Macroeconomic Interdependence and the Transmission Mechanism∗ Simon Lloyd† University of Cambridge

August 2014

This document provides a formal derivation of the two-country, two-good model presented in section 3 of Corsetti, Dedola, and Leduc (2008). The two economies, home H and foreign F , each have an endowment of a good specific to that country, labelled YH and YF respectively. In each country, a continuum of households derive utility from consuming both goods, paving the way for international trade between the two countries. In the baseline model, both goods are tradeable. Within the model, prices are entirely flexible and production is entirely exogenous, given by the endowment of each country.

1

Structure of the Economy

In each country there exists a continuum of infinitely-lived representative households, who seek to maximise the expected discounted value of their lifetime utility. For the home consumers, this is denoted by: Ut = Et

∞ X

β s U (Ct+s ; ζC,t+s ) = Et

s=0

∞ X s=0

β s ζC,t+s

1−σ Ct+s −1 1−σ

(1.1)

where β < 1 is the discount factor, σ > 1 is the inverse elasticity of intertemporal substitution and ζC,t+s denotes a stochastic preference shock at time t + s.1 The corresponding lifetime utility function for foreign households is written in the following manner, where ∗ denotes a foreign country variable: Ut∗ = Et

∞ X

∞ ∗ 1−σ X
−1 Ct+s ∗ ∗ ∗ β ∗ s U Ct+s ; ζC,t+s = Et β ∗ s ζC,t+s 1−σ s=0 s=0

where β ∗ < 1 is the foreign discount factor. For the remainder of this document, the home and foreign discount factors are assumed to be equal, β = β ∗ , as are the home and foreign inverse intertemporal elasticities, σ = σ ∗ . However, the stochastic preference shock realisation can differ across countries. In equation (1.1) Ct represents the total consumption basket of the representative home consumer at time t. Ct∗ denotes the corresponding quantity for the representative foreign consumer. The basket comprises of both the home and the foreign good (CH and CF respectively), represented mathematically ∗ Based on Section 3 of International Risk Sharing and the Transmission of Productivity Shocks by Corsetti, Dedola and Leduc, published in the Review of Economic Studies, 2008. † University of Cambridge. Email: [email protected]. Address: Faculty of Economics, Sidgwick Avenue, Cambridge, CB3 9DD. Thanks go to Riccardo Trezzi, whose notes form the basis of some of the passages in this document. 1 The preference shock represents a change in how households value their current and/or future consumption stream.

1

as a CES aggregator: φ  φ−1  φ−1 φ−1 1/φ 1/φ φ + aF CF,tφ Ct = aH CH,t

(1.2)

where φ > 0 is the elasticity of substitution between home and foreign goods, more commonly referred to as the trade elasticity. CH,t represents domestic consumption of the home H good, while CF,t is the label for domestic consumption of the foreign F good. If aF = 1 − aH , then aH denotes the share of the domestically produced good in the consumption of the home consumer. If aH > 1/2, it is said that domestic consumers exhibit home bias.2 Analogously, the consumption aggregator for the foreign consumer is: φ h i φ−1 φ−1 φ−1 ∗ ∗ φ φ Ct∗ = a∗H 1/φ CH,t + a∗F 1/φ CF,t

∗ ∗ where CH,t denotes foreign consumption of the home H good and CF,t foreign consumption of the foreign

F good. Foreign consumers exhibit home bias if a∗F = 1 − a∗H > 1/2. If aH = a∗F , then home and foreign consumption baskets are said to be symmetric. If aH = a∗F > 1/2, home and foreign baskets exhibit symmetric degrees of home bias. If aH = a∗H , home and foreign consumption baskets are said to be identical.3

1.1

Price of Consumption and Demand for Goods

Let Pt and Pt∗ denote the price of domestic consumption in the home and foreign country respectively. These indexes are consumption-based, defined such that they represent the price of a single unit of the aggregate consumption basket: Pt =

min

CH,t ,CF,t

Zt

subject to

Ct = 1

(1.3)

where Zt = PH,t CH,t + PF,t CF,t denotes the total expenditure on the basket. The foreign price index, Pt∗ , is similarly defined. In the subsequent passage, I derive the price index for the home country Pt . The steps for deriving the foreign price index Pt∗ follow symmetrically. To solve for the home price index, first form the Lagrangian corresponding to equation (1.7): φ #   φ−1 φ−1 φ−1 1/φ 1/φ φ φ = PH,t CH,t + PF,t CF,t + λ 1 − aH CH,t + aF CF,t

"

L=

min

CH,t ,CF,t

where λ is the Lagrange multiplier. Taking the first-order condition with respect to CH,t yields: 1   φ−1 φ−1 φ φ − 1 1/φ −1/φ 1/φ φ−1 1/φ φ φ PH,t − λ aH CH,t aH CH,t + aF CF,t φ−1 φ | {z } 1/φ

Ct 2 Home

bias in consumption has been ratified empirically. baskets are identical and aH > 1/2, then the foreign consumption basket cannot exhibit home bias, a greater relative weight on the domestically produced good. 3 If

2

which simplifies in the following manner: 1/φ

−1/φ

1/φ

PH,t

=

λaH CH,t Ct

−φ PH,t

−1 + λ−φ a−1 H CH,t Ct

Solving for CH,t , this yields: −φ CH,t = λφ aH Ct PH,t

(1.4)

By taking the first-order condition of the Lagrangian with respect to CF,t , a similar result is attained: −φ CF,t = λφ aF Ct PF,t

(1.5)

To solve for λ, the Lagrange multiplier, substitute (1.4) and (1.5) into the equation for Ct , (1.2): Ct

=

φ  φ−1  φ−1 φ−1 1/φ 1/φ φ φ aH CH,t + aF CF,t

 =

aH λ 

= =

φ−1

φ−1

λ

φ−1 φ

Ct

φ−1 φ

Ct



1−φ PH,t

φ−1

+ aF λ

1−φ aH PH,t

+

φ−1 φ

Ct

1−φ aF PF,t

1−φ PF,t

φ  φ−1

φ  φ−1

  φ 1−φ 1−φ φ−1 λφ Ct aH PH,t + aF PF,t

Rearranging this final expression for λφ gives:   φ 1−φ 1−φ 1−φ λφ = aH PH,t + aF PF,t

(1.6)

To solve for the price index Pt , use its definition along with equations (1.4) and (1.5) (line 2), and equation (1.6) (line 4): Pt

=

Zt |Ct =1 = (PH,t CH,t + PF,t CF,t )|Ct =1

=

−φ −φ PH,t λφ aH PH,t + PF,t λφ aF PF,t   1−φ 1−φ λφ aH PH,t + aF PF,t

= =



1−φ 1−φ aH PH,t + aF PF,t

φ  1−φ 

1−φ 1−φ + aF PF,t aH PH,t



which yields the home price index:   1 1−φ 1−φ 1−φ Pt = aH PH,t + aF PF,t

(1.7)

The corresponding foreign price index is:  1  ∗ 1−φ ∗ 1−φ 1−φ Pt∗ = a∗H PH,t + a∗F PF,t

(1.8)

Together, equation (1.6) and (1.7) imply that λφ = Ptφ . Using this fact in equations (1.4) and (1.5)

3

provides two expressions for home consumer demand for the home and foreign good respectively: −φ PH,t Ct Pt  −φ PF,t Ct = aF Pt 

CH,t CF,t

= aH

(1.9) (1.10)

Similar demand expressions can be derived for the foreign consumer. 1.1.1

Complementarity and Substitutability

The two goods, CH and CF , are substitutes if the marginal utility of one good is decreasing in the quantity of the other. Mathematically this is defined as: ∂U 2 ∂U 2 = <0 ∂CH ∂CF ∂CF ∂CH The two goods are complements if the opposite is true; if the marginal utility from consuming one good is increasing in the consumption of the other: ∂U 2 ∂U 2 = >0 ∂CH ∂CF ∂CF ∂CH Using equations (1.1) and (1.2), I show that the two goods are • Substitutes when σφ > 1, • Complements when σφ > 1. To show this, first write expected discounted lifetime utility for the home consumer as:

Ut = Et

∞ X

φ  φ−1  (1−σ) φ−1 φ−1 1/φ 1/φ φ + aF CF,tφ −1 aH CH,t

β s ζC,t+s

1−σ

s=0

The marginal utility of consuming an extra unit of the home good at time t is:  1−σφ  φ−1 φ−1 φ−1 ∂Ut 1/φ −1/φ 1/φ 1/φ φ φ t = β ζC,t aH CH,t aH CH,t + aF CF,t ∂CH,t The differential of this with respect to CF,t is:   2−φ−σφ φ−1 φ−1 φ−1 ∂Ut2 (1 − σφ) t 1/φ −1/φ 1/φ −1/φ 1/φ 1/φ φ φ = β ζC,t aH CH,t aF CF,t aH CH,t + aF CF,t ∂CH,t ∂CF,t φ | {z } >0

By inspection, the sign of this expression depends entirely on the magnitude of σφ.4 If σφ > 1, then the two goods are substitutes. If σφ < 1, the two goods are complements. 4 Since

φ > 0 and σ > 1, the product of the two is always positive.

4

1.2

Exchange Rates and Relative Prices

The nominal exchange rate Et is defined as the domestic currency price of foreign currency. Therefore, an increase in Et represents a deprecations of home currency value. The terms of trade Tt , for the home country, represents the relative price of imports to the price of exports, where both prices are written in terms of the home currency: Tt =

PF,t ∗ Et PH,t

(1.11)

∗ where PF,t is the home currency price of the foreign good (import) and PH,t is the foreign currency price

of the home good (export). To attain the price of an exported unit of the home good in home currency ∗ terms, PH,t must be pre-multiplied by the nominal exchange rate: Et PH,t . An increase in the terms of trade Tt corresponds to a rise in the price of imports relative to exports for the home consumer in home currency terms; foreign imports are relatively more expensive. As a result, an increase in Tt is defined as a worsening of the terms of trade for the home country. A terms of trade improvement is defined in the opposite manner. The real exchange rate Qt is the relative price of consumption. More specifically, it is written as the price of the foreign consumption basket in domestic currency terms, Et Pt∗ , relative to that of the home basket, Pt : Qt =

Et Pt∗ Pt

(1.12)

An increase in Qt corresponds to an increase in the price of the foreign basket relative to the home basket in home currency terms, and is thus referred to as a real exchange rate depreciation for home consumers. 1.2.1

Purchasing Power Parity (PPP)

Purchasing Power Parity (PPP) is defined as the equalisation of the price of consumption across countries: Pt = Et Pt∗ . In this model PPP will hold if two conditions hold: ∗ 1. The law of one price (LOOP) holds: PH,t = Et PH,t and PF,t = Et PF,t (i.e. the price of the home

(foreign) good in the home country is exactly equal to its price in the foreign country in home currency terms). 2. Consumption baskets are identical: aH = a∗H = 1 − a∗F . Therefore, deviations from PPP can reflect either deviations from the law of one price or differences in preferences across countries.

1.3

World Goods Demand

The aggregate demand for each good, H and F , is the sum of the demand for that good from the home ∗ and the foreign economies. For the home good, this is given by: CH + CH . The corresponding statement

for the foreign good is CF + CF∗ . Equilibrium in the world goods market requires that the supply of each good, YH and YF , implied by the endowment, must equal aggregate demand. The two equilbrium goods market conditions are therefore: YH,t YF,t

∗ = CH,t + CH,t

= CF,t + 5

∗ CF,t

(1.13) (1.14)

for the home and foreign goods respectively. Using equations (1.9) and (1.10) (and the foreign counterparts), these expressions can be written as: −φ  ∗ −φ PH,t PH,t Ct + a∗H Ct∗ Pt Pt∗  −φ  ∗ −φ PF,t PF,t ∗ = aF Ct∗ Ct + aF Pt Pt∗

YH,t

=

∗ CH,t + CH,t = aH

YF,t

=

∗ CF,t + CF,t



∗ If the law of one price holds (PH = EPH and PF = EPF∗ ), then:

−φ  −φ PH,t PH,t ∗ Ct∗ = aH Ct + aH Pt Et Pt∗   −φ −φ PF,t PF,t Ct + a∗F Ct∗ = aF Pt Et Pt∗ 

YH,t YF,t

Using the definition of the real exchange rate, Qt = Et Pt∗ /Pt , this can be rewritten as: −φ  −φ PH,t PH,t ∗ = aH Ct + aH Ct∗ Pt Qt Pt  −φ −φ  PF,t PF,t = aF Ct + a∗F Ct∗ Pt Qt Pt 

YH,t YF,t

which can be further rearranged to express the world demand for home H and foreign F goods as:  YH,t

= 

YF,t

1.4

=

PH,t Pt

−φ h

PF,t Pt

−φ h

aH Ct + a∗H Qφt Ct∗

aF Ct + a∗F Qφt Ct∗

i

i

(1.15) (1.16)

Assumptions, Normalisations and Log-Linearisation

For simplicity, assume that the the law of one price (LOOP) holds such that: ∗ PH = EPH

and

PF = EPF∗

Deviations from PPP can still occur, as consumption baskets are not assumed to be identical. Instead, consumption baskets are symmetric with home bias, such that: aH = a∗F = 1 − a∗H > 1/2 Without loss of generality, the nominal exchange rate is normalised such that: Et = 1

∀t

To solve the model, it is log-linearised around a symmetric equilibrium in which Q = T = 1, where the upper-bar, −, denotes the steady state value of a variable. Percentage deviations from the steady state are denoted by a hat, ˆ. A tilde, ˜, represents deviations of endogenous variables from the first-best equilibrium.

6

2

Bridging the Gap: Real Exchange Rate and Terms of Trade

To attain a relationship between the real exchange rate Q and the terms of trade T under the assumptions and normalisations laid out in section 1.4, first note that with E = 1: 1−φ

Q1−φ = t

(Et Pt∗ )

Pt1−φ

=

Pt∗ 1−φ Pt1−φ

Using the price indexes defined in equations (1.7) and (1.8), the law of one price and E = 1, the above expression can be reformed as: Q1−φ t

=

=

1−φ ∗ 1−φ (1 − a∗F )PH,t + a∗F PF,t ∗ 1−φ + (1 − a )P 1−φ aH PH,t H F,t 1−φ  P (1 − a∗F ) + a∗F P F,t ∗ H,t 1−φ  P aH + (1 − aH ) P F,t ∗ H,t

1−φ where the second line is attained by dividing the numerator and denominator by PH,t . Using the fact

that the terms of trade is defined as Tt =

PF,t ∗ Et PH,t

Q1−φ = t

=

PF,t ∗ PH,t

(with E = 1), then:

(1 − a∗F ) + a∗F Tt1−φ aH + (1 − aH )Tt1−φ

(2.1)

The following steps outline the log-linearisation of equation (2.1) around the symmetric steady state with Q = T = 1. I use the substitution method in the steps below: 1. Rewrite equation (2.1) by multiplying both sides by aH + (1 − AH )Tt1−φ : aH Q1−φ + (1 − aH )Tt1−φ Q1−φ = (1 − a∗F ) + a∗F Tt1−φ t t with the steady state expression: aH Q1−φ + (1 − aH )T 1−φ Q1−φ = (1 − a∗F ) + a∗F T 1−φ bt ), where X denotes the steady state and X ˜ t denotes the 2. Use the substitution Xt = X exp(X percentage deviation of Xt from this steady state. bt ] + (1 − aH )T 1−φ Q1−φ exp[(1 − φ)(Tbt + Q bt )] aH Q1−φ exp[(1 − φ)Q = (1 − a∗F ) + a∗F T 1−φ exp[(1 − φ)Tbt ] Noting that T = Q = 1, then: bt ] + (1 − aH ) exp[(1 − φ)(Tbt + Q bt )] = (1 − a∗F ) + a∗F exp[(1 − φ)Tbt ] aH exp[(1 − φ)Q

7

3. Using that facts that: bt ] ≈ 1 + (1 − φ)Q bt exp[(1 − φ)Q exp[(1 − φ)Tbt ] ≈ 1 + (1 − φ)Tbt bt + Tbt )] exp[(1 − φ)(Q

bt + (1 − φ)Tbt ≈ 1 + (1 − φ)Q

then: bt ] + (1 − aH )[1 + (1 − φ)Q bt + (1 − φ)Tbt ] = (1 − a∗F ) + a∗F [1 + (1 − φ)Tbt ] aH [1 + (1 − φ)Q 4. By applying the steady state expression, we achieve the following simplification: bt ] + (1 − aH )[(1 − φ)Q bt + (1 − φ)Tbt ] = a∗F [(1 − φ)Tbt ] aH [(1 − φ)Q 5. Collecting like terms and using the fact that, in a symmetric model, aH = a∗F , then we attain: bt = (2aH − 1)Tbt Q

(2.2)

By inspection of equation (2.2), it is clear that the co-movement of the real exchange rate Q and the terms of trade T depends on the degree of home bias, aH , in the world economy. The two variables positively co-move when there is home bias, aH > 1/2. There is zero co-movement when aH = 1/2; this is the PPP case. Finally, the co-movement is negative when aH < 1/2 (foreign bias).

3

Complete Markets

The complete market economy is characterised by a full set of state-contingent Arrow-Debreu securities. For an agent at time t in state st , the state of the world at time t + 1 is unknown. There are a continuum of possible time t + 1, st+1 ∈ S, where S denotes the state-space. In a complete market economy at time t, there exists at least one bond for each state st+1 in time t + 1. For instance, an agent can buy the bond BH,t+1 (s1,t+1 ) for the price qH,t (s1,t+1 ) at time t, which pays out a single unit upon realisation of the state s1,t+1 at time t + 1. For markets to be complete, there must exist at least on bond of this character for each of the states in time t + 1. As a result, the representative household inter-temporal budget constraint for the complete market economy can be written as: Z qH,t (st+1 )BH,t+1 (st+1 )dst+1 ≤ BH,t + PH,t YH,t − PH,t CH,t − PF,t CF,t

(3.1)

where PH,t YH,t denotes the revenue accrued by the home consumer from the home endowment and PH,t CH,t + PF,t CF,t is the total consumption expenditure of the home consumer. Since there are no distortions in the complete market economy, by the first fundamental welfare theorem, the flexible price allocation will be first best efficient. Maximising the representative household utility, equation (1.1), subject to equation (3.1) yields the following Euler equation:5 UC (s) 5 This

1 q(s0 |s) = β Pr(s0 |s)UC (s0 |s) P (s) P (s0 |s)

is written in recursive form where a dash ’ denotes time t + 1 and the absence of a dash represents time t.

8

(3.2)

The Euler equation states that, in utility terms, the time t marginal cost of buying an extra unit of a state contingent bond that pays out in state s0 in the next period must be equalised with the discounted expected marginal benefit that it will provide. A similar condition will hold for the representative foreign consumer: UC∗ (s)

q(s0 |s) 1 = β Pr(s0 |s)UC∗ (s0 |s) ∗ 0 ∗ P (s) P (s |s)

(3.3)

assuming β = β ∗ . Combining equations (3.2) and (3.3) yields the key condition for efficient risk sharing: UC (s) P ∗ (s) UC (s0 |s) P ∗ (s0 |s) · = ∗ 0 · ∗ UC (s) P (s) UC (s |s) P (s0 |s)

(3.4)

If countries are initially perfectly symmetric and have identical time t endowments, then: UC∗ (s) P ∗ (s) = UC (s) P (s)

(3.5)

which, when substituted into equation (3.4), yields: UC∗ (s0 |s)

1 P ∗ (s0 |s)

= UC (s0 |s)

1 P (s0 |s)

(3.6)

With complete markets and initial perfect symmetry, the marginal utility of a unit of domestic currency is equalised across countries.

3.1

Complete Markets Transmission

To derive an expression linking the real exchange rate Q to the ratio of marginal utilities, note that: Q(s0 |s)

P ∗ (s0 |s) P (s0 |s) UC∗ (s0 |s) = UC (s0 |s)  σ ∗ 0 C(s0 |s) ζC (s |s) = C ∗ (s0 |s) ζC (s0 |s)

as E = 1

=

from (3.6) from (1.1)

The follow steps outline a ‘shortcut’ log-linearisastion method for this final expression around the symmetric equilibrium yields:   bt = ln Xt , an expression which can 1. The log-linear form of the variable Xt can be defined as X X b be inverted to yield: Xt = X exp[Xt ]. In some situations, log-linearisation can be carried out quickly by merely substituting this latter expression for each variable. In this case, the expression of interest becomes:

h i σ ∗ h i ∗ bt C exp C ζ C exp ζbC,t h i h i = ∗ bt∗ C exp C ζ C exp ζbC,t 

h

bt Q exp Q

i

∗

∗

2. In a symmetric steady state Q = 1, C = C , and ζ C = ζ C , which results in a simplification of the above expression to:

h i σ h i ∗ bt exp C exp ζbC,t h i h i = bt∗ exp C exp ζbC,t 

h

bt exp Q

i

9

3. Taking natural logarithms of this yields:     ˜ f b = σ C˜ f b − C˜ ∗f b + ζbC∗ − ζbC = (2aH − 1)T˜ f b Q

(3.7)

where the second equality follows from equation (2.2). Equation (3.7) implies that, for a given taste shock, home consumption can only increase with a depreciation in the real exchange rate Q and a worsening of the terms of trade T . To write the ratio

YH,t YF,t

use equations (1.15) and (1.16): −φ h i PH,t aH Ct + a∗H Qφt Ct∗ Pt =  −φ h i PF,t aF Ct + a∗F Qφt Ct∗ Pt h i  φ a + a∗ Qφ Ct∗ H t H Ct PF,t h i = φ Ct∗ PH,t ∗ aF + aF Qt Ct 

YH,t YF,t

(3.8)

Using the relationship between the real exchange rate, relative consumption and the preference shocks, solved for

Ct∗ Ct :

Ct∗ = Ct

ζC,t Qt ∗ ζC,t

!−σ−1 (3.9)

Combining equations (3.8) and (3.9) yields, along with the definition of the terms of trade Tt =

PF,t PH,t

given that E = 1 and the law of one price holds:

YH,t YF,t

  −σ−1  −1 ζC,t aH + a∗H Qφ−σ ∗ t ζC,t = Tφ  −σ−1  φ−σ −1 ζC,t ∗ aF + aF Qt ζ∗

(3.10)

C,t

The log-linear form of equation (3.10) is (with symmetry aH = a∗F ): fb fb Y˜H,t − Y˜F,t = φT˜tf b +

3.1.1



   1 ∗ ˜ ft b + 1 (2aH − 1) ζbC,t − ζbC,t − φ (2aH − 1)Q σ σ

(3.11)

The Relation Between the Terms of Trade and Relative Endowment

Substituting the terms of trade T and real exchange rate Q relationship, equation (2.2): ˜ ft b = (2aH − 1)T˜tf b Q into equation (3.11) and solving for T˜tf b results in the expression: T˜tf b =

    fb fb ∗ σ Y˜H,t − Y˜F,t − (2aH − 1) ζbC,t − ζbC,t {[1 − (2aH − 1)2 ] σφ + (2aH − 1)2 }

(3.12)

Since aH ∈ [0, 1], the coefficient on relative output is always positive. With constant preferences, a positive shock to the home endowment, ↑ YH , will unambiguously worsen the home terms of trade, ↑ T . This will benefit foreign consumers, who will face a lower price of imports relative to the price of their 10

exports. 3.1.2

Relative Consumption

  For given preferences, ζbC = ζbC∗ = 0, then substituting equation (3.7), σ C˜ f b − C˜ ∗f b = (2aH − 1)T˜ f b   into equation (3.12) and solving for C˜ f b − C˜ ∗f b yields: 

 C˜tf b − C˜t∗f b =

  2aH − 1 fb ∗f b Y˜H,t − Y˜F,t 2 2 [1 − (2aH − 1) ] σφ + (2aH − 1)

(3.13)

With home bias, aH > 1/2, the coefficient on relative output is always positive. In response to a home  fb ˜ ˜ supply shock, ↑ Y , then consumption will grow more at home than it does abroad, ↑ C − C˜ ∗ , even H

if the terms of trade worsens, ↑ T . Therefore, there is no immiserising growth with home bias under complete markets. The difference between home and foreign consumption falls as goods become more substitutable. As σφ → ∞ (goods more substitutable), then C˜ f b → C˜ ∗f b . 3.1.3

Elasticities and Relative Price Volatility

The major points to note: • The lower the elasticity σφ, the greater the real exchange rate and terms of trade response to an endowment shock (to Y˜ f b ). However, with low elasticities, the trade response is smaller, as goods H

are more complementary. • To fit the empirical real exchange rate volatility within the complete markets, two-good endowment model, it is necessary to calibrate the trade elasticity φ to be small. However, this implies unrealistically small trade variance. • With σφ small enough, the foreign consumption may actually fall in response to a positive home endowment shock, ↑ Y˜ f b , despite full risk sharing. H

4

Incomplete Markets: Financial Autarky

If financial markets are incomplete shocks can drive a wedge between home and foreign wealth, leading to a richer array of results than under the complete market case. General equilibrium wealth effects are integral to the analysis of incomplete markets. Under financial autarky, there are no financial flows and external trade must balance in each and every period. The value of imports into the home country must equal the value of exports in the same period. ∗ Tt CF,t − CH,t =0

(4.1)

The total value of the domestic endowment must equal the total value of home consumption: Pt Ct − PH,t YH,t = 0

(4.2)

Since resources cannot be transferred intertemporally, the household maximisation problem is static and of the form: Lt = max ζC,t Ct ,χt

Ct1−σ + χt [PH,t YH,t − Pt Ct ] 1−σ 11

(4.3)

where χt is the Lagrange multiplier, equal to the marginal utility of an extra unit of income. The first-order condition with respect to Ct is: ζC,t Ct−σ = χt Pt The corresponding foreign first-order condition is: ∗ ζC,t Ct∗ −σ = χ∗t Pt∗

Combining these expressions gives: ∗ ζC,t ζC,t



Ct Ct∗

σ

∗ ζC,t Pt = Pt∗ ζC,t



Ct Ct∗

σ

1 χ∗ = t Qt χt

(4.4)

Under complete markets, the corresponding version of this equation (outlined at the start of section 3.1) is the perfect risk sharing condition: ∗ ζC,t ζC,t



Ct Ct∗

σ

∗ ζC,t Pt = Pt∗ ζC,t



Ct Ct∗

σ

1 =1 Qt

(4.5)

Therefore with incomplete markets, the marginal utility of consumption cannot be expected to be equalised across states of nature. In general a country-specific shock will drive domestic and foreign wealth apart, creating global imbalances. It will leave a gap in the marginal utility of income, which accounts for deviations in perfect risk sharing. These imbalances can be formalised in a demand gap term Dt , defined as: ∗ ζC,t χ∗ Dt = t = χt ζC,t



Ct Ct∗

σ

1 Qt

(4.6)

With this definition,6 if Dt > 1 then the marginal utility of an extra unit of income in the foreign country will exceed that in the home country. Therefore, the shadow value of income will be higher in the foreign country; the foreign country will be relatively poor and the home country will be relatively rich. Equation (4.6) can be log-linearised using the same ‘shortcut’ method outlined in section 3.1, yielding:     ∗ bt = ζbC,t bt − C bt∗ − Q bt D − ζbC,t + σ C

4.1

(4.7)

Consumption Demand in Financial Autarky

To understand the wealth effect of shocks to relative supply (and thus relative prices), it is useful to reconsider the wealth and substitution effects of price changes. The domestic demand for home goods under financial autarky will be pinned down by the trade balance condition, Pt Ct = PH,t YH,t , in equation (4.2) combined with equation (1.9):  CH,t = aH

PH,t Pt

−φ Ct

= =



PH,t Pt

−φ



PH,t Pt

1−φ

aH aH

PH,t YH,t Pt YH,t

6 Note that in this document, the demand gap is defined as the ratio of the foreign marginal utility of income to that of the home country. As a result, a positive demand gap here corresponds to a relatively richer home country. In some situations the demand gap is defined in the opposite manner, so be sure to be clear on the definition used.

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Using the price index definition, equation (1.7): 1−φ

 CH,t

 = aH  

PH,t 1−φ aH PH,t

 =

aH  

+

1−φ aF PF,t

= as Tt =

PF,t PH,t ,

YH,t



1−φ PH,t 1−φ 1−φ aH PH,t + aF PF,t

 =

  1  1−φ

  YH,t 

  1 aH   1−φ     YH,t PF,t aH + aF PH,t aH aH + aF Tt1−φ

YH,t

(4.8)

given that the law of one price holds and Et .

In financial autarky equilibrium, changes in relative prices PH /P modify both the relative prices faced by home households as consumers (substitution effect) and the value of the home endowment relative to the foreign endowment (income effect). 4.1.1

Substitution and Income Effects from International Price Movements

Consider a fall in the relative price of the home good, ↓ 

CH,t

PH,t = aH Pt | {z

PH P :

−φ

PH,t YH,t Pt } | {z } IE

SE

Substitution Effect (SE) The substitution effect of a fall in the relative price of the home good will be positive, raising domestic demand for the home good by power φ. The strength of the substitution effect is increasing in the substitutability of the home and foreign goods, φ. Income Effect (IE) For a constant home endowment YH,t , a fall in the relative price of the home tradeable will lead to a 1:1 fall in the consumption of the domestic good through the income effect. The income effect of a price fall is negative, diminishing the value of the home consumers’ endowment. To appreciate these effects, consider the partial differential of equation (4.8) with respect to the terms of trade T : ∂CH,t =φ ∂Tt |

aH (1 − aH )T −φ [aH + (1 − aH )T 1−φ ] {z

! −

2 YH,t

}

SE

aH (1 − aH )T −φ

! 2 YH,t

[aH + (1 − aH )T 1−φ ] | {z IE

}

By inspection, the substitution effect will exceed the income effect when φ > 1 (i.e. when home and foreign goods are highly substitutable). When φ > 1, a deterioration in the terms of trade (↑ Tt ) will raise the domestic demand for the home good. When φ < 1, the income effect exceeds the substitution effect. Here a deterioration in the terms of trade (↑ Tt ) will reduce domestic demand for the home good, due to a fall in the value of endowments.

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When φ = 1, the income and substitution effects will exactly cancel, and terms of trade movements will leave domestic demand unaffected. For any parameterisation of φ, the income and substitution effects for the foreign consumers of a home terms of trade depreciation will work in the same direction. Therefore a reduction in the relative price of the home good PH /P will increase foreign demand for the home good unambiguously. To show this, foreign demand for the home good can be written as: ∗ CH =

(1 − aH )Ttφ Y∗ aH + (1 − aH )Tt F,t

where:

4.1.2

∗ ∂CH,t >0 ∂Tt

World Demand for Home Goods

∗ The world demand for home goods is given by CH + CH . As long as the negative income effect of a fall

PH /P is not too large for the home consumers, then the world demand for the home good will increase in response to the price change. In other words, the world demand for the home good will a decreasing function of its relative price. However, if the income effect is sufficiently large (φ is a lot below 1) and home bias is sufficiently high (aH is enough about 1/2), then the negative income effect of a fall in the price of the home good (↓ PH ) for home consumers will reduce home demand for the home good by so much that it usurps to rise in foreign demand for the home good. As a result, world demand for the home good will fall in response to a fall in its relative price if income effects are sufficiently strong. This gives rise to the possibility that, in equilibrium, the terms of trade may improve (↓ T ) following a fall in the price of home goods (↓ PH ). This is necessary with strong home bias and income effects, as any surplus must effectively be purchased by the home consumers alone and for this, they require favourable price movements.

4.2

Terms of Trade Response to Supply Shocks

To consider the terms of trade response to supply shocks, derive an expression for the terms of trade T in terms of relative endowments YH /YF . To do this, use the trade balance condition (4.1) along with the consumer demand functions, (1.9) and (1.10):  ∗ −φ P ∗ Ct∗ a∗H PH,t ∗ CH,t t = Tt =  −φ CF,t P aF PF,t Ct t Using the knowledge that home and foreign consumption baskets are symmetric, aH = a∗F and aF = a∗H , as well as equation (4.2) (and the foreign country counterpart) yields:

Tt

=

=

−φ



∗ PH,t Pt∗



PF,t Pt



PF,t PH,t

−φ

∗ PF,t ∗ Pt∗ YF,t

PH,t Pt YH,t

1+φ 

14

Pt∗ Pt

φ−1

∗ YF,t YH,t

where the second line is attained by rearranging the first and applying the law of one price with the ∗ ∗ normalisation E = 1, PH,t = PH,t and PF,t = PF,t .

The log-linear form of this expression is:   ∗ bt + YbF,t −φTbt = (φ − 1)Q − YbH,t Utilising equation (2.2) leads to: Tbt

=

bt Q

=

  1 ∗ YbH,t − YbF,t 1 − 2aH (1 − φ)   2aH − 1 ∗ YbH,t − YbF,t 1 − 2aH (1 − φ)

(4.9) (4.10)

Equations (4.9) and (4.10) imply that, in the financial autarky equilibrium, an increase in home output can either appreciate or depreciate the domestic terms of trade and real exchange rate. Under complete markets, an appreciation was not possible following a positive home endowment shock. Under financial autarky, an appreciation following an increase in the home endowment represents a fall in the world price of the relatively more abundant good. The possibility of appreciation following a positive shock is a direct consequence of the wealth effects that arise under financial autarky. A positive innovation to ↑ YbH,t will worsen the terms of trade, ↑ Tbt , and depreciate the real exchange bt , when: rate, ↑ Q 1 φ > φ(TOT) = 1 − 2aH In this circumstance, a positive home output shock will benefit foreign consumers through favourable price adjustment. The international transmission of the home endowment shock is positive. Moreover, the real exchange rate and terms of trade volatility will be decreasing in φ. However, with home bias and a low enough φ, a positive home endowment shock can appreciate the real exchange rate and the terms of trade. In particular, the region in which this occurs is: 0 < φ < φ(TOT) Here there will be a negative international transmission of the home endowment shock under financial autarky. The positive home shock will not benefit the foreign consumers, as prices move against them to ensure that the higher home endowment is met by enough demand. The income effect is very strong in this region. With low enough trade elasticities, this demand can only come from the home consumers and hence prices must move in their favour.

4.3

Consumption and Real Exchange Rates

To consider the comovement between the real exchange rate Q and relative consumption C/C ∗ , it is necessary to derive an expression relating the two. Returning to the trade balance condition (4.1) and the consumer demand functions, (1.9) and (1.10):  ∗ −φ P ∗ a∗H PH,t Ct∗ ∗ CH,t t = Tt =  −φ CF,t P aF PF,t Ct t

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Using the knowledge that home and foreign consumption baskets are symmetric, aH = a∗F and aF = a∗H , and the law of one price: Tt

φ 

=

Pt∗ Pt

=

Qφt Ttφ



PF,t PH,t

φ

Ct∗ Ct

Ct∗ Ct

Rearranging: Ct = Qφt Ttφ−1 Ct∗ Around the symmetric steady state, the log-linear form of the above expression is: bt − C bt∗ = φQ bt + (φ − 1)Tbt C bt , this becomes: Using equation (2.2) and solving for Q   bt − C b∗ bt = 2aH − 1 C Q t 2aH φ − 1

(4.11)

With home bias in consumption (aH > 1/2), the correlation between relative consumption and the real exchange rate will be negative for some values of: φ < φ(CORR) =

1 2aH

In particular: • If φ > φ(CORR), then Corr(C/C ∗ , Q) > 0 and the home terms of trade will worsen (↑ T ) following a positive home endowment shock. Positive international transmission reduces endowment risk. • If φ < φ(TOT), then Corr(C/C ∗ , Q) > 0, but the home terms of trade must appreciate (↓ T ) following a positive home endowment shock. A negative international transmission arises because of income effects under financial autarky, raising endowment risk. • If φ(TOT) < φ < φ(CORR), then the home terms of trade will worsen (↑ T ) following a positive home endowment shock. However, the transmission is excessively positive and the terms of trade depreciates by such a large amount that the correlation of relative consumption and the real exchange rate turns negative: Corr(C/C ∗ , Q) < 0.

4.4

The Degree of Risk Sharing Under Financial Autarky

The equilibrium financial autarky and complete market allocations will differ in general, but turn out to coincide under specific circumstances. The demand gap D is a useful metric to compare the two allocations. Under complete markets, the demand gap (defined in equation (4.7)) is always zero and the equation (3.7) holds with equality:     ˜ = σ C˜ − C˜ + ζbC∗ − ζbC Q

16

Under financial autarky, the synthesis of equations (4.10) and (4.11) yields:    σ(2aH φ − 1) − (2aH − 1)  b ∗ ∗ − ζbC,t − ζbC,t YH,t − YbF,t 1 − 2aH (1 − φ)

(4.12)

This expression demonstrates that in general, with endowment and preference shocks, there is no combination of parameters for which the demand gap is zero under financial autarky. However, there are different combinations of parameters for which the complete markets and financial autarky outcome coincide, conditional upon a single shock in isolation. One such occurrence of this is outlined in Cole and Obstfeld (1991) where: ∗ • There are only endowment shocks, but no preference shocks (ζbC,t = ζbC,t = 0).

• Consumption baskets are identical aH = 1/2, such that purchasing power parity (PPP) holds. • φ = 1, so consumption baskets are given a Cobb-Douglas aggregator. • There a zero initial net foreign assets. Under these four conditions, the complete markets and financial autarky outcomes coincide. In the Cole and Obstfeld (1991) case, terms of trade movements automatically pool endowment risk. When the endowment of the home country rises, the price of the home good (the terms of trade for home consumers) will depreciate exactly proportionally, leaving the total wealth of home consumers unaffected. Domestic consumers will be better off due to their greater endowment, while the proportional price decrease will also make foreign consumers better off. Equation (4.12) displays a more general condition for which the demand gap will be zero under financial autarky without preference shocks: φσ =

1 (2aH + σ − 1) aH

References Cole, H. L. and M. Obstfeld (1991): “Commodity trade and international risk sharing : How much do financial markets matter?” Journal of Monetary Economics, 28, 3–24. Corsetti, G., L. Dedola, and S. Leduc (2008): “International Risk Sharing and the Transmission of Productivity Shocks,” Review of Economic Studies, 75, 443–473.

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