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CM
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Cambridge Lent 2016, 220
Demand and Real Exchange Rate Determination in Open Economies: Foundations of Misalignments and Global Imbalances Two-Good Endowment Economies Giancarlo Corsetti and Simon P. Lloyd
[email protected] [email protected] University of Cambridge
February 2016 Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Questions and goals Think of incomplete markets and financial frictions as a key reason for misalignment and imbalances Financial market structure determines both the determination of the real exchange rate and the international transmission of macroeconomic shocks.
Analyze the international transmission of efficient shocks in a two-Country, Two-Good Real Endowment Economy Draw on international (real) business cycle literature,
No role for price rigidities here: focus on financial frictions Exchange rate misaligned even when works as shock absorber
Focus on: asset market structure, persistence of shocks, trade elasticities (Corsetti et al., 2008).
Corsetti and Lloyd
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The Model Two countries {H, F }, each endowed with a specific good, YH and YF respectively, which are traded internationally (see Corsetti et al., 2008, section 3). Three international asset/financial market structures: complete markets (CM); financial autarky (FA); and a bond economy (BE). To keep focus, omit: production, varieties and nominal rigidities. Key Results: 1
The real exchange rate Qt is determined by the households’ Euler equations, hence it varies with the structure of financial market and risk sharing.
2
The propagation of international macroeconomic shocks thus crucially differs depending on the financial market structure.
Corsetti and Lloyd
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Model
Goods Markets
Asset Market: international
CM
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The Setup of the Model Households enjoy consuming both home H and foreign F goods. Households use the international asset market to (attempt to) insure against risk, emanating from uncertainty about their endowment and preferences. Three markets must equilibrate for a global equilibrium: 1
Home Country Goods Market, YH Supply of YH exogenously given by H’s endowment. Demand for YH from both home and foreign consumers.
2
Foreign Country Goods Market, YF Supply of YF exogenously given by F ’s endowment. Demand for YF from both home and foreign consumers.
3
International Financial Market Nature of equilibrium in this market depends on the prevailing financial market structure.
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Households
In each country, a continuum of households derive utility from consuming both home and foreign goods, summarised by the CES aggregator Ct . For the Home utility (Foreign is analogous, with starred variables and functions): Ut = Et
∞ X
β j U(Ct+j , ζC ,t+j ) = Et
j=0
∞ X j=0
β j ζC ,t+j
1−σ Ct+j −1
1−σ
where β = β ∗ ∈ (0, 1), σ = σ ∗ > 1 and ζC ,t+j (ζC∗ ,t+j ) is a home (foreign) stochastic preference shock at time t + j.
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Consumption Aggregator Ct is the total consumption basket of the representative home consumer at time t. The basket comprises CH and CF , represented by: ω h ω−1 i ω−1 ω−1 1/ω 1/ω Ct = aH CH,tω + aF CF ,tω
ω > 0 is the elasticity of substitution between home and foreign goods = trade elasticity. ∗ CH,t (CH,t ) represents home (foreign) consumption of the home good. CF ,t
(CF∗,t ) represents home (foreign) consumption of the foreign good.
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Price of Consumption and Demand for Goods Pt (Pt∗ ) is the consumption-based price index for the home (foreign) economy, the price of a single unit of the aggregate consumption basket in the domestic economy, defined such that: Pt = min Zt = PH,t CH,t + PF ,t CF ,t CH,t ,CF ,t
subject to
Ct = 1
From this, one can derive the price index: 1 1−ω 1−ω + aF PF1−ω Pt = aH PH,t ,t as well as expressions for home consumer demand for the home and foreign good respectively: CH,t
= aH
CF ,t Corsetti and Lloyd
= aF
PH,t Pt
−ω
PF ,t Pt
−ω
Endowment Economy Model
Ct Ct February 2015
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Complementarity and Substitutability
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 They are complements if the opposite is true. One can show that: When σω > 1, the two goods are substitutes. When σω < 1, the two goods are complements.
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Exchange Rates and Relative Prices Nominal Exchange Rate Et : domestic currency price of foreign currency ⇒ ↑ Et ⇔ Depreciation of Home Currency. Terms of Trade Tt : the relative price of imports to the price of exports, where both price are written in terms of the home currency: Tt =
PF ,t ∗ Et PH,t
↑ Tt ⇔ ↑ Relative Price of Imports ⇔ Worsening of Home Terms of Trade. Real Exchange Rate Qt : the relative price of consumption, the price of the foreign currency basket in domestic currency terms relative to the home basket price:
Et Pt∗ Pt ↑ Qt ⇔ Depreciation of Home Real Exchange Rate. Qt =
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Purchasing Power Parity
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 = aH = 1 − aF∗ .
Therefore, deviations from PPP can reflect either deviations from the law of one price or differences in preferences across countries.
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Assumptions and Normalisations For simplicity, assume that the the law of one price (LOOP) holds: 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 = aF∗ = 1 − aH > 1/2
Without loss of generality, the nominal exchange rate is normalised such that: Et = 1
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∀t
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A Relationship Between Tt and Qt Using the definition of the real exchange rate Qt and the normalisation of the nominal exchange rate Et = 1: 1−ω
Q1−ω = t
(Et Pt∗ ) Pt1−ω
=
Pt∗ 1−ω Pt1−ω
Using definition of price indices: Q1−ω t
=
=
∗ 1−ω (1 − aF∗ )PH,t + aF∗ PF1−ω ,t ∗ 1−ω + (1 − a )P 1−ω aH PH,t H F ,t 1−ω P (1 − aF∗ ) + aF∗ PF∗,t H,t 1−ω PF ,t aH + (1 − aH ) P ∗ H,t
Using the definition of the terms of trade Tt : Q1−ω = t Corsetti and Lloyd
(1 − aF∗ ) + aF∗ Tt1−ω aH + (1 − aH )Tt1−ω
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World Goods Market Equilibrium Equilibrium in home and foreign goods market equilibrium requires: ∗ YH,t = CH,t + CH,t
YF ,t = CF ,t + CF∗,t
One can show that world goods demand can be re-expressed in the following manner, yielding conditions for equilibrium in the home and foreign goods markets respectively:
Corsetti and Lloyd
YH,t
=
YF ,t
=
PH,t Pt
−ω
PF ,t Pt
−ω
∗ ω ∗ [aH Ct + aH Qt Ct ] ∗ [aF Ct + aF∗ Qω t Ct ]
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Asset Market: domestic To close the model, we need to specify the asset markets: which assets can agents trade within and across borders Within borders, we have a representative agent—we can think that markets are complete domestically. Hence by the Euler equation of the agents, we know that the Home and Foreign real interest rate will obey the following −σ Ct+1 Ct ∗ −σ Ct+1 = βRt∗ Et Ct∗
1 1
= βRt Et
(1) (2)
Before proceeding, it is worth summarising what we know about the equilibrium before specifying international asset markets.
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Equilibrium Conditions So Far There are 12 endogenous variables, ∗ {Ct , Ct∗ , CH,t , CH,t , CF ,t , CF∗,t , Yt , Yt∗ , Rt , Rt∗ , Tt , Qt }, and, so far, 11 equilibrium
conditions. Abstracting from preference shocks:
=
CF ,t + CF∗,t
=
Ct∗
=
1 Tt
=
1 Tt Yt Yt∗ Corsetti and Lloyd
=
ω h 1 ω−1 ω−1 i ω−1 1 aHω CH,tω + (1 − aH ) ω CF ,tω ω h 1 ω−1 i ω−1 ∗ ∗ ω−1 ∗ ω1 ∗ ω aHω CH,tω + (1 − aH ) CF ,t ω1 − ω1 aH CH 1 − aH CF ∗ ω1 ∗ − ω1 aH CH ∗ 1 − aH CF∗ ∗ CH,t + CH,t
Ct
=
Endowment Economy Model
(3) (4) (5) (6) (7) (8)
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Yt
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= ρYt−1 + (1 − ρ)Y + εt
(9)
Yt∗
∗ = ρYt−1 + (1 − ρ)Y ∗ + ε∗t −σ Ct+1 1 = βRt Et Ct ∗ −σ C t+1 1 = βRt∗ Et Ct∗ 1 ∗ ∗ aH + (1 − aH )Tt1−ω 1−ω Qt = 1 aH + (1 − aH )Tt1−ω 1−ω
(10) (11) (12) (13)
The last expression links the real exchange rate to the terms of trade. Its determination and dynamics is analysed next.
Corsetti and Lloyd
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Goods Markets
Asset Market: international
CM
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International Transmission
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Financial Market Structure and the International Transmission Mechanism The structure of international asset markets will influence the determination of the real exchange rate Qt and the transmission of international shocks in the global macroeconomic equilibrium. The types of financial market structure considered here: 1
Complete Markets (CM) - There exists a complete set of Arrow-Debreu securities for each state of nature in the future. The real exchange rate must adjust to ensure perfect risk sharing in each period of time.
2
Financial Autarky (FA) - International trade in assets is not permitted whatsoever. The real exchange must adjust to ensure that a country’s trade is balanced in every period.
3
Bond Economy (BE) - There exists a single non-contingent bond.
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Complete Markets Complete Markets ⇒ Full set of state-contingent Arrow-Debreu securities. Time t state known: st . Time t + 1 state unknown: st+1 ∈ S where S is the state-space.
At time t, there exists at least one bond for each state st+1 in time t + 1, BH,t+1 (s1,t+1 ). It can be purchased at the price qH,t (s1,t+1 ) at time t, and pays out a single unit upon realisation of the state s1,t+1 at time t + 1. Representative household inter-temporal complete market budget constraint: 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
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Maximising the representative household utility subject to the budget constraint yields the following Euler equation: UC (s)
1 q(s 0 |s) = β Pr(s 0 |s)UC (s 0 |s) P(s) P(s 0 |s)
Combining this with a similar expression for the foreign consumer results in a condition for efficient risk sharing under complete markets: UC (s) P ∗ (s) UC (s 0 |s) P ∗ (s 0 |s) · = ∗ 0 · ∗ UC (s) P(s) UC (s |s) P(s 0 |s) Important note: there are no distortions in the real complete market economy: by the first fundamental welfare theorem, the flexible price allocation will be first best efficient.
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Complete Markets: Asset Market Equilibrium If countries are initially perfectly symmetric and have identical time t endowments, then:
UC∗ (s) P ∗ (s) = UC (s) P(s)
Real exchange rate determination relies on this condition: Q(s 0 |s)
P ∗ (s 0 |s) P(s 0 |s) UC∗ (s 0 |s) = UC (s 0 |s) σ ∗ 0 C (s 0 |s) ζC (s |s) = C ∗ (s 0 |s) ζC (s 0 |s) =
as E = 1 from above since utility is CRRA
where ζC is a taste/preference shock. This guarantees equilibrium in asset markets under complete markets. Corsetti and Lloyd
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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
The total value of the domestic endowment must equal the total value of home consumption: Pt Ct − PH,t YH,t = 0 Corsetti and Lloyd
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Since resources cannot be transferred intertemporally, the household maximisation problem is static and of the form: Lt = max ζC ,t Ct ,χt
Ct1−σ − 1 + χt [PH,t YH,t − Pt Ct ] 1−σ
where χt is the Lagrange multiplier, equal to the marginal utility of an extra unit of income. Combining the FOC of this problem for the home and foreign consumers yields:
ζC∗ ,t ζC ,t
Ct Ct∗
σ
ζC∗ ,t Pt = ∗ Pt ζC ,t
Ct Ct∗
σ
1 χ∗ = t Qt χt
This defines global market equilibrium under financial autarky.
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Compare with Complete Markets Under complete markets, the corresponding version of this equation is the efficient risk sharing condition: ζC∗ ,t Ct σ Pt ζC∗ ,t Ct σ 1 =1 = ζC ,t Ct∗ Pt∗ ζC ,t Ct∗ Qt 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 i.e. an inefficient allocation of demand across border. It will leave a gap in the marginal utility of income, which accounts for deviations in perfect risk sharing.
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Bond Economy Allow for cross-border trade in a one period nominal discount bond BH,t+1 , 1 1+i
⇒ incomplete markets. R R ∗ In world equilibrium, bond market clears: BH,t+1 (j)dj + BH,t+1 (j ∗ )dj = 0.
with price QH,t =
The budget constraint of home households is: PH,t YH,t + BH,t = Pt Ct + QH,t BH,t+1 so the maximisation problem of the household is: " ∞ X C 1−σ − 1 Lt = max ζC ,t+s t+s Ct ,BH,t+1 ,χt 1−σ s=0 +χt+s (PH,t+s YH,t+s + BH,t+s − Pt+s Ct+s − QH,t+s BH,t+s+1 )]
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The problem yields the F.O.C.: " QH,t = βEt
−σ ζC ,t+1 Ct+1 Pt −σ P ζC ,t Ct t+1
#
∗ Given that the bond is internationally traded, QH,t = QH,t , such that: " # " # ∗−σ −σ ζC∗ ,t+1 Ct+1 ζC ,t+1 Ct+1 Pt Pt∗ Et = Et −σ ∗ Pt+1 ζC ,t Ct−σ Pt+1 ζC∗ ,t Ct∗
The equivalent complete markets condition is: −σ
∗ −σ ζC∗ ,t+1 Ct+1 ζC ,t+1 Ct+1 Pt Pt∗ = −σ −σ ∗ ∗ ∗ Pt+1 Pt+1 ζC ,t Ct ζC ,t Ct
With international trade in a non-contingent bond, households cannot insure ex ante against country-specific shocks. They can only smooth consumption over time, in response to shocks. Hence the two terms above are equalized only in expectations, not in every state of nature, as is the case under compete markets. Corsetti and Lloyd
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With trade in a bond, indeed, shocks create ex post wealth effects, since not all risk is traded. In the bond economy, ex post: −σ
∗ −σ ζC∗ ,t+1 Ct+1 ζC ,t+1 Ct+1 Pt Pt∗ = t+1 − ∗ ∗−σ −σ ∗ Pt+1 Pt+1 ζC ,t Ct ζC ,t Ct
where t+1 is i.i.d. mean zero. Ex post, at the time the shock hits, non-traded risk opens a wedge between domestic and foreign wealth. Note that this is a function of the endogenous response of real exchange rates, determining the relative valuation of a country’s output.
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Remaining Equilibrium Condition Complete Markets Qt =
Ct∗ Ct
−σ (13.CM)
Financial Autarky Yt = CH,t + Tt CF ,t
(13.FA)
Bond Economy 1 = βRt Et
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∗ Ct+1 Ct∗
−σ
Endowment Economy Model
Qt Qt+1
(13.BE)
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The Relationship Between Qt and Tt : linearization Q1−ω = t
∗ ∗ aH + (1 − aH )Tt1−ω aH + (1 − aH )Tt1−ω
One can show that the log-linear form of this equation with symmetry, around a steady state with Q = T = 1, is: bt = (2aH − 1)Tbt Q
(14)
The co-movement of the real exchange rate Q and the terms of trade T depends on the degree of home bias, aH . The two variables positively co-move when there is home bias, aH > 1/2. Zero co-movement when aH = 1/2; this is the PPP case. The co-movement is negative when aH < 1/2 (foreign bias). Corsetti and Lloyd
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Complete Markets Earlier, it was shown that asset market equilibrium in the two good endowment model with complete markets is defined by: σ ∗ 0 Ct ζC (s |s) Qt = Ct∗ ζC (s 0 |s) It can be shown that its log-linear form is: ˜ fb = σ C˜fb − C˜∗fb + ζbC∗ − ζbC = (2aH − 1)T˜ fb Q bt = (2aH − 1)Tbt . where the second equality follows from Q 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 . Corsetti and Lloyd
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(15)
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Complete Markets Transmission Aim To analyse the effect of endowment and preference shocks on international demand and prices. Using the world goods market and complete asset market equilibrium conditions, it can be shown that: −σ−1 −1 ζC ,t ∗ ω−σ aH + aH Qt ζC∗,t YH,t = Tω −σ−1 YF ,t −1 ζC ,t aF + aF∗ Qω−σ ∗ t ζ C ,t
Its log-linear form is: 1 1 fb fb fb b b∗ ˜ ˜ ˜ ˜ fb YH,t − YF ,t = ω Tt + − ω (2aH −1)Q (16) t + (2aH −1) ζC ,t − ζC ,t σ σ Corsetti and Lloyd
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Tt and the Relative Endowment Substituting the T -Q relationship, equation (14), into equation (16) and solving for T˜tf b yields: T˜tfb =
fb σ Y˜H,t − Y˜Ffb,t − (2aH − 1) ζbC ,t − ζbC∗ ,t {[1 − (2aH − 1)2 ] σω + (2aH − 1)2 }
(17)
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 exports.
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Relative Consumption and Endowments For given preferences, ζbC = ζbC∗= 0, then substituting equation (15) into fb ∗fb equation (17) and solving for C˜ − C˜ yields:
C˜tfb − C˜t∗fb =
2aH − 1 fb ˜H,t ˜F∗fb Y − Y ,t [1 − (2aH − 1)2 ] σω + (2aH − 1)2
(18)
With home bias, aH > 1/2, the coefficient on relative output is always positive. fb In response to a home supply shock, ↑ Y˜H , then consumption will grow more ∗ at home than it does abroad, ↑ C˜ − C˜ , even if the ToT worsen, ↑ T .
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 σω → ∞ (goods more substitutable), then C˜fb → C˜∗fb . Corsetti and Lloyd
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Financial Autarky
Earlier, it was shown that under financial autarky there exists a demand gap: ζC∗ ,t Ct σ 1 χ∗ Dt = t = χt ζC ,t Ct∗ Qt It can be shown that its log-linear form is: bt = ζb∗ − ζbC ,t + σ Cbt − Cb∗ − Q bt D C ,t t
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Consumption Demand in Financial Autarky
Aim To understand the wealth effects of shocks to the relative endowment and their effects on the international transmission of shocks. Domestic demand for home goods under financial autarky is pinned down by the trade balance condition Pt Ct = PH,t YH,t , such that: CH,t = aH
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PH,t Pt
−ω
Ct = aH
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PH,t Pt
−ω
PH,t YH,t Pt
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In FA, 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).
−ω PH,t PH,t YH,t = aH Pt Pt | {z } | {z }
CH,t
SE
IE
Substitution Effect (SE) The SE of a fall in the relative price of the home good will be positive, raising domestic demand for the home good by power ω. The strength of the substitution effect is increasing in the substitutability of the home and foreign goods, ω. 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 IE. The IE of a price fall is negative, diminishing the value of the home consumers’ endowment. Corsetti and Lloyd
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To appreciate these effects, it can be shown that: CH,t =
aH YH,t aH + aF Tt1−ω
Consider the partial differential of this equation with respect to the terms of trade T : ∂CH,t =ω ∂Tt |
aH (1 − aH )T −ω [aH + (1 − aH )T 1−ω ] {z
! −
2 YH,t
SE
}
aH (1 − aH )T −ω
! 2 YH,t
[aH + (1 − aH )T 1−ω ] | {z IE
}
ω > 1: SE exceeds IE. A deterioration in the ToT (↑ Tt ) will raise the domestic demand for the home good. ω < 1: IE exceeds SE. A deterioration in the ToT (↑ Tt ) will reduce domestic demand for the home good, due to a fall in the value of endowments. ω = 1: IE and SE will exactly cancel, and terms of trade movements will leave domestic demand unaffected. Corsetti and Lloyd
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World Demand for Home Goods World demand for home goods: CH + CH∗ . As long as the negative IE 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. The world demand for the home good will a decreasing function of its relative price.
If the IE is sufficiently large (ω is a lot below 1) and home bias is sufficiently high (aH is enough about 1/2), then the negative IE 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. World demand for the home good will fall in response to a fall in its relative price if income effects are sufficiently strong. In equilibrium, the terms of trade may improve (↓ T ) following a fall in the price of home goods (↓ PH ). Corsetti and Lloyd
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Tt Response to Supply Shocks ∗ Using the trade balance condition Tt CF ,t − CH,t = 0 and the consumer
demand functions, it can be shown that: 1+ω ∗ ω−1 ∗ YF ,t Pt PF ,t Tt = PH,t Pt YH,t the log-linear form of which is: bt + YbF∗,t − YbH,t −ω Tbt = (ω − 1)Q Utilising equation (14) leads to: Tbt
=
bt Q
=
1 YbH,t − YbF∗,t 1 − 2aH (1 − ω) 2aH − 1 YbH,t − YbF∗,t 1 − 2aH (1 − ω)
(20) (21)
Under FA an increase in the home endowment can either appreciate or depreciate the domestic Tt and Qt , a direct consequence of income effects. Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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bt , when: Positive innovation to ↑ YbH,t will ↑ Tbt and ↑ Q ω > ω(TOT) = 1 −
1 2aH
Positive international transmission: Positive home output shock will benefit foreign consumers through favourable price adjustment. Qt and Tt volatility will be decreasing in ω.
With home bias and a low enough ω, a positive home endowment shock can appreciate the Qt and Tt . The region in which this occurs is: 0 < ω < ω(TOT)
Negative international transmission: Positive home shock will not benefit foreign consumers, as prices move against them to ensure that the higher home endowment is met by enough demand. IE strong in this region. With low enough trade elasticities, demand can only come from the home consumers and hence prices must move in their favour. Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Figure: IM - Impact Response of Home T to Positive Domestic Endowment Shock
x-axis: ω
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Consumption and Qt Consider the co-movement between Q and C /C ∗ , by writing: Ct ω−1 = Qω t Tt Ct∗ which can be log-linearised to attain: bt = 2aH − 1 Cbt − Cb∗ Q t 2aH ω − 1
(22)
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: ω < ω(CORR) =
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Endowment Economy Model
1 2aH
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
ω > ω(CORR): 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. ω < ω(TOT): 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. ω(TOT) < ω < ω(CORR): 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.
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Formalising Imbalances: Demand Gap The imbalances due to incomplete markets can be formalised in a demand gap term Dt :
ζC∗ ,t χ∗ Dt = t = χt ζC ,t
Ct Ct∗
σ
1 Qt
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. With trade in bonds, it is easy to see Et Dt + 1 = Dt In other words, agents optimally choose to borrow or lend in international markets as to ‘close the Dt in expectations’ (since they cannot do this state by state, as is the case with complete markets) Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Formalising Misalignments: Real Exchange Rate Gap Under complete markets, the real exchange rate Q will adjust the equalise ratios of marginal utilities. With incomplete markets, the real exchange rate Q will adjust to ensure external trade balance (under financial autarky) or equalisation of ratios of marginal utilities in expectations (in the bond economy). Thus, under incomplete markets, real exchange rates will be misaligned relative to the efficient benchmark. The real exchange rate gap formalises this concept: Qgap = t
QIM t QCM t
where IM = FA, BE . Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
MATLAB Code
Solution Method: Perturbation (Schmitt-Groh´e and Ur´ıbe, 2004). Code produces: 1
Approximated Policy Functions
2
Impulse Responses
3
Impact Responses
4
Simulations
Download RealCode.zip. There are five folders: Complete Markets (cm), Financial Autarky (fa), Bond Economy - No Uzawa (be), Bond Economy - Uzawa (be uz) and SGU PerturbCode.
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
The four model folders — Complete Markets (cm), Financial Autarky (fa), Bond Economy - No Uzawa (be) and Bond Economy - Uzawa (be uz) — have the following files: XX model run.m: master file, from which the code is run. XX model.m: model file, with model equations. XX model ss.m: steady state file, with model parameters and steady state values. XX model irf.m: impulse response file (adapted from Schmitt-Groh´e and Ur´ıbe, 2004). XX model impact.m: impact response file. XX model ssimpact.m: steady state file for impact responses.
where XX is cm, fa, be or be uz.
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
Impact Responses Parameterisation
Table: Model Parameterisation
Corsetti and Lloyd
Parameter
Value
σ
2
ω
0→2
aH
0.85
ρ
0.99
β
0.96
σY
0.0075
Endowment Economy Model
February 2015
MATLAB Code
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Complete Markets Figure: CM - Impact Response to Positive Domestic Endowment Shock
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Financial Autarky Figure: FA - Impact Response to Positive Domestic Endowment Shock
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Bond Economy Figure: BE - Impact Response to Positive Domestic Endowment Shock
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Bond Economy Computational Point: Non-Stationary Bond Positions
See Schmitt-Groh´e and Ur´ıbe (2003) for more details. In the standard IM model with one non-state-contingent bond, the deterministic steady state of the net foreign asset position derived from linear approximation of the model is not determined. Net foreign asset position not stationary. Absent arbitrage opportunities, the price of the non-state-contingent bond implies that expected marginal utility growth is equalized across countries. In the deterministic steady state, this condition contains no information about the steady state values of the system and the system of equilibrium conditions becomes underdetermined. In particular, any net foreign asset position is compatible with a steady state.
Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Solutions: (i) a model with an endogenous discount factor (Uzawa preferences); (ii) a model with a debt-elastic interest-rate premium; (iii) a model with convex portfolio adjustment costs. Schmitt-Groh´e and Ur´ıbe (2003) find all models deliver similar business cycle frequency dynamics. Here: Uzawa preferences (endogenous discount factor). Let: β(Ct ) = βCt−α where α = 0.01. Agents become more impatient the more they consume. See also Bodenstein (2011) for information on equilibrium multiplicity in large open economy models (distinct from Schmitt-Groh´e and Ur´ıbe, 2003, type non-stationarity), avoided here by careful definition of steady state. Bodenstein (2011) concludes that Uzawa preferences are the preferred equilibrium selection device. Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
BE - No Uzawa Positive Domestic Endowment Shock: Non-Stationary Bond Position
Corsetti and Lloyd
Endowment Economy Model
February 2015
MATLAB Code
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
BE - Uzawa Positive Domestic Endowment Shock: Stationary Bond Position
Corsetti and Lloyd
Endowment Economy Model
February 2015
MATLAB Code
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
News Shocks So far, the endowments of the home and foreign economies evolved in the following manner: Yt = ρYt−1 + (1 − ρ)Y + εt
(7)
Yt∗
(8)
∗
= ρYt−1 + (1 − ρ)Y +
ε∗t
where Y and Y ∗ denote the steady state endowment for the home and foreign economies respectively. ⇒ Productivity shocks, εt and ε∗t , are contemporaneous and persistent. What if at some point in time agents revised their expectations about productivity (endowment) in the future, with no contemporaneous change in endowment? ⇒ ‘News’ Shock. Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
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Replace equilibrium equations (7) and (8) with: Yt = Zt Y Yt =
(7a)
Zt∗ Y ∗
(8a)
where Zt (Zt∗ ) denotes productivity (endowment) in the home (foreign) economy. Let Zt have two subcomponents Z1,t and Z2,t : Zt = Z1,t /Z2,t
(7b)
Z1,t = ρ1 Z1,t−1 + (1 − ρ1 )Z + ξt + εt
(7c)
Z2,t = ρ2 Z2,t−1 + (1 − ρ2 )Z + ξt
(7d)
where ρ1 > ρ2 and Z = 1 is the steady state of the shock processes. Symmetrically for the foreign economy. εt is a contemporaneous productivity shock, while ξt is a ‘news’ shock. Z1,t : Standard autoregressive technology shock. Here, ρ1 = 0.95. Z2,t : Captures ‘news’ shock. Here, ρ2 = 0.9. Corsetti and Lloyd
Endowment Economy Model
February 2015
Model
Goods Markets
Asset Market: international
CM
FA
BE
International Transmission
MATLAB Code
Conclusions Two countries {H, F }, each endowed with a specific good, YH and YF respectively, which are traded internationally. Three international asset/financial market structures: complete markets (CM); financial autarky (FA); and a bond economy (BE). Key Results: 1
Financial market structure directly influences the determination of the real exchange rate Qt through the household Euler equation.
2
The propagation of international macroeconomic shocks differs depending on the financial market structure.
Introduction to MATLAB code. Computational Point: to close bond economy models Schmitt-Groh´e and Ur´ıbe (2003) advocate the use of a stationarity inducing device → Uzawa preferences. Corsetti and Lloyd
Endowment Economy Model
February 2015
References
Bodenstein, M. (2011): “Closing large open economy models,” Journal of International Economics, 84, 160–177. Corsetti, G., L. Dedola, and S. Leduc (2008): “International Risk Sharing and the Transmission of Productivity Shocks,” Review of Economic Studies, 75, 443–473. ´, S. and M. Ur´ıbe (2003): “Closing small open economy models,” Journal of Schmitt-Grohe International Economics, 61, 163–185. ——— (2004): “Solving Dynamic General Equilibrium Models using a Second-Order Approximation to the Policy Function,” Journal of Economic Dynamics and Control, 28, 755–775.
Corsetti and Lloyd
Endowment Economy Model
February 2015
