1

is equal to the effective marginal cost (1 + ξ it) etq∗t . Using this condition, we can derive the firm’s

optimal demand for the intermediate input and profit level.

Iit =

(αPxtAxt θit

(1 + ξ it)etq∗t

) 11−α

, and πot (θit) = (1− α)PxtAxt θit

(αPxtAxt θit

(1 + ξ it)etq∗t

) α1−α− Pxtκ.

The cutoff level of θit for a home firm to enter into an international outsourcing arrangement

is denoted by θot . The firms with θit ∈ [ θ , θot ] choose to produce input I in-house, while those

with θit ∈ ( θot , θ ] prefer to source good I from the foreign country. As will be derived in Sections

2.4 and 2.5, given the timing of events, the outsourcing decisions have to be made based on the

knowledge of θit and κ only, the optimal cutoff level, θot will be determined by

Et [πt(θot )] = Et [πot (θ

ot )] ,

where the expectation operator is conditional on all the information available at the beginning of

period t, before the realization of the current state of the world st is revealed.18

In the foreign country, the foreign entrepreneurs operate firms to produce final good y and/or

intermediate good I, using labor input following the production functions.

Q∗yt = A∗yt l∗ytα∗, (8)

and

Q∗It = φ∗It l∗It. (9)

The output of good y, Q∗yt, depends on the labor input, l∗yt, and the country-wide productivity

parameters, A∗yt and α∗, where A∗yt > 0 and 0 < α∗ < 1. The output of good I, Q∗It, depends on

its labor input, l∗It, and the country-wide productivity parameter, φ∗It, where φ∗It > 0. A foreign

firm will produce good I for a home firm if they have a contractual arrangement. The two goods

are assumed to be produced by the same firm for convenience. The representative foreign firm can

be considered as having two branches, one branch produces good y, and another produces good

I. The decreasing returns production technology given by (8) implies that there will be a positive

profit from producing good y. Given the linear production technology of good I, firms always earn

zero profit from producing good I.

18The value of et will also been known under a fixed exchange rate regime, while it will be determined by therealization of st under a flexible exchange rate regime.

10

Using the production functions (8) and (9), taking as given the foreign prices of good y and

good I, P ∗yt and q∗t , the wage rate in the foreign labor market, w∗t , the nominal interest rate

on foreign-currency-denominated loans, i∗, and the fraction of upfront contract payment, ξ, the

decision problem of the representative foreign entrepreneur is given by,

π∗t ≡ maxb∗t , l

∗yt, l

∗It

P ∗ytA

∗ytl∗ytα∗+ (1− ξ) q∗t φ∗It l∗It +

(b∗t + ξq∗t φ

∗It l∗It − w∗t

(l∗yt + l∗It

))− (1 + i∗t ) b

∗t

,

subject to the liquidity constraint,

w∗t

(l∗yt + l∗It

)≤ b∗t + ξq∗t φ

∗It l∗It, (10)

where ξq∗t φ∗It l∗It = ξq∗tQ

∗It is the upfront contract fee received from the buyers of good I for supplying

Q∗It following the production technology (9). The foreign firm’s working capital need is increasing

in α∗ but decreasing in ξ. By borrowing b∗t units of foreign currency from the foreign financial

intermediary and receiving ξq∗tQ∗It from the home firm to finance its hiring of workers, the firm will

receive the sale revenue from good Y and the remaining contract payment from producing good I

and maximize its profit, π∗t . The first-order condition for l∗yt is given by

α∗P ∗ytA∗ytl∗ytα∗−1 = (1 + i∗t )w

∗t , (11)

the marginal revenue product and the effective marginal cost of labor are equalized, determining

the optimal level of l∗yt.

l∗yt =

(α∗P ∗ytA

∗yt

(1 + i∗t )w∗t

) 11−α∗

.

Using the first-order condition for l∗It, we get

q∗t φ∗It =

(1 + i∗t )w∗t

(1 + ξ i∗t ). (12)

The foreign firm will produce good I only if the marginal benefit of hiring an additional unit of

labor q∗t φ∗It is equal to the effective unit labor cost (1 + i∗t )w

∗t /(1 + ξ i∗t ). It earns zero profit from

producing good I. We can then derive the optimal profit of the foreign firm, π∗t ,

π∗t = (1− α∗)P ∗ytA∗yt

(α∗P ∗ytA

∗yt

(1 + i∗t )w∗t

) α∗1−α∗

.

The home (foreign) firms make profits from producing the final goods when 0 < α < 1 (0 <

α∗ < 1). For simplicity, it is assumed that international trade in equity is prohibited. In order to

eliminate the wealth effects from the realizations of θit among the home households, every home

11

household is assumed to own a share of every home firm so that there is complete risk-sharing within

the home country. In other words, the representative household owns all firms of its country.

2.3 The Monetary Authorities

The monetary authority of each country conducts its monetary and foreign exchange policies

according to its international monetary arrangement. In order to study the effects of a devalua-

tion/revaluation of the foreign currency on the world economy, it is assumed that the monetary

authority of the home country follows an exogenous monetary policy, characterized by an open

market purchase Bt, and does not conduct any foreign exchange policy, while the monetary au-

thority of the foreign country unilaterally maintains an exogenous, fixed exchange rate et = e by

adjusting its sterilized foreign exchange sale Z∗t .19

In period t, the home monetary authority purchases Bt units of home-currency denominated

bonds from the home financial intermediary. The value of Bt is exogenously given by the realization

of the current state of the world st. Given the fixed rate e, in response to the realization of st,

the foreign monetary authority sells Z∗t units of foreign currency to the foreign exchange market

and fully sterilizes its impact on the stock of foreign currency in circulation by selling Z∗t units

of foreign-currency-denominated bonds to the market, leaving the quantity of foreign currency in

circulation unchanged. It is noted that the derivation described below can also be applied to the

case with a flexible exchange rate simply by imposing Z∗t =0 and allowing et to adjust endogenously.

Let Mt denote the aggregate money stock of the home country at the beginning of period t. The

home monetary authority’s open market purchase of Bt units of home-currency-denominated bonds

increases the quantity of home currency in circulation during period t to the level of Mt + Bt. As

the foreign monetary authority uses the home currency purchased, eZ∗t , to buy the home-currency-

denominated bonds from the bond market, it does not affect the home currency in circulation.20

Given that our focus is on the allocations of liquidity in the financial markets, we normalize the

beginning-of-period, aggregate money stock of each country to unity over time, Mt = Mt+1 = 1,

and M∗t = M∗t+1 = 1, ∀ t. At the end of period t, after all the bonds are redeemed, the aggregate

19The modeling of the open market operations in the home loan market follows the literature on the liquidityeffects of monetary policy (See Lucas (1990) and Fuerst (1991)). The modeling of the monetary policy of the foreigncountry tries to capture the characteristics of a managed fixed exchange rate regime of China. As discussed byAizenman (2015), China’s growing trade surplus has been in tandem with its massive international reserve hoardingand sterilization. Chang, Liu, and Spiegel (2015) study China’s optimal monetary policy in a dynamic stochasticgeneral equilibrium model that features a nominal exchange rate peg and and sterilized central bank interventions.

20Under the fixed exchange rate regime, when the foreign country runs a trade surplus in period t, the foreignmonetary authority has to conduct a sterilized official sale of foreign currency, Z∗t > 0, resulting in an increase in itsend-of-period foreign exchange reserve holding by Z∗t (1 + i∗t ) units of home currency.

12

stocks of money in circulation of the home and foreign countries are given by Mt−Btit−eZ∗t (1+ it)

and M∗t +Z∗t (1 + i∗t ), respectively. Hence, the home monetary authority will distribute a lump-sum

transfer of Tt = Btit + eZ∗t (1 + it) units of home currency to each home household, and the foreign

monetary authority will impose a lump-sum tax of T ∗t = Z∗t (1 + i∗t ) units of foreign currency on

each foreign household.

2.4. The Timing of Information and Transactions

The timing of information and transactions are summarized in Figure 1. The values of the set of

parameters (κ, e) are assumed to be exogenously fixed and known to everyone.21 The representative

home household enters period t with cash balances of mht units of home currency and mft units of

foreign currency, and decides on the optimal values of nt and θot .

The household deposits nt units of home currency and mft units of foreign currency in the home

financial intermediary, allocates the remaining mht − nt units of home currency to the shopper.22

After the household members separate and go to different markets to perform different tasks, the

realization of the firm-specific productivity parameter θit is revealed to everyone. As it takes time

to make a contractual arrangement abroad, the home entrepreneur has to make the make-or-buy

decision based on the comparison of θit and θot before the state of the world, st, is observed. The

decision to outsource is irreversible and cannot be changed until the beginning of next period.23

However, the quantity of the intermediate input imported is assumed to be determined by the home

entrepreneur after observing the state of the world, st.

The distribution of st is assumed to be independent and identical across time, with a known

probability density function denoted by G(st). When the state of the world st = (Axt, A∗yt, φ

∗It, Bt )

is revealed; the realizations of the country-wide productivity shocks, Axt, A∗yt, and φ∗It, and the

monetary policy of the home country, Bt, become public knowledge.

Because of the cash-in-advance constraints on transactions, the financial intermediaries play an

important role in the allocation of liquidity in the financial markets. They accept deposits and

make loans to the entrepreneurs to finance their production and to the importers to finance their

21Changes in the fixed cost associated with international outsourcing would not occur very often. Similarly,devaluations/revaluations of a currency under a the fixed exchange rate regime are not supposed to happen frequently.Hence, any changes in these parameters will be treated as unanticipated so that the households would not take intoaccount the possibilities of these changes when making their decisions.

22In order to economize on the notations, each household is assumed to deposit to the financial intermediary of itsown country only.

23The assumption of the sluggish adjustment of firms’ outsourcing decision is to capture the evidence presented byJabbour (2013) that outsourcing is a persistent strategy.

13

imports. In addition, they may participate in trading in the bond markets, buying or selling bonds

denominated in either home or foreign currency. In order to avoid modeling the re-opening of the

foreign exchange market at the end of each period, it is assumed that firms and importers obtain

loans only from the intermediaries of their own country.24

Taking as given the nominal interest rates on the one-period, home-currency-denominated as-

sets, it, and on the one-period, foreign-currency-denominated assets, i∗t , the nominal exchange rate

et = e, the representative home intermediary collects the deposits nht and mft from the home

household, makes loans bit and boit to the home entrepreneurs and Lyt to the representative home

importer, and purchases bht units of home-currency-denominated bonds and b ft units of foreign-

currency-denominated bonds so as to maximize the benefit to its depositors, facing the following

liquidity constraint,

nht + etmft ≥ Lxt + Lyt + bht + etbft, (13)

where

Lxt =

∫ θot

θbit di+

∫ θ

θot

boit di, (14)

denotes the total quantity of loans allocated to the home entrepreneurs. As illustrated respectively

in equations (4) and (6), the home firms use the loans bit and boit to finance their production. At

the end of period t, the intermediary receives the repayments and pays its depositors.25 As will be

shown in Section 2.5, the optimization problem of the financial intermediary can be nested into the

optimization problem of the representative home household.

For a home firm engaging in international outsourcing, its liquidity needs are determined by

the fraction ξ. It borrows boit units of home currency and convert them into foreign currency in

the foreign exchange market so as to make the upfront payment of ξq∗t Iit units of foreign currency

to its foreign supplier of good I. In addition, in order to pay for its remaining balance, the home

firm pre-arranges the purchase of (1 − ξ)q∗t Iit units of foreign currency from the home financial

intermediary to be settled after the production and sale of good x. Hence, the home intermediary

will take the firm’s demand for (1 − ξ)q∗t Iit units of foreign currency into account when choosing

its purchase of bft units of foreign-currency-denominated bonds.

24These assumptions help economize on the notations, while not making any qualitative difference to the results.25It is assumed that the loan market is perfectly competitive, there is no default on loans, and the financial

intermediaries do not face reserve requirements.

14

The requirement of using the sellers’ currency for transactions implies that some economic agents

have to trade for their desired currency. In the foreign exchange market, the home firms sourcing

good I from the foreign country sell∫ θθotboit di, units of home currency, the home importers sell Lyt

units of home currency, and the foreign importers sell L∗xt units of foreign currency. In addition,

the home financial intermediary demands for bft−mft units of foreign currency for its purchase of

bft units of foreign-currency-denominated bonds, while the foreign financial intermediary demands

for b∗ht −m∗ht units of home currency for its purchase of b∗ht units of home-currency-denominated

bonds. In order to maintain the fixed exchange rate e, the monetary authority of foreign country

adjusts the value of its official sale of foreign currency Z∗t .

Labor is assumed to be internationally immobile. The home worker supplies ht units of labor

effort to the market at the nominal wage rate wt. For a home firm producing its intermediate input

in-house, it hires lit units of labor from the home labor market using the home currency borrowed

from the home intermediary bit.

In the goods markets, the home firms sell their output of good x to the home shoppers and

the foreign importers at the price of Pxt units of home currency, while the foreign firms sell their

output of good y to the foreign shoppers and the home importers at the price of P ∗yt units of foreign

currency.

The representative home importer uses the cash balance of Lyt/e units of foreign currency to

purchase IMyt units of good y and then sells them to the home shopper at the price of Pyt units of

home currency so as to maximize the profit,

maxIMyt

Pyt IMyt +

(Lyte− P ∗yt IMyt

)e− Lyt(1 + it)

,

subject to the cash-in-advance constraint,

Lyte≥ P ∗yt IMyt. (15)

The first-order condition indicates that the marginal benefit and marginal cost of imports should

be equalized,

Pyt = (1 + it)eP∗yt. (16)

Taking the home-currency prices of the final goods as given, the representative home shopper

purchases both goods for consumption, facing the cash-in-advance constraint,

mht − nt ≥ PxtCxt + PytCyt. (17)

15

At the end of the period, loan repayments are made, bonds are redeemed, deposits are paid

out, and profits of firms are distributed to their shareholders. After all transactions are completed,

members of each household are reunited, pool their earnings, and consume the final goods purchased

by the shopper. The representative home household then holds the cash balances mht+1 and mft+1

for next period,

mht+1 = [nht + emft − Lxt − Lyt − bht − ebft ] + [mht − nht − PxtCxt − PytCyt ]

+ [Lxt − wtlxt − ξe q∗t QIt] + [PxtQxt − (1− ξ)e q∗t QIt − Lxt(1 + it)]

+ [PytIMyt − Lyt(1 + it) ] + [ (Lxt + Lyt + bt)(1 + it) ] + wt ht + Tt, (18)

and

mft+1 =Lyte− P ∗yt IMyt + bft(1 + i∗t )− (1− ξ)q∗t QIt, (19)

where lxt is the aggregate demand for labor of the home firms producing their intermediate inputs

in-house, QIt is the aggregate demand for good I of the home firms outsourcing abroad, and Qxt

is the aggregate output of good x of all home firms (net of the total fixed cost of outsourcing).

lxt ≡∫ θot

θlit di =

(1− α2− α

)[αPxtAxt

(1 + it)wt

] 11−α [

θot2−α1−α − θ

2−α1−α

], (20)

QIt ≡∫ θ

θot

Iit di =

(1− α2− α

)[αPxtAxt

(1 + ξit) e q∗t

] 11−α

[θ

2−α1−α − θot

2−α1−α

], (21)

Qxt ≡∫ θot

θQixt di+

∫ θ

θot

(Qixt − κ) di

= Axt

(1−α2−α

)[[αPxtAxt(1+it)wt

] α1−α [

θot2−α1−α−θ

2−α1−α]+

[αPxtAxt

(1+ξit)e q∗t

] α1−α

[θ

2−α1−α−θot

2−α1−α

]]−κ

(θ−θo

). (22)

2.5 The Optimization Problem of the Representative Home Household

Given the money holdings of the representative home household at the beginning of the current pe-

riod, (mh,mf ), and based on its knowledge of the probability density function G(s), the household’s

value function V (mh,mf ) is defined as follows.

V (mh,mf ) = maxnh, θo

∫max

Cx, Cy , h, bh, bf , Lx, Ly , IMy

u(Cxt, Cyt, ht) + β V (m′h,m

′f )G(s)ds,

subject to

nh + emf ≥ Lx + Ly + bh + e bf , (13′)

Lx ≥(

1−α2−α

)[w

[αPxAx

(1 + i)w

] 11−α [

θo2−α1−α − θ

2−α1−α

]+ ξe q∗

[αPxAx

(1 + ξi) e q∗

] 11−α[θ

2−α1−α − θo

2−α1−α

]], (14′)

16

Lye≥ P ∗y IMy, (15′)

mh − nh ≥ PxCx + Py Cy, (17′)

m′h = mh + emf − PxCx − PyCy − Ly − e bf + Py IMy + bh i+W h+ T

+

(1 + i

α− 1

)(1−α2−α

)w

[αPxAx

(1 + i)w

] 11−α [

θo2−α1−α−θ

2−α1−α

]

+

(1 + ξi

α− ξ

)(1−α2−α

)e q∗[

αPxAx(1 + ξi) e q∗

] 11−α[θ

2−α1−α−θo

2−α1−α

]− Pxκ

(θ − θo

), (18′)

and

m′f =Lye− P ∗y IMy + bf (1 + i∗)− (1− ξ)

(1−α2−α

)q∗[

αPxAx(1 + ξi) e q∗

] 11−α

[θ

2−α1−α−θo

2−α1−α

]. (19′)

For convenience, the time subscripts of the current-period variables have been dropped, and the

next-period values of the variables are denoted by primes. The first-order conditions are presented

in the Appendix. Solving the constraints and the first-order conditions, we have

PxCx = a(mh − nh), and PyCy = (1− a)(mh − nh), (23)

β w

mh′ − n′h

= v, (24)

E

[uxPx

]= β E [ 1 + i ] E

[ux′

Px ′

], where E

[uxPx

]=

1

mh − nh, (25)

E

[(1− α)PxAxθ

o[αPxAxθ

o

(1+i)w

] α1−α

]

= E

[(1− α)PxAxθ

o[αPxAxθ

o

(1+ξi)eq∗

] α1−α− Px κ+ (1− ξ) eq∗

[αPxAxθ

o

(1+ξi)eq∗

] 11−α

(i∗ − i1 + i∗

)], (26)

where the expectations in equations (25) and (26) are carried out over all possible states of the

world using the probability density function G(s). Equation (23) gives the optimal consumption

allocation of the shopper, a fraction a of the money holding is spent on good x and the remaining

fraction is on good y because the two consumption goods have a unit elasticity of substitution

in the household’s preferences. Equation (24) states that the worker is willing to work if the

household’s discounted expected future marginal utility gain from consuming the additional goods

purchased by the nominal wage, βw[

1m′h−n′

h

], is equal to the current marginal disutility of working,

v. Equation (25) is the intertemporal Euler equation that gives the optimal deposit decision based

17

on the first-order condition for n. The household should deposit the amount n so as to equate the

current marginal utility of consumption and the discounted expected future marginal utility from

consumption purchased by using the gross return from deposits 1+ i. In other words, n is chosen so

that the shadow value of bringing an additional unit of money to the goods market for the purchase

of consumption good and the shadow value of depositing an additional unit of money to the home

loan market for future consumption is expected to be equal. Equation (26) determines the optimal

cutoff value of the firm-specific productivity parameter, θo. The left hand side is the expected

optimal payoff of the home firm with θo choosing to produce good I in house, E [π(θo)]. The right

hand side is the expected optimal payoff if the firm chooses to outsource the production of good

I aboard, E [πo(θo)], which includes the profit net of the fixed cost of outsourcing plus the gain in

interest earning from being able to pay the remaining balance at the end of the period. The optimal

value of θo is where the expected payoffs from the two options, E [π(θo)] and E [πo(θo)], are equal.

Firms with θi > θo would engage in outsourcing, while those with θi < θo would choose in-house

production. Firms with θi = θo are indifferent between the two options. Hence, the number of

home firms sourcing internationally is θ − θo.26

Similarly, solving the optimization problem of the representative foreign household, we get the

following conditions to characterize its optimal choices for C∗x, C∗y , h∗, and n∗f .

P ∗xC∗x = (1− a∗)(m∗f − n∗f ), and P ∗yC

∗y = a∗(m∗f − n∗f ), (27)

β∗w∗

m∗f′ − n∗f

′ = v∗, (28)

E

[u∗x

P ∗x

]= β∗ E [ 1 + i∗ ] E

[u∗x′

P ∗x′

], where E

[u∗x

P ∗x

]=

1

m∗f − n∗f. (29)

2.6 Stationary Rational Expectations Equilibrium

In the stationary rational expectations equilibrium of the world economy, the prices and decision

rules are fixed functions of the current state of the world, s = (Ax, A∗y, φ

∗I , B ). Given that the

deposit decisions, nh and n∗f , and the international outsourcing decision, θo, are made before the

realization of the current state of the world, they are independent of s. The following conditions are

satisfied in the equilibrium. First, the representative household of each country derives its optimal

allocation by solving its optimization problem. Second, the arbitrage conditions hold,

P ∗x =Px (1 + i∗)

eand Py = P ∗y e (1 + i), (30)

26As suggested in the literature on trade, only firms with higher productivity levels engage in international out-sourcing.

18

Finally, all markets clear.

Labor markets: h = lx , h∗ = l∗y + l∗I , (31)

goods markets: QI = Q∗I , (32)

Cx + C∗x = Qx, IM∗x = C∗x, (33)

Cy + C∗y = Q∗y, IMy = Cy, (34)

money markets: mh +m∗h = 1, mf +m∗f = 1, (35)

loan markets: nh + emf = w lx + ξ e q∗I + e P ∗yCy + bh + e bf , (36)

n∗f +m∗he

= w∗l∗y + (w∗l∗I − ξq∗I) +PxC

∗x

e+ b∗f +

b∗he, (37)

bond markets: B + bh + b∗h + eZ∗ = 0, bf + b∗f = Z∗, (38)

foreign exchange market: e P ∗yCy + ξe q∗ I + e (bf −mf ) = PxC∗x + (b∗h −m∗h) + eZ∗. (39)

Solving for the Stationary Rational Expectation Equilibrium

Defining 1 − n ≡ mh − nh and 1 − n∗ ≡ m∗f − n∗f . As the state of the world is independent

and identically distributed over time, households optimize by choosing Vmh = E[uxPx

]= 1

1−n and

V ∗mf = E[u∗xP ∗x

]= 1

1−n∗ in each period. Each household will adjust its deposit decision according to

its money balances carried from the previous period. Each home household allocates 1− n to the

home shopper, and each foreign household will allocate 1 − n∗ to the foreign shopper. With the

assumption that the financial intermediates are allowed to exchange for the currency they need in

the foreign exchange market when allocating their loans, we will simplify the analysis by focusing

on the equilibrium in which mf′ = m∗h

′ = 0, implying bf = (1−ξ) q∗QI1+i∗ and b∗h = 0.

By assuming that the shocks are not too large to push the interest rates to zero, i and i∗ are

always positive. From equations (12), (24), and (28), we obtain the equilibrium values of w, w∗

and q∗,

w =v(1− n)

β, w∗ =

v∗(1− n∗)β∗

, and q∗ =(1 + i∗)v∗(1− n∗)φ∗I(1 + ξ i∗)β∗

.

Using equations (11), (12), (20)−(23), and (27), we get the equilibrium quantities,

l∗y =

(α∗P ∗yA

∗y

1 + i∗

) 11−α∗(v∗(1− n∗)

β∗

)− 11−α∗

, Q∗y = A∗y

(α∗P ∗yA

∗y

1 + i∗

) α∗1−α∗(v∗(1− n∗)

β∗

)− α∗1−α∗

,

19

w∗l∗I =

(1 + ξi∗

1 + i∗

)q∗QI , w∗l∗I − ξq∗QI =

(1− ξ1+i∗

)q∗QI ,

lx =

(v(1−n)

β

)−11−α(αPxAx1 + i

) 11−α(

1−α2−α

)[θo

2−α1−α−θ

2−α1−α

],

QI =

[e(1+i∗)v∗(1−n∗)φ∗I(1+ξ i∗)β∗

] −11−α(αPxAx

1 + ξ i

) 11−α(

1−α2−α

)[θ

2−α1−α−θo

2−α1−α

],

l∗I =1

φ∗I

[e(1+i∗)v∗(1−n∗)φ∗I(1+ξ i∗)β∗

] −11−α(αPxAx

1 + ξ i

) 11−α(

1−α2−α

)[θ

2−α1−α−θo

2−α1−α

],

Qx =Ax

(1−α2−α

)(αPxAx1+ ξ i

)α1−α[e(1+i∗)v∗(1−n∗)φ∗I(1 + ξ i∗)β∗

]−α1−α[θ

2−α1−α−θo

2−α1−α

]−κ(θ− θo

)

+Ax

(1−α2−α

)(αPxAx

1+i

)α1−α[v(1−n)

β

]−α1−α[

θo2−α1−α −θ

2−α1−α

],

Cx =a(1− n)

Px, Cy =

(1− a)(1− n)

P ∗y e(1 + i), C∗y =

a∗(1− n∗)P ∗y

, and C∗x =e(1− a∗)(1− n∗)

Px(1 + i∗).

Substituting these expressions into the market-clearing conditions, we have a simultaneous equation

system of five equations, (40) − (44), to characterize the equilibrium values of the domestic price

of good x, Px, the foreign price of good y, P ∗y , the home and foreign interest rates, i and i∗,

and the foreign monetary authority’s intervention sale of foreign currency in the foreign exchange

market, Z∗.

Ax

(1−α2−α

)(αPxAx1+ ξ i

)α1−α[e(1+i∗)v∗(1−n∗)φ∗I(1 + ξ i∗)β∗

]−α1−α [

θ2−α1−α−θo

2−α1−α

]−κ

(θ− θo

)

+Ax

(1−α2−α

)(αPxAx

1+i

)α1−α[v(1−n)

β

]−α1−α[

θo2−α1−α −θ

2−α1−α]

=1

Px

[a(1− n) +

e(1− a∗)(1− n∗)1 + i∗

], (40)

A∗y

(α∗P ∗yA

∗y

1 + i∗

) α∗1−α∗ (v∗(1− n∗)

β∗

)− α∗1−α∗

=1

P ∗y

[(1− a)(1− n)

e(1 + i)+ a∗(1− n∗)

], (41)

(1−a)(1−n)

e(1 + i)+

1

e

(1 + ξi∗

1 + i∗

)[e(1+i∗)v∗(1−n∗)φ∗I(1+ξ i∗)β∗

]−α1−α(αPxAx

1 + ξ i

) 11−α(

1−α2−α

)[θ

2−α1−α − θo

2−α1−α

]

− (1−a∗)(1−n∗)1 + i∗

= Z∗, (42)

20

n+B =

(1−α2−α

)(αPxAx1 + i

) 11−α

(v(1−n)

β

)−α1−α [

θo2−α1−α−θ

2−α1−α

]+e(1− a∗)(1− n∗)

1 + i∗, (43)

n∗ =(1− n)(1− a)

e(1 + i)+

(α∗P ∗yA

∗y

1 + i∗

) 11−α∗ (v∗(1− n∗)

β∗

)− α∗1−α∗

+1

e

(1 + ξi∗

1 + i∗

)(1−α2−α

)[αPxAx1 + ξi

] 11−α(e(1 + i∗)v∗(1− n∗)φ∗I(1 + ξ i∗)β∗

)−α1−α[

θ2−α1−α − θo

2−α1−α

]. (44)

Equation (40) shows the equilibrium in the market of good x where the total quantity of good

x supplied by all of the home firms, net of the real fixed cost of outsourcing, Qx, is equal to

the total demand for good x from the home and foreign households, Cx + C∗x. Equation (41)

states the equilibrium in the market of good y when the total quantity supplied, Q∗y, is equal to

the total quantity demanded by the home and foreign households, Cy + C∗y . The left hand side

of equation (42) measures the net private demand for foreign currency in the foreign exchange

market, P ∗yCy + w∗l∗I − (PxC∗x/e). In order to maintain a fixed nominal exchange rate, the foreign

monetary authority meets the net private demand for foreign currency by conducting an official sale

of foreign currency in the foreign exchange market, Z∗. The trade balance of the foreign country,

TB∗ ≡ P ∗yCy+q∗QI− (PxC∗x)/e, is the difference between its total values of exports minus its total

values of imports. Using w∗l∗I =(1+ξi∗

1+i∗

)q∗QI , the balance-of-payments equilibrium condition is

TB∗ − i∗[

(1− ξ) q∗QI1 + i∗

]− Z∗ = 0,

where the trade surplus TB∗ is a credit entry to the foreign country’s balance of payments, while

the foreign monetary authority’s interest payment on bonds held by the financial intermediary of

the home country bf = (1−ξ)q∗QI1+i∗ and the increase in its foreign exchange reserve resulting from

the official intervention Z∗ are the debit entries.27

Equations (43) and (44) describe the ultimate allocation of liquidity in each of the home and

foreign loan markets, respectively. Equation (43) demonstrates that n + B = w lx + PxC∗x, the

total supply of liquidity in the home loan market n + B is ultimately used for facilitating the

production of the home firms which have chosen to produce their intermediate inputs in-house,

wlx, and the imports of good x of the foreign importers, PxC∗x. Equation (44) states that n∗ =

w∗ l∗I+w∗l∗y+P ∗yCy, the supply of liquidity in the foreign loan market, n∗, is allocated for facilitating

27In the case of ξ = 1, we have w∗l∗I = q∗QI and TB∗ = Z∗.

21

the home firms’ international outsourcing activities (having the foreign firms producing good I for

the home country), w∗ l∗I , the foreign firms’ production of good y, w∗l∗y, and the home importers’

purchase of good y, P ∗yCy. When 0 ≤ ξ < 1, the liquidity demand w∗ l∗I is negatively related to

the foreign interest rate i∗ because part of the liquidity burden is on the foreign firms, they borrow

w∗l∗I − ξq∗QI from the foreign loan market. An increase in i∗ will increase the equilibrium price

of good I of q∗ and reduce the equilibrium quantity traded QI . In the case of ξ = 1, w∗ l∗I is not

responsive to changes in i∗ as the foreign firms do not need to borrow from the foreign loan market

to finance the production of good I. As reflected by equations (43) and (44), when a home firm

opts for sourcing good I from the foreign country, it changes the liquidity demands in the loan

markets of both countries, decreasing the liquidity demand in the home loan market for financing

the production of good I, w lx, while increasing the liquidity demand in the foreign loan market

for financing the production of good I, w∗ l∗I .

It is noted that the equilibrium values of Px, P ∗y , i, i∗, and Z∗ are functions of n, n∗, θo, and

s. Taking the expectations over s, we can then use the households’ optimal conditions (25), (26),

and (29) to obtain the following three equations and solve for n, θo, and n∗.

1 = βE[1 + i ], (25′)

E

[[αPxAxθ

o]1

1−α

[1−αα

[β

(1+i)v(1−n)

] α1−α

−[

1−αα

+(1−ξ)(i∗−i)(1+ξi)(1+i∗)

][φ∗I(1+ξi∗)β∗

(1+ξi)e(1+i∗)v∗(1−n∗)

] α1−α]]

+ E [Px] κ = 0, (26′)

1 = β∗E[1 + i∗]. (29′)

The Equilibrium Ultimate Allocation of Liquidity in the World Economy

We can now summarize the ultimate allocation of liquidity across the goods and loan markets

in Table 1 to get a clear picture of how the equilibrium real allocation depends on the equilibrium

allocation of liquidity across different markets.

22

Table 1: The Ultimate Allocation of Liquidity in the World Economy

Goods Market Loan Market

1− n = PxCx + PyCy︸ ︷︷ ︸ n+ B = w lx︸︷︷︸ + Px C∗x︸ ︷︷ ︸

Home Home household’s producing exportingCountry consumption spending good I in-house good x

1− n∗ = P ∗xC∗x + P ∗yC

∗y︸ ︷︷ ︸ n∗ = w∗ l∗y︸ ︷︷ ︸ + P ∗yCy︸ ︷︷ ︸ + w∗ l∗I︸ ︷︷ ︸

Foreige Foreign household’s producing exporting producingCountry consumption spending good y good y good I

Recall that at the beginning of each period, the money stock in circulation of each country

is normalized to one. Households’ deposit decisions affect the tightness of liquidity in the goods

and loan markets and therefore the home entrepreneurs’ incentives to outsource abroad. The home

entrepreneurs’ cutoff productivity level for engaging in international outsourcing not only affects the

liquidity flows in the loan markets but also has implications on the households’ liquidity demands

for their consumption purchases in the good markets. The interactions of the decisions of θo, n,

and n∗ determine the general equilibrium allocation of liquidity across different markets and real

activities in the world economy.

3 Analysis

Our analysis will proceed in three parts. First, we will study the effects of monetary shocks of the

home country and various productivity shocks on the world economy, taking as given the presence

of international outsourcing. Second, we will analyze the effects of devaluations/revaluations of

the foreign currency on the home entrepreneurs’ outsourcing decisions, the liquidity allocations

in financial markets, and the trade balances. Third, the effects of a reduction in the fixed cost

associated with international outsourcing on the world equilibrium allocation of production will be

examined.

3.1 Effects of Different Realizations of the Current State of the World, s

Based on their knowledge of the probability density function G(s), the home household chooses n

and θo, while the foreign household chooses n∗, before the realization of the current state of the

world, s. Given the predetermined values of n, n∗, and θo, we can analyze the effects of different

realizations of s on the world economy by performing comparative static exercises on equations

(40)-(44) and deriving the effects on Px, P ∗y , i, i∗, and Z∗. To simplify the analysis, we define the

23

functions for the excess supply of good x, ESx ≡ Qx −Cx −C∗x, and the excess supply of liquidity

in the foreign loan market, ES∗loan ≡ n∗ − w∗l∗I − w∗l∗y − P ∗yCy, based on equations (40) and (44),

respectively. Given the fixed exchange rate e, we then use equations (41) and (43) to eliminate P ∗y

and i from the functions and obtain

ESx(Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗) ≡ Ax

[αPxAx

1+i

] α1−α

Ω +Ax

[αPxAx1+ ξ i

] α1−α[φ∗I(1 + ξ i∗)

1+i∗

] α1−α

Ωo

+ + + + 0 +

−κ(θ− θo

)− 1

Px

[a(1− n) +

e(1− a∗)(1− n∗)1 + i∗

]= 0, (40′)

ES∗loan(Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗) ≡ n∗ −

[1 +

α∗

1 + i∗

](1− a)(1− n)

e(1 + i)− α∗ a∗(1− n∗)

1 + i∗

+ + − + 0 −−1

e

(1 + ξi∗

1 + i∗

)[αPxAx1 + ξi

] 11−α[φ∗I(1 + ξ i∗)

(1 + i∗)

] α1−α

Ωo = 0. (44′)

where Ω ≡(

1−α2−α

)(v(1−n)β

)−α1−α [

θo2−α1−α − θ

2−α1−α

], Ωo ≡

(1−α2−α

)(e v∗(1−n∗)

β∗

)−α1−α

[θ

2−α1−α − θo

2−α1−α

],

1+i =αPxAx Ω 1−α[

n+B − e(1−a∗)(1−n∗)1+i∗

]1−α and 1 + ξi = 1− ξ +ξαPxAx Ω 1−α[

n+B − e(1−a∗)(1−n∗)1+i∗

]1−α .A plus or minus sign underneath an argument in each of the functions denotes the sign of its

respective partial derivative.28

The world economy is in its general equilibrium when equations (40′) and (44′) hold simultane-

ously as they summarize the interactions of various market forces in determining the equilibrium.29

Given the predetermined values of n, n∗, and θo, in response to different realizations of s, the home

price of good x, Px, and the foreign interest rate, i∗, adjust to satisfy these two equations simul-

taneously. The presence of international outsourcing θo < θ and the fractional upfront payment ξ

determine how the foreign loan market will be affected by the home firms’ production decisions.

Hence, we will consider the adjustments of Px and i∗ in three alternate cases. The first case has

no international outsourcing θo = θ. The second case has international outsourcing θo ∈ (θ, θ) and

assumes that the fraction of the contract payment that the home firms have to pay upfront, ξ, is

small. The third case has international outsourcing θo ∈ (θ, θ) and with a large upfront payment

requirement ξ.

28See the Appendix for the derivations.29Recall that the equilibrium is characterized by equations (40)− (44), and that we have used equations (41) and

(43) to rewrite equations (40) and (44) to obtained equations (40′) and (44′). After solving these two equations forthe two unknowns, Px and i∗, the equilibrium values of i, P ∗y and Z∗ can be derived respectively by substituting theequilibrium values of Px and i∗ into equations (41), (42), and (43).

24

In the market for good x, an increase in Px increases the supplies but decreases the demands,

resulting in an excess supply of good x, ESx > 0. An increase in i∗ has two opposing effects on the

market for good x. First, as each foreign shopper has a fixed expenditure on good x, (1−a∗)(1−n∗),

a higher i∗ implies lower demands for good x by the foreign importers, C∗x. Second, an increase in

i∗ would increase the price of good I, q∗, in the cases with 0 ≤ ξ < 1, making outsourcing more

costly to the home firms and reducing their supplies of good x. The first effect would dominate the

second effect so that an increase in i∗ results in an increase in ESx. It is noted that the strength of

the second effect is decreasing in ξ. The larger the value of ξ, the smaller the fractions of the foreign

suppliers’ costs of producing good I are financed by loans from the foreign financial intermediaries,

and the less sensitive the price of good I, q∗, and supply of good x, Qx, respond to changes in i∗.

Hence, the net positive effect of an increase in i∗ on ESx becomes stronger when ξ is larger.

In the market for foreign loans, an increase in i∗ leads to decreases in the demands for loans

from the foreign firms for the production of good y, w∗l∗y, generating an excess supply ES∗loan > 0.

In the case with international outsourcing and ξ < 1, an increase in i∗ will also reduce the foreign

firms’ liquidity demands for the production of good I, w∗l∗I , reinforcing the increase in ES∗loan. The

smaller the value of ξ, the larger the reduction in w∗l∗I will be caused by an increase in i∗. Hence,

the increase in ES∗loan is more responsive to an increase in i∗ when ξ is small.

An increase in Px has two effects on the foreign loan market. An increase in Px causes the home

firms to increase their production; and the increase in their liquidity demand generates an upward

force on i. Firstly, the increase in i leads the home importers to reduce their demands for good

y, Cy, and induces the foreign firms to reduce their production and lower their liquidity demands,

w∗l∗y. Secondly, in the case with ξ < 1, an increase in Px and the induced adjustment in i increase

Px/(1 + ξi) and lead the foreign firms to increase their liquidity demands for the production of

good I, w∗l∗I .30 As the first effect dominates the second effect, an increase in Px results in an excess

supply of liquidity in the foreign loan market, ES∗loan > 0, while the increase in ES∗loan becomes

weaker when ξ gets smaller.

In summary, an excess supply of good x can be eliminated by a decrease in Px and/or a decrease

in i∗, and an excess supply of liquidity in the foreign loan market can be eliminated by a decrease in

i∗ and/or a decrease in Px. As shown in the Appendix, in the case with no international outsourcing,

there is trade in final goods only, the equilibrium adjustments of Px and i∗ to different realizations of

30It is noted that in the case of ξ = 1, the changes in Px and i will leave Px/(1 + i) unchanged.

25

s depend on the parameter values of a, a∗, α and α∗. The parameters a and a∗ measure the degree

of home bias in households’ preferences, determining the strengths of the liquidity flows required

for facilitating international final good trade, PxC∗x and P ∗yCy. The parameters α and α∗ represent

the shares of inputs that are subject to the liquidity constraints, measuring the liquidity needs of

the final good producers, wlx and w∗l∗y. In the case with outsourcing, as there are final-good trade

and intermediate-good trade, the equilibrium adjustments of Px and i∗ to different realizations

of s depend on not only the values of a, a∗, α and α∗ but also the value of ξ. The fraction ξ

determines the foreign firms’ reliance on the foreign loan market in meeting their working capital

needs for the production of good I. The larger the value of ξ, the larger the increase in ESx and

the smaller the increase in ES∗loan will respond to an increase in i∗. In order to better demonstrate

the role of ξ, Figure 2 plots the two linearized relations between Px and i∗ that satisfy respectively

equations (40′) and (44′) to present a graphical illustration of the effects of the shocks.31 Following

the discussion above, it is clear that the two linearized relations are both negatively sloped, and

their relative slopes depend on the value of ξ.

The realization of s determines all the values of B, Ax, A∗y, and φ∗I . To identify the effects of each

of these exogenous shocks, our analysis allows only one of them to vary at a time, while assuming

the rest being constant. We will first derive the equilibrium effects on Px and i∗ by analyzing

equations (40′) and (44′), and then use equations (41), (42), and (43) to obtain the functions,

i(Px, i∗,B, Ax, A∗y, φ∗I , θo, n, n∗), P ∗y (Px, i

∗,B, Ax, A∗y, φ∗I , θo, n, n∗), and TB∗(Px, i∗,B, Ax, A∗y, φ∗I , θo, n, n∗),

+ − − + 0 0 − + + − − 0 − + + − 0 +

and determine the equilibrium effects on i, P ∗y , and TB∗. As each of these shocks generates opposing

economic forces on the utility level of each household, the welfare effects are in general ambiguous.

Some numerical exercises will be presented in Section 4 to illustrate the welfare effects.

3.1.a The effect of a larger realization of B

An increase in the home country’s open market purchase of home-currency-denominated bonds, B,

increases the supply of liquidity in the home loan market and has a downward force on i. Holding

Px and i∗ constant, the ESx equation shows that an increase in B loosens the liquidity in the home

loan market, causing the home firms to increase their supplies of good x; while the ES∗loan equation

shows that an increase in B increases the home country’s demands for both good I and good y

31As these two excess supply functions are nonlinear, we plot the linearized relations between Px and i∗ using thefirst-order Taylor approximations of ESx = 0 and ES∗loan = 0 around the equilibrium. The derivation is presented inthe Appendix.

26

via its impact on i, inducing the foreign firms to increase their demands for loans to expand their

production. Hence, a larger realization of B results in an excess supply of good x, ESx > 0, and

an excess demand for foreign loans, ES∗loan < 0. The required adjustments of Px and i∗ must be

in opposing directions. As shown in Figure 2(a), an increase in B shifts the ESx = 0 schedule

downward and the ES∗loan = 0 schedule rightward.

In the case with no international outsourcing, θo = θ, i and Px will fall, and i∗ will rise to clear

the markets if there is home bias in consumption in each household’s preferences and the liquidity

needs of the final good producers are high (the parameter values of a, a∗, α, and α∗ are sufficiently

high). Both the direct effect of the increase in B and the indirect effects via the decrease in Px and

the increase in i∗ have negative impacts on the home interest rate, i, while having positive impacts

on the foreign price of good y, P ∗y , and the trade balance of the foreign country, TB∗.

In the case with international outsourcing, θo < θ, the equilibrium changes in Px and i∗ depend

not only on the parameter values of a, a∗, α, and α∗, but also the fraction ξ. When the value of ξ

is small, an increase in i∗ will have a weak positive effects on ESx and a strong positive effect on

ES∗loan. The combination of a decrease in Px and an increase in i∗ will help to achieve a general

equilibrium in the world economy, as in the case with no outsourcing. Both the direct effect of an

increase in B and the indirect effects of the induced adjustments in Px and i∗ act to increase the

excess supply of home-currency-denominated loans, leading to a decrease in the equilibrium level

of i. Similarly, both the direct and the indirect effects work to increase P ∗y and to improve TB∗.

When ξ is large, clearing the markets requires a combination of an increase in Px and a decrease

in i∗. An increase in Px encourages the home firms to absorb the extra liquidity in the home loan

markets and expand their production, causing ESx to increase further and turning ES∗loan from

negative to positive. A decrease in i∗ would then help eliminate both of the excess supplies because

it would have a strong negative effect on ESx and a weak negative effect on ES∗loan. The equilibrium

effects on i, P ∗y , and TB∗ are ambiguous as the direct effects of an increase in B are offset by the

indirect effects of the induced adjustments of Px and i∗. If the indirect effects dominate, i will

increase while P ∗y and TB∗ will decrease in the new equilibrium.

3.1.b The effect of a higher realization of Ax

If Px and i∗ remain unchanged, a higher realization of Ax will induce the home firms to expand

their production and increase their demands for home-currency-denominated loans, leading to an

excess supply of good x, ESx > 0 and an increase in the interest rate i. A higher value of Ax affects

27

the foreign loan market directly by increasing the home firms’ demand for good I, and indirectly

via its upward force on i that leads to a decrease in the home importers’ demand for good y and

a decrease in the home firms’s demand for good I. As the decrease in w∗l∗y + P ∗yCy dominates the

increase in w∗l∗I , there is an excess supply of liquidity in the foreign loan market, ES∗loan > 0. Hence,

the ESx = 0 schedule shifts downward and the ES∗loan = 0 schedule shifts leftward in Figure 2(b).

In the case with no outsourcing, θo = θ, in order to reach a new general equilibrium, the world

economy requires only an equal proportional decrease in Px to respond to an increase in Ax, leaving

the home firms’ marginal revenue product schedules αPxAxθiIiα−1 unchanged. The home firms

have no incentive to change their inputs. There are no effects on the allocation in the liquidity

in each loan market and the interest rates i and i∗. Given that each shopper has a constant

expenditure on each good, an increase in Ax and an equal proportional decrease in Px simply cause

Qx, Cx and C∗x to increase proportionally, and they do not affect the production and allocation of

good y, leaving P ∗y , Q∗y, Cy, C∗y , and TB∗ unchanged.

When there is international outsourcing, θo < θ, although an equal proportional decrease in Px

will eliminate the excess supply in the foreign loan market, it remains to have an excess supply of

good x because of the presence of the total real fixed cost of outsourcing, κ(θ− θo). As an increase

in i∗ will generate a small increase in ESx but a large increase in ES∗loan when ξ is small, the

combination of a more than proportional decrease in Px and an increase in i∗ can help to achieve a

general equilibrium. The decrease in Px dominates the increase in Ax and results in a net decrease

in PxAx, leading the home firms to reduce their demands for home-currency-denominated loans

and causing the equilibrium value of i to fall. There will be increases in P ∗y and TB∗ because of

the decrease in PxAx and the increase in i∗.

When ξ is large, a decrease in i∗ can induce a large decrease in ESx and only a small decrease

in ES∗loan. The combination of a less than proportional decrease in Px and a decrease in i∗ can help

to achieve a general equilibrium. As the value of PxAx increases, the home firms will increase their

demands for loans for financing their production, resulting in an increase in the equilibrium value

of i. The increase in PxAx and the decrease in i∗ imply that both P ∗y and TB∗ fall.

3.1.c The effect of a larger realization of A∗y:

As A∗y does not appear in the expressions for ESx and ES∗loan, it has no effect on Px and i∗. From

equations (42) and (43), it is clear that it also has no effect on Z∗ and i. Given that each shopper

has constant expenditure shares of the two goods, an increase in A∗y reduces P ∗y and increases Q∗y,

28

Cy, and C∗y proportionally. It does not affect the production and consumption of good x, and has

no effect on the equilibrium values of Px, i, i∗, and TB∗.

3.1.d The effect of a larger realization of φ∗I

Holding Px and i∗ constant, a larger realization of φ∗I will lower the foreign price of good I, q∗,

and increase the home firms’ purchases of good I. There will be an excess supply of good x and

an excess demand for foreign loans, shifting the ESx = 0 schedule downward and the ES∗loan = 0

schedule rightward in Figure 2(c). Given that ESx > 0 and ES∗loan < 0, the required adjustments

of Px and i∗ must be in opposing directions.

As discussed above, when ξ is small, an increase in i∗ will generate a small increase in ESx

but a large increase in ES∗loan. Hence, a combination of a decrease in Px and an increase in i∗ will

help attaining a new general equilibrium. A decrease in Px reduces both ESx and ES∗loan, while an

increase in i∗ will have a weak positive effects on ESx but a strong positive effect on ES∗loan. The

direct effects of an increase in φ∗I are reinforced by the indirect effects via the adjustments in Px

and i∗, they work together to lower i, increase P ∗y , and improve TB∗.

When ξ is large, a combination of an increase in Px and a decrease in i∗ will help to attain a

general equilibrium. Although an increase in Px raises both ESx and ES∗loan, a decrease in i∗ will

have a strong negative effect on ESx and a weak negative effect on ES∗loan, leading to market clearing.

Given that the direct effect of an increase in φ∗I and its indirect effects via the adjustments in Px

and i∗ result in increases in the demands for home-currency-denominated loans, the equilibrium

level of i will increase. The adjustments in Px and i∗ also imply a decrease in P ∗y . However, the

equilibrium effect on TB∗ is ambiguous. The direct effect of an increase in φ∗I works to improve

TB∗, while the induced adjustments in Px and i∗ act to deteriorate TB∗.

3.1.e The case with a flexible exchange rate regime

It would be of interested to investigate whether the dependence of the effects of the monetary and

productivity shocks on the contractual upfront payment arrangement, ξ, would hold under a flexible

exchange rate regime. The derivation of the ESx and ES∗loan equations in the case with a flexible

exchange rate is presented in the Appendix. Using equations (40)-(44) and setting Z∗ = 0, we get

ESx(Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗) and ES∗loan(Px, i

∗, B, Ax, A∗y, φ∗I , θo, n, n∗).+ + + + 0 + − − + − 0 −

There are two interesting findings. Firstly, the presence of the endogenous adjustment of e alters

the relation of ES∗loan with Px, i∗, and the monetary and productivity shocks qualitatively. Contrary

29

to the quantity adjustments (official sale of foreign currency, Z∗) in the foreign exchange market

under a fixed e that respond to accommodate the changes in the liquidity allocation in the foreign

loan market, the price adjustments (relative price of foreign currency, e) in the foreign exchange

market under a flexible e tend to counteract the impacts of the shocks on the liquidity allocation

in the foreign loan market, resulting in a ES∗loan relation that is qualitatively different from the

one in the case with a fixed e. Secondly, the qualitative differences in the effects of the shocks for

different values of ξ do not prevail in the case of a flexible exchange rate regime.32 It highlights

how the fraction ξ determines the direct impacts of the shocks on the liquidity allocation in the

foreign loan market and then the directions and magnitudes of the official intervention, Z∗, required

to accommodate the fixed nominal exchange rate e. Hence, the payment arrangements of the

outsourcing contracts matter for the transmission mechanism under a fixed e.

3.1.f Discussion

As summarized in Table 2, under a fixed exchange rate regime, the signs of these effects depend

on the presence of international outsourcing activities and the contractual upfront payment ar-

rangements between the home firms and their foreign suppliers. The numerical example in Table 3

confirms these analytical findings. Under a fixed exchange rate regime, the values of θo and ε are

crucial in determining the directions and magnitudes of the effects of the shocks. It also highlights

the stabilization property of a flexible nominal exchange rate in reducing the volatilities generated

by the shocks; and demonstrates that the value of ξ matters for the effects of the shocks only

quantitatively, but not qualitatively, when e is flexible.

Table 2: The Effects of Monetary and Productive Shocks under a Fixed Exchange Rate

4B 4Ax 4A∗y 4φ∗Iθo=θ θo<θ θo<θ θo=θ θo<θ θo<θ θo=θ θo<θ θo<θ θo<θ θo<θ

small ξ large ξ small ξ large ξ small ξ large ξ small ξ large ξ

4Px − − + − − − 0 0 0 − +

4i∗ + + − 0 + − 0 0 0 + −4i − − ? 0 − + 0 0 0 − +

4P ∗y + + ? 0 + − − − − + −4TB∗ + + ? 0 + − 0 0 0 + ?

Some recent studies have examined the effects of financial frictions on firms’ sourcing decisions

and choices of trade modes. Some examples are Antras, Desai, and Foley (2009), Feenstra, Li, and

32See the Appendix for the illustration of how the value of ξ affects the effects quantitatively.

30

Yu (2009), and Manova and Yu (2012). This paper contributes to these discussions by examining

the role of contractual upfront payment arrangements in the determination of liquidity allocations

in financial markets and their implications on firms’ production capacity and the macroeconomy

in a general equilibrium environment. Our results indicate the importance of modeling the finan-

cial aspects involving firms’ intermediate input sourcing decisions in the studies of the effects of

intermediate good trade on the economy.

3.2 Effects of Permanent Changes in e and κ

In order to study the effects of changes in the fixed exchange rate e or the fixed cost associated with

international outsourcing κ on the world economy, we need to examine not only equations (40)-(44)

but also equations (25′), (26′), and (29′) because the households determine their deposit decisions

and outsourcing cutoff productivity level based on the given values of ( e, κ ). As our focus is now

on the effects on n, n∗, and θo, we could simplify the derivation and allow for the analytical results

by assuming no uncertainty in the realization of s.33 This assumption implies that the households

will choose their deposit decisions so that the optimal conditions βE[1 + i] = β(1 + i) = 1 and

β∗E[1 + i∗] = β∗(1 + i∗) = 1 are always satisfied. As expected, the value of ξ does not matter

as much as in the cases with temporary shocks. The intuition is that the value of ξ is crucial

in determining the strengths of the effects on liquidity demands generated by temporary shocks

when the outsourcing and deposit decisions, θo, n, and n∗, are sluggish. However, in the case with

permanent changes in e and κ, the decisions θo, n, and n∗ are allowed to respond so that the liquidity

supplies can adjust to accommodate the changes in liquidity demands, making ξ less important in

the transmissions of these shocks.

3.2.a Effects of a Revaluation of the Foreign Currency

Consider now that at the beginning of the current period, before any decision is made, the for-

eign monetary authority announces a permanent revaluation of its currency. In the case without

international outsourcing, θo = θ, an increase in the fixed exchange rate e will induce the home

importers to reduce their demands for good y, Cy, if n remains unchanged. Facing decreases in

the demands for good y, the foreign firms reduce their demands for foreign loans w∗l∗y. The foreign

interest rate i∗ will fall if n∗ does not adjust. Expecting a decrease in the demand for liquidity in

the foreign loan market, the foreign households reduce their deposits, n∗ falls to satisfy the optimal

33The assumption of no uncertainty in s does not alter the results qualitatively. As shown in the numerical exercisesin Section 4.2, the analytical results discussed in this section remain valid even when there is uncertainty in s.

31

condition β∗E[1 + i∗] = 1. Similarly, if n∗ remains unchanged, an increase in e will increase the

demands for good x from the foreign importers, C∗x, inducing the home firms to increase their loans

w lx from the home loan market. The home interest rate i will rise if n does not change. Expecting

an increase in the demand for liquidity in the home loan market, the home households increase

their deposits, n rises until the optimal condition βE[1+ i] = 1 is satisfied. Although these changes

in n and n∗ will induce further adjustments in the loan markets that would offset some of the initial

changes, it can be shown that an increase in e has a positive equilibrium effect on n and a negative

equilibrium effect on n∗. For the foreign country, as its value of export of good y decreases and

its value of import of good x increases, its trade balance, TB∗ = P ∗yCy − PxC∗x/e, deteriorates,

requiring the foreign monetary authority to conduct a smaller official sale of foreign currency in

the foreign exchange market, Z∗=TB∗ falls.34

When θo = θ,dn∗

de< 0,

dn

de> 0,

dP ∗yCy

de< 0,

d(PxC∗xe

)de

> 0, anddTB∗

de< 0.

In the case with outsourcing, in addition to the effects on trade in final goods described above,

there are adjustments in outsourcing activities at both the intensive and extensive margins. The

additional adjustment at the extensive margin (in the number of home firms sourcing abroad),

θ − θo, could be the dominant force in the determination of the equilibrium effects. If θo and n

remain unchanged, an increase in e will increase the foreign importers’ demands for good x but will

reduce the home firms’ demands for input I (at the intensive margin) and their production of good

x, resulting in a large excess demand for good x and therefore a substantial increase in Px.35 Using

equation (26′), we can identify two effects of an increase in e on θo. The first one is a cost effect

through a reduction in the unit cost difference between producing good I in-house and sourcing

abroad. The second one is a demand effect reflected by an increase in Px as a higher e increases

the foreign households’ demand for good x. Although the two effects have opposing effects on the

home firms’ incentives to outsource, the demand effect dominates and results in an increase in the

number of home firms sourcing abroad (a decrease in θo).36

In the home loan market, some of the home firms switch to outsourcing and reduce their liquidity

demands wlx. Given that the home firms’ liquidity demand w lx is more sensitive to an increase in

34As discussed above, the values of a, a∗, α and α∗ are assumed to be sufficiently high.35Each home firm that has decided to outsource the production of the intermediate input will reduce the quantity

of the intermediate input demanded.36It is noted that a decrease in θo has a strong positive effect on QI but a weak negative effect on Qx.

32

e than the foreign importers’ liquidity demand PxC∗x does, there will be a net decrease in liquidity

demand in the home loan market. The home interest rate i will fall if n does not change. With the

expectation of a decrease in the value of liquidity in the home loan market, the home households

reduce their deposits n to satisfy the optimal condition βE[1 + i] = 1.

In the foreign loan market, if n∗ remains unchanged, an increase in e will lower the home

importers’ demands for good y and cause the foreign firms to reduce their production of good y.

The liquidity demands P ∗yCy and w∗l∗y decrease. However, the effect of the decrease in θo is stronger

than that of the increase in e, resulting in an increase in the liquidity demand for the production

the intermediates, w∗l∗I . As the increase in w∗l∗I dominates, the foreign nominal interest rate i∗

will rise if n∗ does not adjust. Hence, following the optimal condition β∗E[1 + i∗] = 1, the foreign

households increase their deposits n∗ to ease the upward pressure on i∗.

As demonstrated by equations (24) and (28), the decrease in n implies an increase in the home

real wage w; and the increase in n∗ implies decreases in the foreign real wage, w∗, and price of good

I, q∗. Although the increase in e would increase the home-currency price of good I, e q∗, the relative

unit cost of producing to importing good I, w/(e q∗), increases and induces more home firms to

switch to international outsourcing, leading θo to fall further. The interactions of the adjustment

at the extensive margin in outsourcing (the number of firms sourcing abroad) with the households’

deposit decisions are the driving forces of the equilibrium effects. It is shown that a revaluation of

the foreign currency (an increase in e) has positive equilibrium effects on n∗ and w/(e q∗), and a

negative equilibrium effect on θo.

When θo ∈ ( θ, θ ),dn∗

de> 0,

d(we q∗

)de

> 0,dθo

de< 0,

d(PxC∗xe

)de

< 0, anddTB∗

de> 0.

Ifd(1−ne

)de

< 0, thendP ∗yCy

de< 0,

d q∗QIde

> 0, andd (q∗QI + P ∗yCy)

de> 0.

A positive effect on n∗ indicates a negative effect on PxC∗xe . The equilibrium effects on n and

(1−n)/e are ambiguous. If the direct effect of an increase in e dominates, we will haved( 1−n

e )de < 0,

implying a negative effect on P ∗yCy and positive effects on q∗QI and q∗QI + P ∗yCy, while the

net trade in consumer goods, P ∗yCy −PxC∗xe , is ambiguous. The foreign country’s trade balance,

TB∗ = q∗QI +P ∗yCy−PxC∗x/e, improves; the foreign monetary authority needs to conduct a larger

open market sale of the foreign currency in the foreign exchange market to meet the net private

demand for foreign currency, Z∗ = TB∗ rises.

33

Discussions: A revaluation of the foreign currency is expected to induce expenditure switching

and to have an impact on the trade balance. In the Mundell-Fleming model, it will worsen the

trade balance of the foreign country if the Marshall-Lerner conditions are satisfied. As the Mundell-

Fleming model is a static model, the Marshall-Lerner conditions consider intra-temporal elasticities

of demand for home and foreign goods. Devereux (2000) introduces the considerations of inter-

temporal elasticities of consumption by incorporating firms’ pricing-to-market behavior into the

open-economy macroeconomic model of Obstfeld and Rogoff (1995). He shows that when prices

are set in consumer’s currency, the Marshall-Lerner conditions become irrelevant, and a devaluation

of a currency will improve, leave unchanged, or worsen the country’s current account if the elasticity

of inter-temporal substitution is less than, equal to, or greater than one.37 Devereux and Genberg

(2007) analyze the effect of an exchange rate appreciation on the current account by introducing

three asymmetries between the two countries into the model of Obstfeld and Rogoff (1995). In

their model, the foreign country must use imported intermediate inputs in a fixed proportion with

domestic inputs in the production of their exports, sets export prices in the currency of the home

country, and holds a large fraction of nominal bonds denominated in the home currency.38 They

find that with low trade elasticities in the short-run, a nominal appreciation of the foreign currency

may worsen the current account of the home country, and that a nominal appreciation of the foreign

currency can improve the current account of the home country when trade elasticities become higher

in the longer run.39

In contrast, this model emphasizes the high substitutability between intermediates produced

domestically and abroad implied by international fragmentation of production and highlights the

impacts of the firms’ sourcing decisions on the liquidity demand in each loan market. For simplicity,

we assume that the intra-temporal and inter-temporal elasticities of substitution in consumption

are equal to unity, goods prices are flexible, there are no inter-period nominal bonds, and the foreign

country does not use intermediate inputs from the home country in its production. By looking into

the liquidity flows in the foreign exchange market, we can gain some insights into the interactions

at work. The demand for foreign currency is derived from the home country’s demand for imports

37International trade in intermediates has not been included in the study.38The two countries were named as country A (the US) and country B (China) in their paper. However, for easy

comparison, country A is referred to as the home country and country B is referred to as the foreign country in ourdiscussion here.

39Dong (2012) uses a two-country model to illustrate how changes in firms’ pricing behavior and global tradepattern reduce the responsiveness of both US imports and exports to exchange rate movements, explaining whyexchange rate movements fail to facilitate the trade imbalance adjustments.

34

of final good y and intermediate input I, and the supply of foreign currency is derived from the

foreign country’s import of final good x. When θo = θ, there is trade in final goods only, and

changes in e induce adjustments of n and n∗. Hence, P ∗yCy is negatively related to e, and PxC∗x/e

is positively related to e, we have the usual downward sloping demand schedule and upward sloping

supply schedule of foreign currency in the foreign exchange market. An increase in e reduces TB∗,

the foreign monetary authority conducts a smaller official sale of foreign currency Z∗ to meet the

decrease in the net private demand for foreign currency.

The incorporation of international outsourcing activities alters the responses of the demand and

supply of foreign currency to changes in e. When θo < θ, with the adjustments of θo, n, and n∗,

an increase in e effectively increases the demand q∗QI + P ∗yCy and reduces the supply PxC∗x/e of

foreign currency, resulting in an upward sloping demand schedule and a downward sloping supply

schedule in the foreign exchange market. The foreign monetary authority has to conduct a larger

official sale Z∗ to meet the increase in the net private demand for foreign currency due to a larger

TB∗. As a result, a revaluation of foreign currency will only make the home country’s trade deficit

deteriorate further.

This finding provides a rationale for the puzzling evidence of a positive relationship between

China’s currency value and its trade surplus during the period of 1994-2008. Figure 3 presents

the official exchange rate of China’s currency, the Renminbi (RMB), in units of the US dollar and

China’s trade balance in goods and services (TB) as a percentage of its gross domestic product

(GDP) from 1985 to 2014. It also includes the real effective exchange rate (REER) and the decom-

position of the trade balance.40 As the data show, the correlation coefficient between the official

USD/RMB rate and China’s TB/GDP ratio was −0.72 during the period of 1985-1993 as China’s

overall trade balance tended to improve when RMB was devalued. In contrast, the correlation co-

efficient was positive and equal to 0.75 during the period of 1994-2008. If recent years are included,

the correlation coefficient would be 0.17 for the period of 1994-2014. The global trade collapse in

2008-2009 and the slow economic recovery may have contributed to this significant change in the

correlation. Similar patterns of the correlation coefficients between the REER the TB/GDP ratio

are obtained.

40The data are obtained from the World Integrated Trade Solution and the World Development IndicatorsDatabases of the World Bank. The real effective exchange rate is the nominal effective exchange rate (a mea-sure of the value of RMB against a weighted average of several foreign currencies) divided by a price deflator. Dataon the decomposition of the trade balance into the four major product categories (capital goods, consumer goods,intermediate goods, and raw materials) are available from 1992 only.

35

After a substantial devaluation of the RMB in 1994, the official USD/RMB exchange rate had

been kept relatively stable, and it remained steady at 12.08 U.S. cents per RMB from 1997 to 2005.

The surge of China’s trade surplus to 4.5% in 2005 ignited the political pressure from policymakers

of other countries, including those of the U.S., on RMB to revalue. China introduced a new regime

in 2005 to allow for graduate revaluations of the RMB. Its trade surplus not only did not fall but

rose substantially to 6.54% of its GDP in 2008 despite of a more-than 20% cumulative appreciation

of the RMB against the US dollar.41 Although many studies in the literature have explained why

the trade flows might not be responsive to the currency revaluations, they could not explain the

accelerating increase in China’s trade surplus. The decomposition of the trade balance reveals that

the net trade flows of different categories of products responded quite differently to the changes in

the currency value. For the period 1994-2008, the correlation coefficient of the official USD/RMB

rate and the TB/GDP ratio was equal to 0.91 for the intermediate goods and−0.08 for the consumer

goods, while for the period of 1994-2014, they were equal to 0.83 and −0.79, respectively. These

correlations are in line with our model predictions that a revaluation of the foreign currency would

increase the foreign country’s exports of intermediate goods, while having an ambiguous effect on

its trade balance of the consumer goods.

The increases in the differences in the labor costs of China and the U.S. are consistent with

the increases in China’s outsourcing activities and exports of intermediate goods. According to the

Bureau of Labor Statistics of the United States Department of Labor, the hourly labor compensation

in China’s manufacturing sector increased from USD0.60 in 2002 to USD1.74 in 2009, while those

of the U.S. increased from USD27.36 to USD34.19. Although the hourly labor compensation in

China increased from about 2% of the U.S. level in 2002 to 5% in 2009, the level differences were

widened and the labor cost in China remained very low.42

Our model provides a theoretical framework to illustrate how the rise of China as a major host

country of international outsourcing may have altered the response of the overall trade balance to

its currency revaluation, helping to rationalize the relation between its currency revaluation and

trade surplus.

41Although its trade flows and trade surplus have been declining since global trade collapsed in 2008, China hasbecome the world’s largest exporter since 2009. According to the World Trade Report 2013, its share of world exportwas 11% in 2011. The upward trend in the trade balance has been observed in the recent years.

42The hourly labor costs in China and the U.S. are from the website of the Bureau of Labor Statistics of theUnited States Department of Labor, http://www.bls.gov/fls/home.htm, which provides data from 2002 to 2009 only.Some reasons for the increases in China’s labor costs were rising labor productivity, inflation, and the increasedrequirements for social insurance payments by firms.

36

3.2.b Effects of Reductions in the Costs Associated with International Outsourcing

We now consider the effects of a reduction in the fixed cost of international outsourcing κ. In order

to disentangle the direct effects of a reduction in κ and the indirect effects of the induced adjustment

of Z∗ under a fixed nominal exchange rate regime, our analysis will proceed in two steps. Firstly, we

investigate the effects on the households’ decisions and the nominal exchange rate by assuming that

Z∗ is held constant and e is allowed to adjust endogenously to clear the foreign exchange market.

Secondly, maintaining the nominal exchange rate at e requires the foreign monetary authority to

conduct an official sale/purchase of its currency in the foreign exchange market, we identify the

required endogenous adjustment of Z∗ and its effects on the world economy. We can then combine

the effects to obtain the equilibrium impacts of an increase in κ under a fixed exchange rate regime

and have a better understanding of the driving forces behind the results.

Holding Z∗ constant, a reduction in the fixed cost of outsourcing κ encourages some home firms

to switch from producing their intermediate inputs domestically to importing from aboard, resulting

in a fall in θo, a decrease in liquidity demand w lx in the home loan market, and an increase in the

liquidity demand w∗l∗I in the foreign loan market. In the foreign exchange market, the increase in

the home country’s demand for foreign currency for its outsourcing activities, w∗l∗I =(1+ξ i∗

1+i∗

)q∗QI

generates an upward force on e, discouraging the foreign country’s export of good Y , P ∗yCy, and

encouraging the home country’s export of good x, PxC∗x. The nominal exchange rate will increase

to keep the excess demand for foreign currency P ∗yCy +w∗l∗I −PxC∗x/e = Z∗ unchanged. However,

the effects on the households’ deposit decisions, n and n∗, are ambiguous. In the home loan market,

if the decrease in wlx is dominated by the increase in PxC∗x, there will be a net increase in the

demand for home liquidity, and the home households will increase their deposits n until the optimal

condition βE[1 + i] = 1 is met.43 In the foreign loan market, if the increase in w∗l∗I dominates the

decreases in w∗l∗y and P ∗yCy, it will result in a net increase in the demand for liquidity in the foreign

loan market, and the foreign households will increase their deposits n∗ until the optimal condition

β∗E[1 + i∗] = 1 is satisfied. It can be shown that

de

dκ

∣∣∣∣dZ∗=0

< 0,dθo

dκ

∣∣∣∣dZ∗=0

> 0,d(1−ne

)dκ

∣∣∣∣∣∣dZ∗=0

> 0, anddn∗

dκ

∣∣∣∣dZ∗=0

> 0.

To fix the exchange rate unilaterally, the foreign monetary authority is required to adjust Z∗

endogenously to meet the net private demand for foreign currency in the foreign exchange market

43It is shown that a decrease in κ reduces (1− n)/e. Hence, P ∗yCy = (1− a)(1− n)/(e(1 + i)) also decreases.

37

implied by the foreign trade balance, TB∗. Keeping the exchange rate at e in face of a lower κ

is like conducting a devaluation of the foreign currency to bring the exchange rate from the new

market equilibrium level down to its original fixed level e. As discussed in Section 3.2, a reduction

in e has a positive effect on θo and a negative effect on TB∗. Hence, the foreign monetary authority

must reduce its official sale of foreign currency Z∗ in response to an increase in κ.

As the indirect effect on θo resulting from the decrease in Z∗ required to keep the exchange rate

at e is in the opposing direction of the direct effect from the reduction in κ, the equilibrium effect

is ambiguous. The numerical example presented in the next section shows that the indirect effect

from the endogenous adjustment in Z∗ could dominate and result in an increase in θo.

4 Welfare Analysis

Using the labor market clearing conditions and the equilibrium expressions of Cx, Cy, lx, C∗x, C∗y ,

l∗y, and l∗I in section 2.6, we can rewrite the instantaneous utility functions as the indirect utility

functions of i, i∗, Px, P ∗y , θo, n, and n∗.

u( i, i∗, Px, P∗y , θ

o, n, n∗) = a ln

(a(1− n)

Px

)+ (1− a) ln

((1− a)(1− n)

eP ∗y (1 + i)

)

+ v

1−(v(1−n)

β

)−11−α

(αPxAx1 + i

) 11−α

(1−α2−α

)[θo

2−α1−α−θ

2−α1−α

] ,u ∗( i, i∗, Px, P

∗y , θ

o, n, n∗) = a∗ ln

(a∗(1− n∗)

P ∗y

)+ (1− a∗) ln

(e(1− a∗)(1− n∗)

Px(1 + i∗)

)

+ v∗

1−[

α∗P ∗yA∗y β∗

(1+i∗)v∗(1−n∗)

] 11−α∗

− 1

φ∗I

[φ∗I(1+ξ i∗)β∗

e(1+i∗)v∗(1−n∗)

] 11−α(αPxAx1 + ξ i

) 11−α(

1−α2−α

)[θ

2−α1−α−θo

2−α1−α

].The equilibrium effects of the exogenous changes on the welfare of each household are generally

ambiguous as the direct and indirect effects are in opposing directions. We will construct some

numerical examples to illustrate the welfare impacts of various exogenous changes. It should be

noted that the equilibrium welfare effects depend on the specifications of the utility functions, and

that the aim of these exercises is to illustrate the theoretical properties of the model rather than

to match the observed data.

4.1 Welfare effects of different realizations of s

Given e and κ, holding θo, n, and n∗ constant, we can use our findings of changes in i, i∗, Px, and

38

P ∗y to determine the impacts on households’ consumption and leisure levels and pin down the welfare

effects of different realizations of s. The results of the numerical exercises are presented in Table 3.

First, no matter whether there are international outsourcing activities, an improvement in the

production technology of a final good (an increase in Ax or A∗y) benefits the households of both

countries. The increases in consumption of the final good that experiences a positive productivity

shock play dominant roles in improving the welfare levels of the households.

Second, the welfare effects of domestic monetary policy depend on the fraction ξ because it

determines the adjustments of i, i∗, Px, and P ∗y in achieving a new general equilibrium. When ξ

is small, an increase in the open market purchase of home-currency-denominated bonds B reduces

Px and i but increases P ∗y and i∗, leading to increases in Cx, Cy, and h but decreases in C∗x, C∗y ,

and h∗. Because of home bias in consumption, households experience substantial welfare gains

(losses) from the increases (decreases) in the consumption of the domestically produced final good

when its price falls (rises). In addition, a decrease (an increase) in the domestic interest rate lowers

(raises) the domestic-currency price of the imported final good, leading to an increase (a decrease)

in the consumption of the imported final good. As the welfare changes from the adjustments

in consumption dominate the impacts from the adjustments in work effort, u increases and u∗

decreases.44 When ξ is large, an increase in B increases Px and i but decreases P ∗y and i∗, reducing

Cx, Cy, and h and increasing C∗x, C∗y , and h∗. When the welfare effects from the adjustments in

consumption dominate, u decreases and u∗ increases.

Third, given that the fraction ξ determines the adjustments of i, i∗, Px, and P ∗y in response

to an improvement in the foreign firms’ productivity in producing intermediate good I, φ∗I , the

welfare effects also depend on the value of ξ. When ξ is small, an increase in φ∗I reduces Px and

i but increases P ∗y and i∗, leading to increases in Cx, Cy, and h but decreases in C∗x, C∗y , and h∗,

resulting an increase in u and a decrease in u∗. When ξ is large, an increase in φ∗I increases Px and

i but decreases P ∗y and i∗, reducing Cx, Cy, and h and increasing C∗x, C∗y , and h∗, causing u to fall

and u∗ to rise.

4.2 Welfare effects of changes in e or κ

The adjustments of θo, n, and n∗ in response to the changes in e or κ determine the allocation of

liquidity in the world economy. As demonstrated by the numerical results in Table 4, given that

44These effects are similar to those in the case without international outsourcing (θo = θ) where an increase in Breduces Px and i but increases P ∗y and i∗, leading to increases in Cx, Cy, and h and decreases in C∗x , C∗y , and h∗, andresulting in an increase in u and a decrease in u∗.

39

transactions are facilitated by liquidity, the adjustments of deposit decisions n and n∗ are good

indicators of the welfare effects.

4.2.a Welfare Effects of an Increase in e

As shown in Cases II and III of Table 4, holding κ constant, a revaluation of the foreign currency

(an increase in e) leads some home firms to switch to sourcing the intermediate inputs from abroad

(θo falls); and the reallocations of liquidity in the loan markets induce a decrease in n and an

increase in n∗. A decrease in n implies that higher consumption expenditures are available for the

home shoppers while fewer loans are available for the hiring of workers in the home country, Cx

and Cy increase and h falls. An increase in n∗ indicates that the foreign shoppers’ consumption

expenditures are lower and that the foreign firms obtain more loans to hire workers in the foreign

country, C∗x and C∗y decrease and h∗ rises. Hence, a revaluation of the foreign currency leads to an

increase in E[u] and a decrease in E[u∗]. It is found that E[u] is increasing and E[u∗] is decreasing in

e, and the total expected utility level of the two counties, E[u] + E[u∗], can be increasing in e when

the utility function is linear in leisure.45 In contrast, when there is no international outsourcing, a

revaluation of the foreign currency results in an increase in n and a decrease in n∗, improving the

welfare of the foreign households, while making the home households worse off. Case I of Table 4

illustrates that when θo = θ, E[u] is decreasing and E[u∗] is increasing in e, and the total expected

utility E[u]+E[u∗] displays an inverted U relationship as e increases, reaching its maximum when the

exchange rate is at its equilibrium without any intervention. These results highlight the important

role of international outsourcing in the determination of the welfare effects of currency revaluations

on the world economy.

4.2.b Welfare Effects of a Reduction in κ

A reduction in the fixed cost associated with outsourcing activities improves the welfare of the

foreign households but makes the home households worse off under a flexible exchange rate regime.

As illustrated by the comparison of columns I.2, II.3, and III.2 in Table 4, when the exchange rate

is flexible (Z∗ = TB∗ = 0), a decrease in κ increases e and n but reduces θo and n∗.46 The changes

45It is noted that the increase in E[u]+E[u∗] cannot go unbounded as the foreign labor effort, h∗, will reach its upperbound eventually. When there is diminishing marginal utility of leisure, the total utility level of the two counties,u+u∗, may not be monotonic in e. For example, in the case with u(Cx, Cy, h) = a lnCx+(1−a) lnCy+ 1

1+σv [ 1−h ]1+σ,

and u∗(C∗x , C∗y , h∗) = (1 − a∗) lnC∗x + a∗ lnC∗y + 1

1+σ∗ v∗ [ 1 − h∗ ]1+σ

∗, where σ = σ∗ = −0.2, it is found that

E[u] + E[u∗] increases as e rises, reaches its maximum when e is somewhere above its equilibrium level that is free ofany intervention, and becomes decreasing in e thereafter.

46The numerical example assumes ξ = 1. the qualitative effects on the welfare levels of a reduction in κ are notsensitive to the value of ξ.

40

in the households’ consumption expenditures result in decreases in Cx and Cy and increases in C∗x

and C∗y . Although h falls and h∗ rises when more production activities are relocated to the foreign

country, the welfare changes from the adjustments in consumption dominate the impacts from the

adjustments in work effort, leading to a decrease in E[u] and an increase in E[u∗] and resulting in

a small net increase in the total expected utility level E[u] + E[u∗].

The presence of a fixed exchange rate regime reinforces the welfare effects. The foreign exchange

intervention that prevents the nominal exchange rate from increasing and keeps it at the level

e works like a policy to devalue the foreign currency, reducing the home country’s welfare and

improving the foreign country’s welfare further. Consider the original equilibrium given by column

II.3. Under a flexible exchange rate, a reduction in the value of κ from 0.2 to 0.1 results in the new

equilibrium presented in column III.2. Keeping the exchange rate at the original level would lead

to the new equilibrium given by column III.1. Overall, the reallocations of liquidity in the loan

markets result in an increase in n and a decrease in n∗. The home households are worse off as their

consumption levels Cx and Cy decrease and work effort h increases. The foreign households are

better off as C∗x and C∗y increase and work effort h∗ decreases. Holding e constant, the increase in

E[u∗] is dominated by the decrease in E[u], resulting in a net decrease in the total expected utility

level E[u] + E[u∗].

4.2.c Discussion

As demonstrated above, a decrease in κ can have substantial distributional effects between countries,

while the total utility gain of the world economy is insignificant or may even be negative under a

fixed e. Using the results from the changes in e and κ, we can conjecture that when a reduction in

κ is accompanied by a revaluation of the foreign currency (an increase in e), it is possible to result

in an increase in outsourcing activities (a decrease in θo) and improvements in the welfare levels

of both countries (increases in u and u∗). However, due to the welfare conflict between the two

countries generated by the foreign exchange policy, the foreign country would resist the revaluation

of its currency. The numerical exercise confirms the validity of this conjecture. Suppose that the

world is in the initial equilibrium with κ = 0.2 and e = 1.07 (column II.4). There exists some

fixed exchange rate levels that can benefit both countries and allow for larger total welfare gains

when the value of κ is decreased to 0.1 (see for example columns III.4 and III.5). As the foreign

country would be better off in the equilibrium with κ = 0.1 and e = 1.07 (given by column III.3),

41

it would have no incentive to revalue its currency.47

5 Conclusions

The paper complements the existing macroeconomic literature on international trade in intermedi-

ates by modeling explicitly firms’ make-or-buy decisions and highlighting the role of financial flows

required to facilitate the flows of goods and services related to international outsourcing activities.

A two-country, monetary model with segmented financial markets is constructed to examine the in-

terdependence of firms’ international outsourcing decisions and governments’ conducts of monetary

and foreign exchange policies.

Our results show that international outsourcing and its associated contractual upfront payment

arrangements between the domestic firms and their foreign suppliers determine the responsiveness

of the liquidity allocation in the foreign loan market to the monetary and productivity shocks, and

have an important role in the international transmission of the effects of the shocks when the nom-

inal exchange rate is fixed. The paper also examines how the presence of international outsourcing

alters the responses of the demand and supply of foreign currency in the foreign exchange market to

currency revaluations. As firms can choose between producing their intermediate inputs in-house

or buying them from foreign suppliers, the perfect substitution between the intermediate goods

produced domestically and abroad implies that the adjustment at the extensive margin of interna-

tional outsourcing would be significant and can have a dominant role in determining the effects on

production and trade. This finding helps explain qualitatively why a foreign currency revaluation

may increase the foreign country’s exports of intermediate goods and worsen the trade deficit of the

home country even in the time frame in which firms are able to adjust their outsourcing decisions.

Finally, the model demonstrates the distributional welfare effects of international outsourcing and

highlights the role of a fixed exchange rate regime in welfare redistribution when there is interna-

tional outsourcing, suggesting the needs to further explore the gains from international monetary

policy coordination by modeling explicitly firms’ sourcing decisions of their intermediate inputs.

47Instead, the foreign country would want to devalue its currency so as to capture an even larger welfare gain.

42

Appendix

The first-order condition of the representative home household’s problem

Let ρ1, ρ2, ρ3, and ρ4 be the multipliers associated with constraints (13′)− (15′) and (17′), respectively.

We can then derive the first-order conditions,

n :

∫ρ1 G(s) ds =

∫ρ4 G(s) ds,

θo :

∫ρ2

e ξ q∗

[αPxAxθ

o

(1+ξi)eq∗

] 11−α

− w[αPxAxθ

o

(1+i)w

] 11−αG(s) ds

+

∫βVm′

h

[w

[1 + i

α−1

][αPxAxθ

o

(1+i)w

] 11−α

− e q∗[

1 + ξi

α− ξ][αPxAxθ

o

(1+ξi)eq∗

] 11−α]

+ Pxκ

G(s) ds,

+

∫βVm′

f(1− ξ) q∗

[αPxAxθ

o

(1+ξi)e q∗

] 11−α

G(s) ds = 0

Cx, Cy :uxPx

=uyPy

= ρ4 + β Vm′h, uj ≡

∂ u(Cx, Cy, l)

∂ Cj, j = x, y,

h : v = β Vm′hw,

bh : ρ1 = β Vm′hi,

bf : e (ρ1 + β Vm′h) = β Vm′

f(1 + i∗)

Lx : ρ1 = ρ2,

Ly : ρ1 + β Vm′h

=[ρ3 + β Vm′

f

] 1

e,

IMy :[ρ3 + β Vm′

f

]P ∗y = β Vm′

hPy,

where Vmh≡∂ V (mh, mf )

∂ mh

=

∫[ ρ4 + β Vm′

h] G(s)ds =

∫uxPxG(s)ds = E

[uxPx

], and

Vmf≡∂ V (mh, mf )

∂ mf

=

∫[ e(ρ1 + β Vm′

h) ]G(s)ds = E

[ρ3 + β Vm′

f

]= β E

[Vm′

f(1 + i∗)

].

The comparative static exercises in Section 3

Taking as given the predetermined values of n, n∗, and θo, we totally differentiate the two excess supply

equations (40′) and (44′) and obtain the following two differential equations.

dESx =

[PxC

∗x −

(1 + ξi)qQI1− α

(1− ξ

1 + ξi∗

)]di∗

1 + i∗+

[(1 + i)wlx + (1 + ξi)qQI

α(1− α)− Pxκ

(θ− θo

)] dPxPx

+

[(1 + i)wlx + (1 + ξi)qQI

α(1− α)

]dAxAx

+(1 + ξi)qQI

1− αdφ∗Iφ∗I

−[

(1 + i)(wlx + ξqQI)

1− α

]di

1 + i−[PxC

∗x +

(1 + ξi)qQI1− α

]de

e

43

dES∗loan =

[α∗

1 + i∗[P ∗yCy + P ∗yC

∗y

]+

w∗l∗I1− α

(1− ξ

1 + ξi∗

)]di∗

1 + i∗− w∗l∗I

1− α

[dPxPx

+dAxAx

]− αw∗l∗I

1− αdφIφI

+

[P ∗yCy

[1 +

α∗

1 + i∗

]+

w∗l∗I1− α

ξ(1 + i)

1 + ξi

]di

1 + i+

[P ∗yCy

[1 +

α∗

1 + i∗

]+

w∗l∗I1− α

]de

e.

Under a fixed exchange rate regime, we have de = 0 anddi

1 + i=−dB − PxC∗x di∗

1+i∗ + wlx1−α

[dPxPx

+ dAxAx

](wlx1−α

)from equation (43). We then get dESx = λ1Px dPx + λ1i∗ di

∗ + λ1B dB + λ1Ax dAx + λ1A∗y dA∗y + λ1φ∗

Idφ∗I

and dES∗loan = λ∗2Px dPx + λ∗2i∗ di∗ + λ∗2B dB + λ∗2Ax dAx + λ∗2A∗y dA

∗y + λ∗2φ∗

Idφ∗I , where

λ1Px ≡∂ESx∂Px

=1

1− α[PxCx + PxCx + αPxκ(θ − θo)− [w lx + ξ q QI ] (1 + i)

] 1

Px> 0,

λ1i∗ ≡∂ESx∂i∗

=

[PxC

∗x

[1 + (1 + i)

[1 +

ξ q QIwlx

]]− (1 + ξi)q QI

α

(1− ξ

1 + ξi∗

)]1

1 + i∗> 0,

λ1B ≡∂ESx∂B

=

[w lx + ξ q QI

wlx

](1 + i) > 0,

λ1Ax ≡∂ESx∂Ax

=1

1− α[[PxCx + PxC

∗x + Pxκ(θ − θo)

]− [w lx + ξ q QI ] (1 + i)

] 1

Ax> 0,

λ1A∗y ≡∂ESx∂A∗y

= 0,

λ1φ∗I≡ ∂ESx

∂φ∗I=

(1 + ξi

1− α

)q QI

1

φ∗I> 0,

λ∗2Px ≡∂ES∗loan∂Px

=

P ∗yCy

[1 +

α∗

1 + i∗

]− w∗l∗I

1− α

[1− ξ1 + ξi

]1

Px> 0,

λ∗2i∗ ≡∂ES∗loan∂i∗

=

w∗l∗I1− α

[1− ξ

1 + ξi∗

]+[P ∗yCy + P ∗yC

∗y

] α∗

1 + i∗

−[w∗l∗I1− α

[ξ(1 + i)

1 + ξi

]+ P ∗yCy

[1 +

α∗

1 + i∗

]](1− α)PxC

∗x

wlx

1

1 + i∗> 0,

λ∗2B ≡∂ES∗loan∂B

= −w∗l∗I1− α

[ξ(1 + i)

1 + ξi

]+ P ∗yCy

[1 +

α∗

1 + i∗

](1− α)

wlx< 0,

λ∗2Ax ≡∂ES∗loan∂Ax

=

P ∗yCy

[1 +

α∗

1 + i∗

]− w∗l∗I

1− α

[1− ξ1 + ξi

]1

Ax> 0,

λ∗2A∗y ≡∂ES∗loan∂A∗y

= 0,

λ∗2φ∗I≡ ∂ES∗loan

∂φ∗I= − αw∗l∗I

(1− α)

1

φ∗I< 0,

44

where we have assumed that a, a∗, α and α∗ are sufficiently high such that λ1Pxλ∗2i∗ − λ∗2Pxλ1i∗ > 0 when

θo = θ, and λ1i∗ > 0 and λ∗2i∗ > 0 when θo < θ.

With outsourcing, θo<θ, as∂2ESx∂ξ ∂Px

=∂λ1Px∂ξ

< 0,∂2ESx∂ξ ∂i∗

=∂λ1i∗

∂ξ> 0,

∂2ES∗loan∂ξ ∂Px

=∂λ∗2Px∂ξ

> 0, and

∂2ES∗loan∂ξ ∂i∗

=∂λ∗2i∗

∂ξ< 0, we find that λ1Pxλ

∗2i∗−λ∗2Pxλ

∗1i∗> 0 when ξ is small, and λ1Pxλ

∗2i∗−λ∗2Pxλ1i∗< 0

when ξ is large.

Let (i∗, Px) be the general equilibrium values of i∗ and Px that solve ESx = 0 and ES∗loan = 0 si-

multaneously, given the values of the exogenous variables B, Ax, A∗y, and φ∗I , the linear approximations of

ESx(i∗, Px) and ES∗loan(i∗, Px) about (i∗, Px) are given by

ESx(i∗, Px) ≈ λ1Px(i∗, Px)(Px − Px) + λ1i∗(i∗, Px)(i∗ − i∗), dPxdi∗

∣∣∣∣ESx(i

∗, Px) = 0

≈ − λ1i∗

λ1Px< 0,

ES∗loan(i∗, Px) ≈ λ∗2Px(i∗, Px)(Px − Px) + λ∗2i∗(i∗, Px)(i∗ − i∗), dPx

di∗

∣∣∣∣ES∗

loan(i∗, Px) = 0

≈ − λ∗2i∗

λ∗2Px< 0.

Figure 2 plots these two linearized relations between i∗ and Px. Given that λ1Px λ∗2i∗ − λ∗2Px λ1i∗ > 0 when

ξ is small, and λ1Px λ∗2i∗ − λ∗2Px λ1i∗ < 0 when ξ is large, the linearized relation that satisfies ESx = 0 will

be flatter (steeper) than the one satisfying ES∗loan = 0 if ξ is small (large).

Deriving the Results of the Comparative Static Exercises in Section 3.1

The equilibrium effects on Px and i∗ are given by

dPxdJ

= −λ1J λ

∗2i∗ − λ∗2J λ1i∗

λ1Px λ∗2i∗ − λ∗2Px λ1i∗

, anddi∗

dJ= −

λ1Px λ∗2J − λ∗2Px λ1J

λ1Px λ∗2i∗ − λ∗2Px λ1i∗

, where J = B, Ax, A∗y, φ∗I .

Using equations (41), (42), and (43), we can derive the following results and summarize the effects of the

temporary shocks in Table 2.

di

1 + i=dAxAx− 1− α

wlx

[dB + PxC

∗x

di∗

1 + i∗

]+dPxPx

⇒ i (Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗ ), ,

+ − − + 0 0

dP ∗yP ∗y

= −dA∗yA∗y

+ α∗di∗

1 + i∗−[

(1− α)P ∗yCy

P ∗yCy + P ∗yC∗y

]di

1 + i, ⇒ P ∗y (Px, i

∗, B, Ax, A∗y, φ∗I , θo, n, n∗ ),

− + + − − 0

dTB∗ =

[PxC

∗x

e− α q∗QI

1− α

[1− ξ

1 + ξi∗

]]di∗

1 + i∗+q∗QI1− α

[αdφ∗Iφ∗I

+dPxPx

+dAxAx

]−[P ∗yCy +

q∗QI1− α

[ξ(1 + i)

1 + ξi

]]di

1 + i.

⇒ TB∗(Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗ ),

− + + − 0 +

45

The case with a flexible exchange rate regime

Similar to the derivation in the case with a fixed exchange rate regime, we can examine the dESx and

dES∗loan equations in the case with a flexible exchange rate by using equation (42) and (43),

de

e= − di

1 + i− n∗(1 + i∗)

α∗P ∗yCy

di∗

1 + i∗and

di

1 + i=−dB − PxC∗x

[1 + n∗(1+i∗)

α∗P∗yCy

]di∗

1+i∗ + wlx1−α

[dPxPx

+ dAxAx

]wlx1−α + PxC∗x

.

We obtain

dESx =

[PxC

∗x −

(1 + ξi)qQI1− α

(1− ξ

1 + ξi∗

)]+

[(1 + i)wlx − (1− ξ)qQI

1− α− PxC∗x

] PxC∗x [1 + n∗(1+i∗)α∗P∗yCy

]wlx1−α + PxC∗x

+

[PxC

∗x +

(1 + ξi)qQI1− α

][n∗(1 + i∗)

α∗P ∗yCy

]di∗

1 + i∗

+

[(1 + i)wlx + (1 + ξi)qQI

α(1− α)−Pxκ

(θ− θo

)]−[

(1 + i)wlx − (1− ξ)qQI1− α

− PxC∗x] wlx

1−αwlx1−α + PxC∗x

dPxPx

+

[(1 + i)wlx + (1 + ξi)qQI

α(1− α)

]−[

(1 + i)wlx − (1− ξ)qQI1− α

− PxC∗x] wlx

1−αwlx1−α + PxC∗x

dAxAx

+

[(1 + ξi)qQI

1− α

]dφ∗Iφ∗I

+

[(1 + i)wlx − (1− ξ)qQI

1− α− PxC∗x

]dB

wlx1−α + PxC∗x

and

dES∗loan = −−[

α∗

1 + i∗[P ∗yCy + P ∗yC

∗y

]+

w∗l∗I1− α

(1− ξ

1 + ξi∗

)]+

w∗l∗I1− α

(1− ξ1 + ξi

) [ wlx1−α

[n∗(1+i∗)α∗P∗yCy

]− PxC∗x

]wlx1−α + PxC∗x

+

[P ∗yCy

[1 +

α∗

1 + i∗

]+

w∗l∗I1− α

ξ(1 + i)

1 + ξi

][n∗(1 + i∗)

α∗P ∗yCy

]di∗

1 + i∗

−

[w∗l∗I1− α

+w∗l∗I1− α

(1− ξ1 + ξi

) wlx1−α

wlx1−α + PxC∗x

]dPxPx−

[w∗l∗I1− α

+w∗l∗I1− α

(1− ξ1 + ξi

) wlx1−α

wlx1−α + PxC∗x

]dAxAx

−αw∗l∗I

1− αdφIφI

+w∗l∗I1− α

(1− ξ1 + ξi

)dB

wlx1−α + PxC∗x

.

Hence, we have ESx(Px, i∗, B, Ax, A∗y, φ∗I , θo, n, n∗) and ES∗loan(Px, i

∗, B, Ax, A∗y, φ∗I , θo, n, n∗).

+ + + + 0 + − − + − 0 −

The linear approximation of ESx(i∗, Px) = 0 and ES∗loan(i∗, Px) = 0 around the general equilibrium under a

flexible exchange rate regime are both downward sloping, and the linearized relation that satisfies ESx = 0

will be flatter than the one satisfying ES∗loan = 0. The value of ξ does not affect the derivatives in a way

that can change the relative slope of the two linearized relations qualitatively.

46

Deriving the Results of the Comparative Static Exercises in Section 3.2

For simplicity, we assume no uncertainty in s and have β(1 + i) = β∗(1 + i∗) = 1.

The case with no outsourcing (θo = θ):

Using equations (42), (43), and (44), we get

n+ B − αa(1− n)

1 + i=

[1 +

α

1 + i

]e(1− a∗)(1− n∗)

1 + i∗,

n∗ − α∗a∗(1− n∗)1 + i∗

=

[1 +

α∗

1 + i∗

](1− a)(1− n)

e(1 + i),

(1− a)(1− n)

1 + i=

e(1− a∗)(1− n∗)1 + i∗

+ eZ∗.

Assuming that the values of a, a∗, α and α∗ are sufficiently high. We can show that

dn

de> 0,

dn∗

de< 0 and

dZ∗

de=dTB∗

de< 0. Hence, we get

dP ∗yCy

de< 0, and

d(PxC∗xe

)de

> 0.

The case with international outsourcing (θo < θ):

Let p∗x ≡ Px/e, N ≡ (1− n)/e, N ∗ ≡ (1− n∗), and b ≡ B/e. Substituting equation (42) into equation (44)

yields

1−[1 +

αa∗

1 + i∗+

1− a∗

1 + i∗

]N ∗ −

(α

1 + i∗

)(1− a)N

1 + i− Z∗ = 0, (44′′)

Equation (43) can be written as follows by using equations (26′), (42), and (44).[Z∗ +

(1− a∗)N ∗

1 + i∗− (1− a)N

1 + i

]1 + i∗

1 + ξi∗

·

θ2−α1−α− θ

2−α1−α −

[θo

2−α1−α − θ

2−α1−α + (2− α)

(θ − θo

)θo

11−α

]1−

(1+i∗)v∗N∗φ∗I(1+ξ i∗)β∗

[1+ξi1+i

]vNβ

α

1−α

=α

1 + ξi

[aN +

(1− a∗)N ∗

1 + i∗

] [θ

2−α1−α − θo

2−α1−α

]. (A1)

Using an expression ofαp∗xAx1+i from equation (42), we rewrite equations (26′) and (43) respectively as follows.

κ

Ax

[1 + ξi∗

1 + i∗

]α [(1 + i∗)v∗N ∗

φ∗I(1 + ξ i∗)β∗

]α [θ

2−α1−α − θo

2−α1−α

]α

=(2− α)

[Z∗+

(1− a∗)N ∗

1 + i∗− (1− a)N

1 + i

]α[1− α2− α

]1−αθo

11−α

1−

(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

[1+ξi1+i

]vNβ

α

1−α, (A2)

47

[1

e−N+b− (1− a∗)N ∗

1 + i∗

] ( (1+i∗)v∗N∗φ∗I(1+ξ i∗)β∗

)−α1−α[θ

2−α1−α − θo

2−α1−α

](vNβ

)−α1−α[θo

2−α1−α − θ

2−α1−α

] [1 + ξi∗

1 + i∗

] [1 + i

1 + ξi

] 11−α

= Z∗ +(1− a∗)N ∗

1 + i∗− (1− a)N

1 + i. (A3])

Define δ1 ≡(

α∗

1 + i∗

)1− a1 + i

, δ2 ≡ 1 +α∗a∗

1 + i∗+

1− a∗

1 + i∗, V ≡

1−

(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

[1+ξi1+i

]vNβ

α

1−α ,

Θ ≡ θ2−α1−α− θ

2−α1−α , and Ψ ≡

[θo

2−α1−α − θ

2−α1−α + (2− α)

(θ − θo

)θo

11−α

].

Totally differentiating equations (44′), (A1), (A2), and (A3), we get

dN = d

(1− ne

)= − 1

δ1[ δ2 dN ∗ + dZ∗] .

A1N∗dN ∗

N ∗+A1θo

dθo

θo+A1Z∗

dZ∗

Z∗= 0,

A2N∗dN ∗

N ∗+A2θo

dθo

θo+A2Z∗

dZ∗

Z∗+A2κ

dκ

κ= 0,

A3N∗dN ∗

N ∗+A3θo

dθo

θo+A3Z∗

dZ∗

Z∗+A3e

de

e= 0,

where

A1N∗ ≡

[1−a∗1+i∗ +

(1−a1+i

)δ2δ1

]N ∗

Z∗ + (1−a∗)N∗1+i∗ − (1−a)N

1+i

+

[a δ2δ1− 1−a∗

1+i∗

]N ∗

aN + (1−a∗)N∗1+i∗

+Ψ[

α1−α

][(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

] α1−α[vNβ

]−α1−α[1+ξi1+i

] α1−α [δ1N+δ2N∗]

δ1N

Θ−ΨV

A1θ∗ ≡

(2−α1−α

)θo

2−α1−α

θ2−α1−α − θo

2−α1−α

−V (2− α)θo

11−α

[θ

1−α − θo]

Θ−ΨV

A1Z∗ ≡Z∗ + (1−a)

1+i[1−δ1N−δ2N∗]

δ1

Z∗ + (1−a∗)N∗1+i∗ − (1−a)N

1+i

+

a [1−δ1N−δ2N∗]δ1

aN + (1−a∗)N∗1+i∗

+Ψ[

α1−α

][(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

] α1−α[vNβ

]−α1−α[1+ξi1+i

] α1−α [1−δ1N−δ2N∗]

δ1N

Θ−ΨV

A2N∗ ≡Z∗ − (1−a)

1+i[δ1N+δ2N∗]

δ1

Z∗ + (1−a∗)N∗1+i∗ − (1−a)N

1+i

+

11−α

[(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

] α1−α[vNβ

]−α1−α[1+ξi1+i

] α1−α

[[δ1N+δ2N∗]

δ1N

]V

A2θo ≡ −

(

2−α1−α

)θo

2−α1−α

θ2−α1−α − θo

2−α1−α

+1

α(1− α)

A2Z∗ ≡ −Z∗ + (1−a)

1+i[1−δ1N−δ2N∗]

δ1

Z∗ + (1−a∗)N∗1+i∗ − (1−a)N

1+i

+

11−α

[(1+i∗)v∗N∗φ∗I(1+ξi∗)β∗

] α1−α[vNβ

]−α1−α[1+ξi1+i

] α1−α

[[1−δ1N−δ2N∗]

δ1N

]V

48

A2κ ≡1

α

A3N∗ ≡α

1− α

[δ1N + δ2N ∗

δ1N

]−

[δ2δ1− 1−a∗

1+i∗

]N ∗[

1e−N+b− (1−a∗)N∗

1+i∗

] +

[1−a∗1+i∗ + 1−a

1+i

(δ2δ1

)]N ∗

Z∗ + (1−a∗)N∗1+i∗ − (1−a)N

1+i

A3θo ≡

2−α1−α θ

o2−α1−α

[θ

2−α1−α − θ

2−α1−α

][θ

2−α1−α − θo

2−α1−α

] [θo

2−α1−α − θ

2−α1−α

]

A3Z∗ ≡α

1− α

[1− δ1N − δ2N ∗

δ1N

]−

1−δ1N−δ2N∗δ1[

1e−N+b− (1−a∗)N∗

1+i∗

] +

[Z∗ + 1−a

1+i

[1−δ1N−δ2N∗

δ1N

]]Z∗ + (1−a∗)N∗

1+i∗ − (1−a)N1+i

A3e ≡1e[

1e−N+b− (1−a∗)N∗

1+i∗

]

The Effects of Changes in e

Holding κ constant, the derivation of the effects of a change in e as follows. Using equation (A3), we have

A3e +

[A3Z∗ + A3N∗

Z∗

N ∗dN ∗

dZ∗+A3θo

Z∗

θodθo

dZ∗

]e

Z∗dZ∗

de= 0.

Using equations (A1) and (A2), we can show that

sign [A1N∗ A2θo −A2N∗ A1θo ] = sign [A1Z∗ A2θo −A2Z∗ A1θo ] = sign [A1N∗ A2Z∗ −A2N∗ A1Z∗ ] = − sign δ1.

δ1 > 0 ⇒ Z∗

N ∗dN ∗

dZ∗= − A1Z∗ A2θo −A2Z∗A1θo

A1N∗ A2θo −A2N∗A1θo< 0, and

Z∗

θodθo

dZ∗= −A1N∗ A2Z∗ −A2N∗A1Z∗

A1N∗ A2θo −A2N∗A1θo< 0.

In addition, A3e > 0 and

[A3Z∗ + A3N∗

Z∗

N ∗dN ∗

dZ∗+A3θo

Z∗

θodθo

dZ∗

]

=A3Z∗ [A1N∗ A2θo −A2N∗ A1θo ]− A3N∗ [A1Z∗ A2θo −A2Z∗ A1θo ]− A3θo [A1N∗A2Z∗ −A2N∗ A1Z∗ ]

A1N∗ A2θo −A2N∗ A1θo< 0.

Hence, we havedZ∗

de> 0,

dθo

de< 0,

dN ∗

de< 0,

dn∗

de> 0, and

d

(PxC∗xe

)de

< 0

[1− Z∗

Z∗Z∗

N ∗dN ∗

dZ∗+ 1

]< 0 ⇒ Z∗

N ∗dN ∗

dZ∗− Z∗

NdNdZ∗

=Z∗

δ1N

[1− Z∗

Z∗Z∗

N ∗dN ∗

dZ∗+ 1

]< 0.

49

Using P ∗yCy =(1− a)N

1 + i, q∗QI

[1 + ξi∗

1 + i∗

]= 1−

[1 +

a∗α∗

1 + i∗

]N ∗−

[1 +

α∗

1 + i∗

](1− a)N

1 + i, and

Z∗

N ∗dN ∗

dZ∗− Z∗

NdNdZ∗

< 0, we haved q∗QIde

> 0, andd(q∗QI + P ∗yCy)

de> 0.

The Effects of Changes in κ:

Using equation (A3), we get

A3ede

e+

[A3Z∗ + A3N∗

Z∗

N ∗∂N ∗

∂Z∗+A3θo

Z∗

θo∂θo

∂Z∗

]dZ∗

Z∗+

[A3N∗

κ

N ∗∂N ∗

∂κ+A3θo

κ

θo∂θo

∂κ

]dκ

κ= 0.

In the case with a flexible exchange rate regime, dZ∗ = 0,

κ

N ∗dN ∗

dκ

∣∣∣∣dZ∗ = 0

=κ

N ∗∂N ∗

∂κ=

A2κA1θo

A1N∗ A2θo −A2N∗ A3θo,

dN ∗

dκ

∣∣∣∣dZ∗ = 0

< 0.

κ

θodθo

dκ

∣∣∣dZ∗ = 0

=κ

θo∂θo

∂κ= − A2κA1N∗

A1N∗ A2θo −A2N∗ A3θo,

dθo

dκ

∣∣∣dZ∗ = 0

> 0.

A2κ > 0, A3e > 0, sign [A3N∗A1θo −A3θoA1N∗ ] = sign [A1N∗ A2θo −A2N∗ A1θo ] = − sign δ1

⇒[A3N∗

dN ∗

N ∗κ

dκ

∣∣∣∣dZ∗=0

+A3θodθo

θoκ

dκ

∣∣∣dZ∗=0

]=A2κ [A3N∗A1θo −A3θoA1N∗ ]

A1N∗ A2θo −A2N∗ A1θo> 0 and

κ

e

de

dκ

∣∣∣dZ∗=0

= − 1

A3e

A3N∗

dN ∗

N ∗κ

dκ

∣∣∣∣dZ∗=0

+A3θodθo

θoκ

dκ

∣∣∣dZ∗=0

= −A2κ

A3e

[ A3N∗A1θo −A3θoA1N∗

A1N∗ A2θo −A2N∗ A1θo

]< 0.

When e is fixed at e by allowing Z∗ to adjust, using

[A3N∗

κ

N ∗∂N ∗

∂κ+A3θo

κ

θo∂θo

∂κ

]> 0 and

[A3Z∗ + A3N∗

Z∗

N ∗∂N ∗

∂Z∗+A3θo

Z∗

θo∂θo

∂Z∗

]< 0, we have

κ

Z∗dZ∗

dκ

∣∣∣∣de=0

= −[A3N∗

κN∗

∂N∗∂κ

+A3θoκθo

∂θo

∂κ

][A3Z∗ + A3N∗

Z∗N∗

∂N∗∂Z∗ +A3θo

Z∗θo

∂θo

∂Z∗

] > 0.

50

Figure 1 : The timing of transactions of the representative home household

Period t

• The representative household of the home country enters period t with cash balances of mht units

of home currency and mft units of foreign currency.

• Everyone learns if there are any unexpected changes in the set of parameters (κ, e).

• Deposit and sourcing decisions (nt, θot ) are made.

• The home household allocates its cash balances across its members and then separates.

∗ nt units of home currency are deposited to the intermediary;

∗ the shopper takes mht − nt units of home currency;

∗ the importer takes mft units of foreign currency;

∗ the worker and the entrepreneur obtain no cash.

• The firm-specific productivity parameter, θit, is revealed to everyone, θit ∈ [ θ, θ ]. The en-

trepreneur makes the make-or-buy decision of intermediate input by comparing θit to θot .

• International outsourcing contract is signed if θit ∈ (θot , θ ].

• The realization of the state of the world st = (Axt, A∗yt, φ

∗It,Bt ) is revealed to everyone.

• All markets are opened. Production and trade take place.

∗ loan/bond markets

∗ foreign exchange market

∗ labor markets

∗ goods markets

• All markets are closed.

• Loan repayments and bond redemptions are settled, deposits are paid out, and dividends are

distributed.

• The household is reunited, receives the lump-sum transfer Tt, consumes Cxt and Cyt, and takes

1− ht units of leisure.

• The household carries the cash balances of mht+1 units of home currency and mft+1 units of

foreign currency over to period t+ 1.

Period t+ 1

•••

51

.

Figure 2: The effects of various temporary shocks in Section 3.1

The solid lines show the original linearized relations between Px and i∗ around the general equilibrium

point and the dotted lines reflect the effects of the shocks on the relations.

Under a fixed e, the linearized relation ofES∗loan=0 is flatter than that ofESx =0 when ξ is sufficiently high.

(a) An increase in B shifts the ESx = 0 relation downward and the ES∗loan = 0 relation rightward.

fixed e, θo = θ

-

6

0r

Px

i∗

BBBBBBBB

ES∗loan

=0

PPPPPP

ESx=0 q qfixed e, θo< θ, and low ξ

-

6

0r

Px

i∗

AAAAAAAA

ES∗loan

=0

HHHH

HH

ESx=0

qqfixed e, θo< θ, and high ξ

-

6

0r

Px

i∗

AAAAAAAA

ESx=0

HHHH

HHES∗loan

=0

q qflexible e and θo < θ

-

6

0r

Px

i∗

AAAAAAAA

ES∗loan

=0

HHHHHH

ESx=0 qq

(b) An increase in Ax shifts the ESx = 0 relation downward and the ES∗loan = 0 relation leftward.

fixed e and θo = θ

-

6

0r

Px

i∗

BBBBBBBB

ES∗loan

=0

PPPPPP

ESx=0 qq

fixed e, θo< θ and low ξ

-

6

0r

Px

i∗

AAAAAAAA

ES∗loan

=0

HHHHHH

ESx=0

qqfixed e, θo< θ, and high ξ

-

6

0r

Px

i∗

AAAAAAAA

ESx=0

HHHHHHES∗

loan=0

q qflexible e and θo< θ

-

6

0r

Px

i∗

AAAAAAAA

ES∗loan

=0

HHHH

HH

ESx=0

qq

(c) An increase in φ∗I shifts the ESx = 0 relation downward, while shifting the ES∗loan = 0 relation rightward

under a fixed exchange rate regime but leftward under a flexible exchange rate regime.

fixed e, θo< θ and low ξ

-

6

0r

Px

i∗

AAAAAAA

ES∗loan

=0

HHHH

HH

ESx=0 q qfixed e, θo< θ, and high ξ

-

6

0r

Px

i∗

AAAAAAA

ESx=0

HHHHHH

ES∗loan

=0qq

flexible e and θo< θ

-

6

0r

Px

i∗

AAAAAAAA

ES∗loan

=0

HHHHHH

ESx=0

q q

52

53

Figure 3: China’s Exchange Rates and Trade Balance to GDP Ratios

-6.00

-4.00

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

1994

1995

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

2013

2014

TB/GDP (%)

60.00

70.00

80.00

90.00

100.00

110.00

120.00

130.00

140.00

Exchange

Rate

(2010=100)

All Products

Consumer goods

Intermediate goods

Real effective exchange rate index (2010 = 100)

Adjusted Official USD/RMB Rate (2010 = 100)

Source: The World Development Indicators and the World Integrated Trade

Solution Databases, the World Bank.

Trade Balance of Capital goods/ GDP (%)

Trade Balance of Consumer goods/ GDP (%)

Trade Balance of Intermediate goods/ GDP (%)

Trade Balance of Raw Material/ GDP (%)

Trade Balance of All Products/ GDP (%)

Real Effective Exchange Rate Index (REER) (2010 =100)

Official Nominal Exchange Rate, USD per RMB

1985 -4.97 169.92 0.3405

1986 -4.04 123.72 0.2896

1987 -1.19 107.15 0.2687

1988 -1.91 116.83 0.2687

1989 -1.43 135.20 0.2656

1990 2.19 99.06 0.2091

1991 1.96 87.03 0.1879

1992 -4.65 8.96 -4.10 0.67 1.02 83.48 0.1813

1993 -7.07 9.12 -5.46 0.55 -2.76 88.86 0.1736

1994 -6.10 9.86 -3.50 0.57 0.96 69.63 0.1160

1995 -3.85 8.53 -2.29 -0.17 2.28 77.57 0.1197

1996 -3.09 7.55 -2.91 -0.24 1.42 85.25 0.1203

1997 -1.83 8.43 -2.22 -0.30 4.22 91.78 0.1206

1998 -1.56 8.04 -2.27 -0.17 4.25 96.66 0.1208

1999 -1.92 7.68 -2.66 -0.51 2.68 91.50 0.1208

2000 -1.93 8.25 -2.89 -1.58 2.00 91.45 0.1208

2001 -2.08 7.64 -2.65 -1.36 1.69 95.40 0.1208

2002 -2.09 8.08 -2.83 -1.30 2.08 93.20 0.1208

2003 -2.18 8.62 -3.14 -2.06 1.54 87.08 0.1208

2004 -1.65 8.93 -2.45 -3.56 1.65 84.73 0.1208

2005 0.05 9.83 -1.78 -4.10 4.50 84.26 0.1220

2006 1.01 9.90 -0.52 -4.53 6.50 85.57 0.1254

2007 2.40 9.33 -0.05 -4.84 7.49 88.93 0.1314

2008 3.29 7.86 0.80 -5.86 6.54 97.11 0.1439

2009 2.37 6.33 -0.97 -4.23 3.88 100.41 0.1464

2010 2.60 6.50 -0.76 -5.48 3.01 100.00 0.1477

2011 2.69 6.19 -0.34 -6.20 2.07 102.70 0.1548

2012 2.89 6.23 -0.18 -5.72 2.73 108.43 0.1584

2013 2.71 6.19 -0.08 -5.29 2.73 115.30 0.1614

2014 2.68 6.17 0.04 -4.71 3.71 119.01 0.1628

Correlation Coefficient with REER

1985-1993 -0.72

1994-2008 0.57 -0.63 0.32 -0.25 0.34

1994-2014 0.76 -0.84 0.65 -0.60 0.12

Correlation Coefficient with USD/RMB Rate

1985-1993 -0.72 0.93

1994-2008 0.86 -0.08 0.91 -0.78 0.75 0.44

1994-2014 0.85 -0.79 0.83 -0.80 0.17 0.86

Tab

le3:

Nu

mer

ical

exer

cise

son

the

effec

tsof

tem

pora

rysh

ock

sin

Sec

tion

s3.1

an

d4.1

a

fixed

e,e

=1.0

5,

Ba

=0.0

001,

%4

(M+B

)=

0.0

1%

flex

iblee,

Z∗

=0,Ba

=0.0

001,

%4

(M+B

)=

0.0

1%

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

ξ0

0.2

0.4

0.6

0.8

0.9

1ξ

00.2

0.4

0.6

0.8

0.9

1n

0.5

24

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13n

0.5

03

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

n∗

0.4

84

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95n∗

0.5

03

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

θo

1.0

00

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93θo

1.0

00

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

Z∗ a

−0.0

10−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

05

−0.0

16−

0.0

09e a

1.0

001

1.0

659

1.0

659

1.0

659

1.0

659

1.0

659

1.0

659

1.0

659

Z∗ b

−0.0

38−

0.0

08−

0.0

08−

0.0

08−

0.0

08−

0.0

10

0.0

01−

0.0

06e b

1.0

000

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

4i

−8.1

64−

0.1

53−

0.1

97−

0.2

78−

0.4

78−

1.7

50

5.0

41

1.0

284i

−0.0

17−

0.0

18−

0.0

18−

0.0

17−

0.0

17−

0.0

17−

0.0

17−

0.0

17

4i∗

7.5

26

0.1

26

0.1

67

0.2

44

0.4

33

1.6

37

−4.7

90−

0.9

924i∗

0.0

00

0.0

01

0.0

01

0.0

01

0.0

01

0.0

00

0.0

00

0.0

00

%4Px

−6.0

53−

0.1

09−

0.1

43−

0.2

04−

0.3

56−

1.3

15

3.9

15

0.7

91

%4Px−

0.0

09−

0.0

09−

0.0

09−

0.0

09−

0.0

08−

0.0

08−

0.0

08−

0.0

08

%4P

f y5.9

29

0.0

99

0.1

31

0.1

89

0.3

34

1.2

63

−3.5

98−

0.7

53

%4P

f y0.0

00

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

0.0

00

0.0

00

%4C

x6.4

43

0.1

10

0.1

43

0.2

05

0.3

57

1.3

33

−3.7

67−

0.7

85

%4C

x0.0

09

0.0

09

0.0

09

0.0

09

0.0

08

0.0

08

0.0

08

0.0

08

%4C

y2.1

04

0.0

48

0.0

59

0.0

78

0.1

25

0.4

26

−1.1

70−

0.2

30

%4C

y0.0

00

0.0

03

0.0

03

0.0

02

0.0

02

0.0

01

0.0

01

0.0

01

%4l x

4.9

09

0.1

12

0.1

38

0.1

87

0.3

07

1.0

75

−2.9

56−

0.5

94

%4l x

0.0

22

0.0

25

0.0

25

0.0

24

0.0

24

0.0

24

0.0

23

0.0

23

%4u

3.5

62

0.0

35

0.0

52

0.0

83

0.1

60

0.6

51

−1.9

42−

0.4

18

%4u

−0.0

16−

0.0

16−

0.0

17−

0.0

17−

0.0

17−

0.0

17−

0.0

17−

0.0

17

%4C

∗ x−

0.9

79−

0.0

11−

0.0

18−

0.0

30−

0.0

59−

0.2

48

0.7

62

0.1

64

%4C

∗ x0.0

25

0.0

20

0.0

21

0.0

21

0.0

22

0.0

22

0.0

22

0.0

22

%4C

∗ y−

5.5

97−

0.0

99−

0.1

30−

0.1

89−

0.3

33−

1.2

47

3.7

32

0.7

59

%4C

∗ y0.0

00−

0.0

01−

0.0

01−

0.0

01−

0.0

01−

0.0

01

0.0

00

0.0

00

%4l∗ y

−4.3

11−

0.0

66−

0.0

91−

0.1

35−

0.2

46−

0.9

45

2.8

47

0.5

88

%4l∗ y

0.0

00

0.0

01

0.0

01

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ I

0.0

00−

0.6

88−

0.6

92−

0.6

98−

0.7

14−

0.8

16

−0.2

70−

0.5

94

%4l∗ I

0.0

00−

0.0

68−

0.0

57−

0.0

47−

0.0

37−

0.0

29−

0.0

24−

0.0

20

%4u∗

−2.1

11−

0.0

24−

0.0

37−

0.0

61−

0.1

20−

0.4

96

1.5

25

0.3

26

%4u∗

0.0

19

0.0

16

0.0

16

0.0

16

0.0

16

0.0

17

0.0

17

0.0

17

%4Z

∗−

73.2

31−

5.6

78−

7.5

79−

10.9

88−

18.8

76−

56.9

95−

1344.7

54

64.3

76

%4e

0.0

16

0.0

13

0.0

13

0.0

13

0.0

14

0.0

14

0.0

14

0.0

15

%4TB

∗−

73.2

31−

6.0

50−

7.9

71−

11.4

04−

19.3

30−

57.5

32−

1296.3

89

64.3

76

%4TB

∗0.0

00−

0.0

40−

0.0

34−

0.0

28−

0.0

23−

0.0

17−

0.0

15

0.0

00

fixed

e,e

=1.0

5,

Axa

=4.4,

%4A

x=

10%

flex

iblee,

Z∗

=0,

Axa

=4.4,

%4A

x=

10%

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

ξ0

0.2

0.4

0.6

0.8

0.9

1ξ

00.2

0.4

0.6

0.8

0.9

1n

0.5

24

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13n

0.5

03

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

n∗

0.4

84

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94n∗

0.5

03

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

θo

1.0

00

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93θo

1.0

00

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

Z∗ a

−0.0

24−

0.0

08−

0.0

08−

0.0

08−

0.0

08−

0.0

08

−0.0

09−

0.0

08e a

1.0

00

1.0

666

1.0

666

1.0

666

1.0

666

1.0

666

1.0

666

1.0

666

Z∗ b

−0.0

24−

0.0

08−

0.0

08−

0.0

08−

0.0

08−

0.0

08

−0.0

07−

0.0

08e b

1.0

00

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

4i

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00−

0.0

02

0.0

05

0.0

014i

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

4i∗

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

02

−0.0

05−

0.0

014i∗

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4Px

−0.0

45−

0.0

45−

0.0

45−

0.0

45−

0.0

45−

0.0

46

−0.0

43−

0.0

45

%4Px−

0.0

46−

0.0

46−

0.0

46−

0.0

46−

0.0

46−

0.0

46−

0.0

46−

0.0

46

%4P

f y0.0

00

0.0

10

0.0

13

0.0

19

0.0

33

0.1

24

−0.3

65−

0.0

75

%4P

f y0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4C

x10.0

00

10.0

13

10.0

17

10.0

23

10.0

40

10.1

45

9.5

80

9.9

15

%4C

x10.0

00

10.0

04

10.0

04

10.0

04

10.0

04

10.0

04

10.0

04

10.0

04

%4C

y0.0

00

0.0

05

0.0

06

0.0

08

0.0

12

0.0

42

−0.1

18−

0.0

23

%4C

y0.0

00

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

%4l x

0.0

00

0.0

08

0.0

10

0.0

15

0.0

27

0.1

02

−0.3

03−

0.0

62

%4l x

0.0

00−

0.0

01−

0.0

01−

0.0

01−

0.0

02−

0.0

02−

0.0

02−

0.0

02

%4u

13.1

54

11.8

59

11.8

61

11.8

64

11.8

72

11.9

23

11.6

47

11.8

11

%4u

10.9

16

11.4

72

11.4

72

11.4

72

11.4

72

11.4

72

11.4

72

11.4

72

%4C

∗ x10.0

00

10.0

00

9.9

99

9.9

98

9.9

95

9.9

74

10.0

84

10.0

19

%4C

∗ x10.0

00

10.0

07

10.0

07

10.0

07

10.0

07

10.0

07

10.0

07

10.0

07

%4C

∗ y0.0

00−

0.0

10−

0.0

13−

0.0

19−

0.0

33−

0.1

24

0.3

67

0.0

75

%4C

∗ y0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ y

0.0

00−

0.0

07−

0.0

09−

0.0

13−

0.0

24−

0.0

93

0.2

81

0.0

58

%4l∗ y

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ I

0.0

00−

0.0

71−

0.0

72−

0.0

72−

0.0

74−

0.0

84

−0.0

30−

0.0

62

%4l∗ I

0.0

00−

0.0

20−

0.0

18−

0.0

15−

0.0

13−

0.0

11−

0.0

11−

0.0

10

%4u∗

6.2

08

6.6

79

6.6

78

6.6

75

6.6

69

6.6

31

6.8

38

6.7

15

%4u∗

7.1

89

6.8

64

6.8

64

6.8

64

6.8

64

6.8

65

6.8

65

6.8

65

%4Z

∗0.0

00−

0.5

50−

0.7

42−

1.0

93−

1.9

52−

7.2

50

25.2

89

4.7

39

%4e

0.0

00

0.0

02

0.0

02

0.0

02

0.0

03

0.0

03

0.0

03

0.0

03

%4TB

∗0.0

00−

0.5

85−

0.7

79−

1.1

35−

2.0

01−

7.3

39

25.4

69

4.7

39

%4TB

∗0.0

00−

0.0

12−

0.0

10−

0.0

09−

0.0

08−

0.0

07−

0.0

06

0.0

00

54

fixed

e,e

=1.0

5,

A∗ ya

=4.4,

%4A

∗ y=

10%

flex

iblee,

Z∗

=0,

A∗ ya

=4.4,

%4A

∗ y=

10%

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo

=θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

ξ0

0.2

0.4

0.6

0.8

0.9

1ξ

00.2

0.4

0.6

0.8

0.9

1n

0.5

24

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

n0.5

03

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

n∗

0.4

84

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95

0.4

95

n∗

0.5

03

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

θo

1.0

00

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

θo

1.0

00

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

Z∗ a

−0.0

24−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07

e A1.0

000

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

Z∗ b

−0.0

24−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07

e B1.0

000

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

1.0

658

4i

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

004i

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

4i∗

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

004i∗

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4Px

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4Px

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4P

f y−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91

%4P

f y−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91−

9.0

91

%4C

x0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4C

x0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4C

y10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

%4C

y10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

%4l x

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l x

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4u

8.7

69

7.8

84

7.8

84

7.8

84

7.8

84

7.8

84

7.8

84

7.8

84

%4u

7.2

78

7.6

39

7.6

39

7.6

39

7.6

39

7.6

39

7.6

39

7.6

39

%4C

∗ x0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4C

∗ x0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4C

∗ y10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

%4C

∗ y10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

10.0

00

%4l∗ y

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ y

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ I

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4l∗ I

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4u∗

9.3

13

10.0

37

10.0

37

10.0

37

10.0

37

10.0

37

10.0

37

10.0

37

%4u∗

10.7

83

10.2

94

10.2

94

10.2

94

10.2

94

10.2

94

10.2

94

10.2

94

%4Z

∗0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4e

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4TB

∗0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4TB

∗0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

fixed

e,e

=1.0

5,

φ∗ Ia

=1.0

02,

%4φ∗ I

=0.2

%fl

exib

lee,

Z∗

=0,

φ∗ Ia

=1.0

02,

%4φ∗ I

=0.2

%

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

θo<θ

ξ0

0.2

0.4

0.6

0.8

0.9

1ξ

00.2

0.4

0.6

0.8

0.9

1n

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13

0.5

13n

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

0.5

09

n∗

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94

0.4

94n∗

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

0.4

98

θo

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93

0.9

93θo

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

0.9

87

Z∗ a

−0.0

08−

0.0

08−

0.0

08−

0.0

07−

0.0

06−

0.0

14−

0.0

09e A

1.0

667

1.0

667

1.0

667

1.0

667

1.0

667

1.0

667

1.0

667

Z∗ b

−0.0

08−

0.0

08−

0.0

08−

0.0

08−

0.0

10−

0.0

02−

0.0

07e B

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

1.0

665

4i

−0.0

01−

0.0

01−

0.0

02−

0.0

03−

0.0

12

0.0

35

0.0

074i

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

4i∗

0.0

01

0.0

01

0.0

02

0.0

03

0.0

11−

0.0

33−

0.0

074i∗

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

0.0

00

%4Px

−0.0

75−

0.0

98−

0.1

39−

0.2

41−

0.8

89

2.6

68

0.5

35

%4Px

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

0.0

01

%4P

f y0.0

70

0.0

91

0.1

31

0.2

29

0.8

52−

2.4

77−

0.5

09

%4P

f y−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07−

0.0

07

%4C

x0.0

75

0.0

98

0.1

39

0.2

42

0.8

97−

2.5

99−

0.5

32

%4C

x−

0.0

01−

0.0

01−

0.0

01−

0.0

01−

0.0

01−

0.0

01−

0.0

01

%4C

y0.0

25

0.0

32

0.0

45

0.0

76

0.2

79−

0.8

13−

0.1

64

%4C

y−

0.0

19−

0.0

19−

0.0

18−

0.0

18−

0.0

17−

0.0

17−

0.0

17

%4l x

0.0

57

0.0

75

0.1

08

0.1

89

0.7

04−

2.0

54−

0.4

21

%4l x

−0.0

17−

0.0

17−

0.0

16−

0.0

16−

0.0

16−

0.0

16−

0.0

16

%4u

0.0

41

0.0

53

0.0

74

0.1

26

0.4

57−

1.3

22−

0.2

67

%4u

0.0

04

0.0

04

0.0

04

0.0

04

0.0

04

0.0

04

0.0

04

%4C

∗ x−

0.0

12−

0.0

16−

0.0

24−

0.0

44−

0.1

71

0.5

15

0.1

06

%4C

∗ x0.0

24

0.0

23

0.0

23

0.0

23

0.0

23

0.0

22

0.0

22

%4C

∗ y−

0.0

69−

0.0

91−

0.1

31−

0.2

28−

0.8

45

2.5

40

0.5

11

%4C

∗ y0.0

07

0.0

07

0.0

07

0.0

07

0.0

07

0.0

07

0.0

07

%4l∗ y

−0.0

52−

0.0

69−

0.0

99−

0.1

73−

0.6

45

1.9

35

0.3

91

%4l∗ y

−0.0

04−

0.0

04−

0.0

04−

0.0

04−

0.0

03−

0.0

03−

0.0

03

%4l∗ I

0.1

14

0.1

12

0.1

07

0.0

97

0.0

29

0.3

96

0.1

77

%4l∗ I

0.5

65

0.5

57

0.5

50

0.5

44

0.5

38

0.5

35

0.5

32

%4u∗

−0.0

31−

0.0

40−

0.0

56−

0.0

96−

0.3

48

1.0

26

0.2

06

%4u∗

−0.0

02−

0.0

02−

0.0

02−

0.0

02−

0.0

02−

0.0

02−

0.0

02

%4Z

∗−

4.1

05−

5.3

57−

7.6

23−

12.9

71−

41.0

86

654.1

06

36.8

80

%4e

0.0

19

0.0

19

0.0

19

0.0

18

0.0

18

0.0

18

0.0

18

%4TB

∗−

5.3

63−

6.5

89−

8.8

53−

14.2

58−

42.7

26

729.1

64

36.8

80

%4TB

∗0.4

16

0.4

13

0.4

10

0.4

07

0.4

05

0.4

03

0.0

00

aT

he

nu

mer

ical

exer

cise

sass

um

etw

op

oss

ible

state

sof

the

worl

d,s

=a

orb,

each

has

pro

bab

ilit

yeq

ual

to0.5

.T

he

para

met

ervalu

esareβ

=β∗

=0.9

6,a

=a∗

=0.6

,

v=

1,v∗

=0.9

,α

=α∗

=2/3,κ

=0.1

,θ

=0,θ

=1,e

=1.0

5,A

xa

=A

xb

=A

∗ ya

=A

∗ yb

=4,Ba

=Bb

=0,

an

dφ∗ Ia

=φ∗ Ib

=1,

exce

pt

oth

erw

ise

men

tion

edin

the

exer

cise

s.F

or

any

vari

ab

leχ

,4χ≡χa−χb

an

d%4χ≡

(χa−χb)/χb·1

00%.

55

Tab

le4:Numerical

exercisesof

chan

gesineandκforSections3.2

and4.2

a

CaseI:κ

isprohibitivelyhigh

(θo=θ)

CaseII:κ

=0.2

(θo<θ)

CaseIII:κ

=0.1

(θo<θ)

I.1

I.2

I.3

I.4

II.1

II.2

II.3

II.4

II.5

II.6

II.7

II.8

II.9

III.1

III.2

III.3

III.4

III.5

e0.9

000

1.0000b

1.1

000

1.2

000

1.0

300

1.0

500

1.0507b

1.0

700

1.0

705c

1.0

715c

1.0

720

1.2

000

1.3

800

1.0

507

1.0662b

1.0

700

1.0

920d

1.0

925d

n0.4

620

0.5034

0.5

449

0.5

863

0.5

128

0.5

078

0.5077

0.5

028

0.5

027

0.5

025

0.5

024

0.4

701

0.4

234

0.5

162

0.5088

0.5

079

0.5

026

0.5

025

n∗

0.5

495

0.5034

0.4

657

0.4

343

0.4

943

0.4

994

0.4995

0.5

042

0.5

043

0.5

046

0.5

047

0.5

321

0.5

634

0.4

947

0.4984

0.4

993

0.5

043

0.5

045

θo

1.0

000

1.0000

1.0

000

1.0

000

0.9

983

0.9

908

0.9905

0.9

832

0.9

830

0.9

827

0.9

825

0.9

331

0.8

583

0.9

931

0.9872

0.9

857

0.9

772

0.9

770

i a0.0

417

0.0417

0.0

417

0.0

417

0.0

422

0.0

422

0.0417

0.0

421

0.0

421

0.0

421

0.0

421

0.0

420

0.0

418

0.0

419

0.0417

0.0

419

0.0

419

0.0

419

i b0.0

417

0.0417

0.0

417

0.0

417

0.0

411

0.0

412

0.0417

0.0

412

0.0

412

0.0

412

0.0

412

0.0

413

0.0

415

0.0

414

0.0417

0.0

414

0.0

414

0.0

414

i∗ a0.0

417

0.0417

0.0

417

0.0

417

0.0

411

0.0

412

0.0417

0.0

412

0.0

412

0.0

412

0.0

412

0.0

413

0.0

413

0.0

414

0.0417

0.0

414

0.0

414

0.0

414

i∗ b0.0

417

0.0417

0.0

417

0.0

417

0.0

422

0.0

422

0.0417

0.0

421

0.0

421

0.0

421

0.0

421

0.0

421

0.0

420

0.0

419

0.0417

0.0

419

0.0

419

0.0

419

Pxa

0.2

706

0.2583

0.2

454

0.2

318

0.2

558

0.2

582

0.2582

0.2

606

0.2

607

0.2

608

0.2

608

0.2

762

0.2

974

0.2

564

0.2583

0.2

588

0.2

615

0.2

615

Pxb

0.2

841

0.2712

0.2

577

0.2

434

0.2

683

0.2

709

0.2711

0.2

734

0.2

735

0.2

736

0.2

737

0.2

899

0.3

122

0.2

691

0.2712

0.2

716

0.2

745

0.2

745

P∗ ya

0.1

504

0.1593

0.1

662

0.1

717

0.1

607

0.1

592

0.1592

0.1

576

0.1

576

0.1

575

0.1

575

0.1

489

0.1

392

0.1

603

0.1591

0.1

588

0.1

572

0.1

572

P∗ yb

0.1

504

0.1593

0.1

662

0.1

717

0.1

609

0.1

593

0.1592

0.1

578

0.1

577

0.1

576

0.1

576

0.1

490

0.1

393

0.1

603

0.1591

0.1

589

0.1

573

0.1

573

Cxa

1.1

930

1.1535

1.1

128

1.0

708

1.1

430

1.1

438

1.1443

1.1

446

1.1

446

1.1

447

1.1

447

1.1

509

1.1

635

1.1

405

1.1410

1.1

409

1.1

413

1.1

413

Cxb

1.1

362

1.0986

1.0

598

1.0

198

1.0

895

1.0

901

1.0897

1.0

909

1.0

909

1.0

909

1.0

910

1.0

965

1.1

081

1.0

866

1.0867

1.0

869

1.0

874

1.0

874

Cya

1.5

264

1.1973

0.9

561

0.7

709

1.1

294

1.1

303

1.1305

1.1

313

1.1

313

1.1

313

1.1

314

1.1

383

1.1

524

1.1

111

1.1115

1.1

116

1.1

121

1.1

121

Cyb

1.5

264

1.1973

0.9

561

0.7

709

1.1

297

1.1

306

1.1305

1.1

315

1.1

315

1.1

315

1.1

316

1.1

384

1.1

522

1.1

112

1.1115

1.1

117

1.1

122

1.1

122

ha

0.5

464

0.6046

0.6

733

0.7

558

0.6

161

0.5

966

0.5962

0.5

775

0.5

770

0.5

760

0.5

756

0.4

611

0.3

196

0.6

080

0.5932

0.5

895

0.5

688

0.5

683

hb

0.5

464

0.6046

0.6

733

0.7

558

0.6

166

0.5

970

0.5962

0.5

778

0.5

773

0.5

764

0.5

759

0.4

614

0.3

198

0.6

082

0.5932

0.5

897

0.5

690

0.5

685

C∗ xa

0.5

754

0.7382

0.9

197

1.1

246

0.7

824

0.7

822

0.7822

0.7

820

0.7

820

0.7

820

0.7

820

0.7

807

0.7

783

0.7

952

0.7951

0.7

951

0.7

950

0.7

950

C∗ xb

0.5

480

0.7031

0.8

759

1.0

710

0.7

450

0.7

448

0.7449

0.7

446

0.7

446

0.7

446

0.7

446

0.7

433

0.7

408

0.7

573

0.7572

0.7

572

0.7

571

0.7

571

C∗ ya

1.7

974

1.8708

1.9

291

1.9

768

1.8

875

1.8

872

1.8865

1.8

870

1.8

870

1.8

870

1.8

870

1.8

852

1.8

817

1.8

915

1.8910

1.8

914

1.8

912

1.8

912

C∗ yb

1.7

974

1.8708

1.9

291

1.9

768

1.8

860

1.8

858

1.8865

1.8

856

1.8

856

1.8

856

1.8

856

1.8

841

1.8

809

1.8

908

1.8911

1.8

907

1.8

906

1.8

906

h∗ a

0.7

575

0.6717

0.6

126

0.5

693

0.6

598

0.6

806

0.6811

0.7

011

0.7

016

0.7

026

0.7

031

0.8

268

0.9

845

0.6

685

0.6838

0.6

876

0.7

090

0.7

095

h∗ b

0.7

575

0.6717

0.6

126

0.5

693

0.6

594

0.6

802

0.6811

0.7

006

0.7

011

0.7

022

0.7

027

0.8

262

0.9

842

0.6

682

0.6838

0.6

874

0.7

088

0.7

093

ua

0.7

286

0.5531

0.3

729

0.1

812

0.5

127

0.5

330

0.5338

0.5

529

0.5

534

0.5

544

0.5

549

0.6

751

0.8

280

0.5

130

0.5282

0.5

319

0.5

530

0.5

535

ub

0.6

994

0.5239

0.3

436

0.1

519

0.4

836

0.5

039

0.5044

0.5

238

0.5

243

0.5

253

0.5

257

0.6

457

0.7

985

0.4

838

0.4989

0.5

027

0.5

238

0.5

243

u∗ a

0.3

490

0.5498

0.7

094

0.8

434

0.5

892

0.5

703

0.5695

0.5

517

0.5

512

0.5

503

0.5

498

0.4

373

0.2

930

0.5

892

0.5752

0.5

718

0.5

524

0.5

520

u∗ b

0.3

295

0.5303

0.6

899

0.8

239

0.5

695

0.5

506

0.5500

0.5

320

0.5

316

0.5

307

0.5

302

0.4

178

0.2

733

0.5

696

0.5556

0.5

523

0.5

329

0.5

325

E[u

]0.7

140

0.5385

0.3

583

0.1

665

0.4

982

0.5

184

0.5191

0.5

383

0.5388

0.5398

0.5

403

0.6

604

0.8

132

0.4

984

0.5136

0.5

173

0.5384

0.5389

E[u

∗]

0.3

392

0.5401

0.6

996

0.8

337

0.5

793

0.5

604

0.5598

0.5

418

0.5414

0.5405

0.5

400

0.4

275

0.2

831

0.5

794

0.5654

0.5

621

0.5427

0.5422

E[u

]+E

[u∗]

1.0

532

1.0786

1.0

579

1.0

002

1.0

775

1.0

789

1.0789

1.0

802

1.0

802

1.0

803

1.0

803

1.0

880

1.0

964

1.0

778

1.0790

1.0

793

1.0

811

1.0

811

P∗ yaC

ya

0.2

296

0.1907

0.1

589

0.1

324

0.1

815

0.1

799

0.1799

0.1

783

0.1

783

0.1

782

0.1

782

0.1

695

0.1

604

0.1

781

0.1769

0.1

766

0.1

749

0.1

748

P∗ ybC

yb

0.2

296

0.1907

0.1

589

0.1

324

0.1

817

0.1

801

0.1799

0.1

785

0.1

785

0.1

784

0.1

783

0.1

696

0.1

605

0.1

782

0.1769

0.1

766

0.1

750

0.1

749

q∗ aQ

∗ Ia

0.0

000

0.0000

0.0

000

0.0

000

0.0

023

0.0

119

0.0122

0.0

212

0.0

214

0.0

219

0.0

221

0.0

744

0.1

326

0.0

086

0.0157

0.0

174

0.0

272

0.0

274

q∗ bQ

∗ Ib

0.0

000

0.0000

0.0

000

0.0

000

0.0

023

0.0

119

0.0122

0.0

212

0.0

214

0.0

219

0.0

221

0.0

743

0.1

326

0.0

086

0.0157

0.0

174

0.0

271

0.0

273

P∗ yaCya+q∗ aQ

∗ Ia

0.2

296

0.1907

0.1

589

0.1

324

0.1

838

0.1

918

0.1922

0.1

996

0.1

997

0.2

001

0.2

003

0.2

439

0.2

930

0.1

867

0.1926

0.1

940

0.2

020

0.2

022

P∗ ybCyb+q∗ bQ

∗ Ib

0.2

296

0.1907

0.1

589

0.1

324

0.1

840

0.1

920

0.1922

0.1

997

0.1

999

0.2

002

0.2

004

0.2

439

0.2

930

0.1

867

0.1926

0.1

940

0.2

021

0.2

023

PxaC

∗ xa/e

0.1

730

0.1907

0.2

052

0.2

172

0.1

943

0.1

923

0.1922

0.1

905

0.1

904

0.1

903

0.1

903

0.1

797

0.1

677

0.1

941

0.1926

0.1

923

0.1

904

0.1

903

PxbC

∗ xb/e

0.1

730

0.1907

0.2

052

0.2

172

0.1

941

0.1

922

0.1922

0.1

903

0.1

903

0.1

902

0.1

901

0.1

796

0.1

676

0.1

940

0.1926

0.1

922

0.1

903

0.1

902

TB

∗ a0.0

566

0.0000−

0.0

463−

0.0

849−

0.0

105−

0.0

005

0.0000

0.0

091

0.0

093

0.0

098

0.0

100

0.0

642

0.1

253−

0.0

074

0.0000

0.0

017

0.0

117

0.0

119

TB

∗ b0.0

566

0.0000−

0.0

463−

0.0

849−

0.0

101−

0.0

002

0.0000

0.0

094

0.0

096

0.0

101

0.0

103

0.0

643

0.1

254−

0.0

072

0.0000

0.0

018

0.0

118

0.0

120

aT

hes

enu

mer

ical

exer

cise

sass

um

etw

op

oss

ible

state

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the

worl

d;G

(s)

=0.5

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=a,b

.T

he

para

met

ervalu

esareβ

=β∗

=0.9

6,a

=a∗

=0.6

,α

=α∗

=2/3,v

=1,v∗

=0.9

,

θ=

0,θ

=1,ξ

=1,A

xa

=4.2

,A

xb

=A

∗ ya

=A

∗ yb

=4,Ba

=Bb

=0,

an

dφ∗ Ia

=φ∗ Ib

=1.

bT

he

equ

ilib

riu

mu

nd

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isfi

xed

exch

an

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ratee

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oth

eeq

uilib

riu

mu

nd

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=1

an

dκ

isp

roh

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igh

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ally,

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efit

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ual

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om

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ake

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ual

to0.1

.

56

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