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Practical Information
• email : [email protected]
• Course website: Link PhD course at
http://econ.sciences-po.fr/nicolas-coeurdacier/nicolas-coeurdacier
• References: reading list for articles. Obstfeld and Rogoff "Foundations
of International Macroeconomics".
• Assignment: Mid-term take home exam (handed out at lecture 3 due to
lecture 5); Students presentations; End-of-term exam.
Road-map for the course
Three main topics
• International Real Business Cycles and the Transmission of Shocks
• International Portfolios and Risk-Sharing
• Global Imbalances and the Valuation Channel of External Adjustment
Part 1- International Real business Cycles and the Transmission of Shocks
- A simple two-country/two-good/two period endowment economy
Complete markets and Financial Autarky
International transmission of output shocks / Backus-Smith Puzzle
- Two country/two good RBC model with complete markets
Backus, Kehoe, Kydland, AER 1994;
Quick reminder about the one good IRBC model
Facts and anomalies:
- ’The quantity Puzzle’: in the data, cross-country correlation of output is
larger than cross-country correlation of consumption.
Most models predict the opposite: with complete markets (perfect risk-sharing),
consumption is perfectly correlated across countries (if utility separable between
consumption and leisure), output is not = ’Quantity Puzzle’.
Very difficult to solve. Market incompleteness helps if strong wealth effects (for
instance, very persistent technology shocks; see Kollmann (1996), Baxter and
Crucini (1995)).
’The quantity puzzle’
Quick reminder about the one good IRBC model
Facts and anomalies:
- Very high cross-country correlation of investment in the data but not in the
models where capital moves to the most productive location.
- Similarly, high cross country correlation of employment in the data but not in
the model.
- Countercyclical trade balance in the data: key role for investment. Intuition?
Relaxing the assumption of a single perfectly tradable good?
One good model has nothing to say about real exchange rate and relative prices.
Main price in international macro!
What do we expect in terms of risk sharing implications from a two-good model?
Real exchange rate will introduce a wedge between marginal utilities of con-
sumption in the risk sharing condition - hence, one may expect that this model
predicts lower cross-country consumption correlation
International transmission through relative prices might reduce the negative
correlation of investment/hours/output across countries
Risk-sharing in a two country/two good endowment economy with com-
plete markets
- Two countries Home (H) and Foreign (F ) perfectly symmetric ex-ante.
- Two periods t = 0; 1.
Output in i at t = 0 is y0,i. Uncertainty in period t = 1. Finite number of
states s (proba π(s)). Output y1,i(s).
- Each country produces one differentiated good.
- Complete markets implemented with AD securities: AD security s pays 1
in state s at t = 1; price at t = 0 is pb(s)
I abstract from time indices for simplicity.
Aggregate consumption index
CH =[a1/φ (cHH)(φ−1)/φ + (1− a)1/φ (cHF )
(φ−1)/φ]φ/(φ−1)
CF =[a1/φ (cFF )
(φ−1)/φ + (1− a)1/φ (cFH)(φ−1)/φ]φ/(φ−1)
with φ elasticity of substitution between the two goods, and cij = consumptionof good j by country i, i = H,F
Home bias in preferences a ≥ 12. With a = 1
2, identical preferences ⇒ ahomogenous to goods market integration. With “Cobb-Douglas” preferences,(φ = 1), a = share of consumption spending devoted to local good
Price indices
PH =[a (pH)1−φ + (1− a) (pF )
1−φ]1/(1−φ)
PF =[(1− a) (pH)1−φ + a (pF )
1−φ]1/(1−φ)
Terms of trade: q = relative price of Home goods over Foreign goods
q ≡pHpF
Real Exchange Rate
RER =PHPF
=
(aq1−φ + (1− a)
(1− a)q1−φ + a
)1/(1−φ)
Intertemporal Utility
Ui = u(C0,i) + βE0u(C1,i) = C0,i + β∑
sπ(s)u(C1,i(s))
Ct,i=aggegate consumption at date t in country i et β=discount factor; u(c) =c1−σ
1−σ with σ = CRRA coefficient. Finite number of states s
Budget constraints:
P0,iC0,i = p0,iy0,i −∑
spb(s)bi(s) λi0
P1,iC1,i(s) = p1,iy1,i(s) + bi(s) βλi1(s)
Ressource constraints (at both dates and in all states at t=1)
cHH + cFH = yH ; cFF + cHF = yF
with yi endowment of country i
Asset market clearing condition∑
i
bi(s) = 0
Euler equation for Arrow-Debreu Securities
u′(C0,i) = λi0P0,i π(s)u′(C1,i(s)) = λ
i1(s)P1,i
⇒ pb(s)u′(C0,i)P0,i
= βπ(s)u′(C1,i(s))
P1,i⇒ pb(s) = βπ(s)
u′(C1,i)/P1,i
u′(C0,i)/P0,i
This is again true for both countries (abstracting from indices s); this leads
to the fundamental risk-sharing condition in presence of real exchange rate
fluctuations
u′(C1,H)/P1,H
u′(C0,H)/P0,H=u′(C1,F )/P1,F
u′(C0,F )/P0,F⇒
(C1,H
C0,H
)−σ
=P1,H/P1,F
P0,H/P0,F
(C1,F
C0,F
)−σ
Consumption growth rates are no more equal across countries. But changes in
consumption are linked to changes in the real exchange rate.
The wedge introduced by the real exchange rate might potentially help address-ing the problem of (too) high cross-country consumption correlations (’QuantityPuzzle’)
Is it good news? Not really: this is the consumption-real exchange rate anomaly(see Kollmann (1995) and Backus and Smith (1993))
Note that this equation shows up in any complete markets model with realexchange rate fluctuations, CRRA preferences & separability between leisureand consumption.
Remark 1: If countries are symmetric ex-ante (q0 = 1):(C1,H
C1,F
)−σ
=P1,H
P1,F= RER
Remark 2: extension to a multi-period model?
"The consumption-real exchange rate anomaly"
Note: in the table real exchange rate is 1/RERt compared to our definition. Source: Corsetti et al.
Intratemporal allocation across goods (abstracting from time/state indices)
cii = a
(piPi
)−φ
Ci cij = (1− a)
(pj
Pi
)−φ
Ci
Demand of Home over Foreign goods (with market clearing conditions)
q−φΩa
[(PFPH
)φCF
CH
]=yHyF
Ωu(x) is a continuous function of (u, x) with Ωu(x) =1+x(1−u
u)
x+(1−uu
)
Remark: if a = 1/2, then Ω1/2(x) = 1 for all x and yHyF
= q−φ. Terms oftrade decreases with an elasticity 1/φ with respect to increase in output atHome relative to Foreign.
Log-linearization of the model
To have analytical expression and shed light on the transmission mechanism
of endowment (productivity) shocks, we log linearize the model around the
symmetric equilibrium (assuming that ex-ante at t=0 countries are symmetric;
assuming small shocks).
We write y ≡ yHyF
to denote relative outputs in both countries.
We log-linearize the model around the symmetric equilibrium where y equal
unity, and use Jonesian hats (x ≡ log(x/x)) to denote the log deviation of a
variable x from its mean value x.in a given state
The international transmission mechanism
Terms-of-trade and relative output
Asssume symmetric countries:(CH
CF
)−σ
=PHPF
⇒yHyF
= q−φΩa
[(PFPH
)φ−1/σ
](1)
Home country’s real exchange rate RER ≡ PHPF
:
RER =PHPF
= (2a− 1)q. (2)
Log-linearizing (1) and using (2) implies:
y = −φq + (2a− 1)(φ−
1
σ
)RER = −λq
where λ ≡ φ(1− (2a− 1)2
)+
(2a−1)2
σ . Note that λ > 0 as 1/2 < a < 1.
A relative increase in the supply of the home good (y > 0) is always associatedwith a worsening of the terms of trade (q < 0) with an elasticity −1/λ. Note
that without home bias in preferences (a = 12), λ is simply the elasticity of
substitution between Home and Foreign goods φ
=⇒ With complete markets, the international transmission of supply (produc-
tivity) shocks is always positive : a positive supply shock at Home unambigu-
ously worsens Home terms-of-trade.
The international transmission mechanism
Relative consumption and relative output
CH − CF = −1
σ
PHPF
= −(2a− 1)
σq ⇒ CH − CF =
(2a− 1)
σλy
With Home Bias, the coefficient above is always positive. In response to a
Home supply shock, consumption grows more at Home than abroad. Even if
the Home terms of trade fall, it will never be the case that their adverse move-
ments cause ‘immiserizing growth’. In response to a positive supply shocks,
domestic consumption will never fall either in absolute level, or relative to For-
eign Consumption. The consumption growth difference tends to fall with the
elasticity of substitution among goods (φ) (as φ increases, we approach the
one good model).
Net exports and relative output
Log-linearization of Net Exports as a share of GDP (NXH). Use foreign good
as a numeraire: pF = 1 ;q = pH
NXH = pHyH − PHCH = −NXF ; NXF = pFyF − PFCF
NXH =1
2
[qy −
(PHCH − PFCF
)]=q
2
[(1− λ)− (2a− 1)(1−
1
σ)]
NXH = q(1− a)[1− φ+ (2a− 1) (
1
σ− φ)
]
NXH =y
λ(1− a)
[φ− 1 + (2a− 1) (φ−
1
σ)]
Net exports are procyclical unless the elasticity of substitution (φ) between
Home and Foreign goods is very low. What does it mean? Is it realistic?
[recall that we do not have investment]
Terms-of-trade volatility and Net exports volatility
- When goods are highly substitutable (high φ), the elasticity of prices to
quantities is small - small price changes and large quantity changes
- When goods are poor substitutes (low φ), the elasticity of prices to quantities
is large - large price changes for small quantity changes
This price-quantity relationship turns out to be a problem... in the data, terms-
of-trade are very volatile around 2 times more volatile than output (low φ?)...but
so are quantities traded (at least more than what is predicted by simple two
goods model for low values of φ)
- Relative price adjustment tells us that Home terms-of-trade should depreciate
following an increase in productivity at Home (due to the relative scarcity of
Foreign goods). Data?
Terms-of-trade volatility and Net exports volatility
Volatility of Net Exports and terms of trade (% of volatility of output)
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0.8
1 1.2
1.4
1.6
1.8
2 2.2
2.4
2.6
2.8
3 3.2
3.4
3.6
3.8
4 4.2
4.4
4.6
4.8
5 5.2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Volatility of Terms of Trade (left-axis) Volatility of Net exports (right-axis)
A benchmark two-country/two good IRBC model with production
Reference: Backus, Kehoe and Kydland, American Economic Review, 1994[slightly modified version]
What do we expect compared to the one good BKK model?
Real exchange rate introduces wedge between marginal utilities of consumptionin the risk sharing condition - hence, one may expect that this model predictslower cross-country consumption correlation
International transmission through relative prices might reduce the negativecorrelation of investment/hours/output across countries.
A benchmark two-country/two good IRBC model with production
Two symmetric countries, Home (H) and Foreign (F ), each with a represen-tative household.
Each country i produces one good using labor and capital.
Markets are complete.
There is trade in goods and in Arrow-Debreu securities
All markets are perfectly competitive
Preferences
Country i is inhabited by a representative household who lives in periods
t = 0, 1, 2, .... The household has the following life-time utility function:
Ui = E0
∞∑
t=0
βtu(Cit, lit) =∞∑
t=0
∑
stπ(st)βtu(Cit, lit)
where Ci,t is i’s aggregate consumption and lit is labor supplied by the repre-
sentative household in country i
Consumption and price index
CH,t =[a1/φ
(cHH,t
)(φ−1)/φ+ (1− a)1/φ
(cHF,t
)(φ−1)/φ]φ/(φ−1)
CF,t =[a1/φ
(cFF,t
)(φ−1)/φ+ (1− a)1/φ
(cFH,t
)(φ−1)/φ]φ/(φ−1)
where cij,t is country i′s consumption of the good produced by j; 12 < a < 1.
PH,t =[a(pH,t
)1−φ+ (1− a)
(pF,t
)1−φ]1/(1−φ)
PF,t =[(1− a)
(pH,t
)1−φ+ a
(pF,t
)1−φ]1/(1−φ)
where pH,t and pF,t are the prices of goods H and F , respectively.
Technologies
In period t, country i produces yi,t units of good i according to the production
function
yi,t = zi,t(li,t)1−θ(ki,t)
θ
with 0 < θ < 1. li,t is the labor supply in country i at date t. ki,t is the
country’s stock of capital. Total factor productivity zi,t > 0 is an exogenous
random variable.
Capital is derived from physical investment in previous periods:
ki,t+1 = (1− δ)ki,t + Ii,t
where 0 < δ < 1 is the depreciation rate of capital. Ii,t is gross investment in
country i at date t.
Aggregate Investment
In both countries, investment goods are generated using Home and Foreign
inputs:
IH,t =[a1/φ
(iHH,t
)(φ−1)/φ+ (1− a)1/φ
(iHF,t
)(φ−1)/φI]φ/(φ−1)
IF,t =[a1/φ
(iFF,t
)(φ−1)/φ+ (1− a)1/φ
(iFH,t
)(φ−1)/φ]φ/(φ−1)
where iij,t is the quantity of the good produced by country j used for investment
in country i. The associated (ideal) price indices of investment goods are the
same than for consumption
P IH,t = PH,t ; P I
F,t = PF,t
Household decisions
E0
∞∑
t=0
βtu(Cit, lit) =∞∑
t=0
∑
stπ(st)βtu(Cit, lit)
The country i household maximizes life-time utility (selects Cit and lit, and
buy Arrow-Debreu securities) subject to the following BC for t ≥ 0 :
Pi,t(Ci,t + Iit) +∑
st+1
Q(st, st+1)BH(st, st+1)
= wi,tli,t + ri,tki,t +BH(st−1, st)
where wi,t = wage in country i, ri,t = return to capital in country i and
Q(st, st+1) = price of the Arrow-Debreu securities in state st (at date t) that
pays one unit of the numeraire in state st+1
Household decisions
Intra-temporal allocation across goods:
cHH,t = a
(pH,t
PH,t
)−φ
CH,t, cHF,t = (1− a)
(pF,t
PH,t
)−φ
CH,t
Aggregate consumption (with λH,t =Lagrange-multiplier of BC):
uc(Cit, lit) = λH,tPit
Labor supply decision:
−ul(Cit, lit)
uc(Cit, lit)= (wH,t
PH,t)
Household decisions
Euler equations for Arrow-Debreu securities:
λH,tQ(st, st+1) = βπ(s
t)λH,t+1
⇒uc(Cit, lit)
PitQ(st, st+1) = βπ(s
t)uc(Cit+1, lit+1)
Pit+1
Symmetric expressions hold for the country F household
Risk-sharing condition
Last equation for countryH and F gives the fundamental risk sharing condition
(assuming that countries are ex-ante symmetric : λH,0 = λF,0)
λH,t = λF,t for all t⇒uc(CHt, lHt)
uc(CFt, lFt)=PH,t
PF,t= RERt
Remark: With separable utility:(CH,t
CF,t
)−σ
=PH,t
PF,t= RERt
Market clearing conditions
Market-clearing in goods market:
cHH,t + cFH,t + i
HH,t + i
FH,t = yH,t , cFF,t + c
HF,t + i
FF,t + i
HF,t = yF,t ,
Asset market-clearing condition
BH(st, st+1) +BF (st, st+1) = 0
Firms’ decisions
Firms maximize profits, taking goods and factor prices as given. Due to the
Cobb-Douglas technology, a share (1− θ) of output is paid to workers.
Thus, labor income in country i is given by:
wi,tli,t = (1− θ)pi,tyi,t,
where pi,t is the price of the country i good and wi,t is the wage in country i.
A share θ of country i output is paid a capital
ri,tki,t = θpi,tyi,t
Firms’ decisions
Investment decisions have two dimensions: firms choose aggregate invest-
ment spending Pi,tIi,t, and they decide how to allocate that spending over
Home and Foreign inputs.
For country H firms, the allocation over the two inputs must satisfy the fol-
lowing first-order conditions:
iHH,t
IH,t= a
(pH,t
PH,t
)−φ
iHF,t
IH,t= (1− a)
(pF,t
PH,t
)−φ
The symmetric applies to the Foreign country
Investment decisions (intertemporal)
Investment spending at date t must equalize the expected future marginal gain
of investment to the marginal cost at date t. So at time time t, the first-order
condition for investment in country i is:
Pi,t = Etit,t+1[pi,t+1θzi,t+1(li,t+1)
1−θ(ki,t+1)θ−1 + (1− δ)Pi,t+1] (3)
where it,t+1 is a pricing kernel (SDF) used by the firm at date t to value date
t + 1 payoffs (that are expressed in units of the country i final consumption
good) [note that the pricing kernel is the same in both countries due to perfect
risk-sharing; using Home and Foreign pricing kernel does not affect investment
decisions.
Future TFP zi,t+1 induces higher investment, while a higher price of investment
goods Pi,t discourages investment.
Backus, Kehoe and Kydland, American Economic Review, 1994
Calibration (based on quarterly data, as in BKK 1992):
Two symmetric countries (Europe and US)
Instantaneous utility: u(Cit, lit) =(Cµ
it(1−lit)1−µ)
1−σ
1−σ
c/y = 0.75; β = 0.99; δ = 0.025; θ = 0.36; σ = 2; µ = 0.34
New parameters compared BKK 1992: Import share: 15% - this matches US
but not Europe. This calibrates the home bias in preferences a = 0.85.
Elasticity of substitution between Home and Foreign goods = φ = 1.5
Backus, Kehoe and Kydland, American Economic Review, 1994
Calibration (based on quarterly data, as in BKK 1992):
Productivity shocks:
(ln(zHt+1)ln(zFt+1)
)=
(0.906 0.0880.088 0.906
)(ln(zHt)ln(zFt)
)+
(εHtεFt
)
Correl (εHt; εFt) = 0.25; var(εHt) = 0.008522
No adjustment costs.
Note: Terms-of-trade are defined as p = pF/pH
Main findings...
Following a postive Home productivity shock, Home terms-of-trade depreciate.
The depreciation of the real exchange rate coincides with a deficit of the balance
trade as the domestic investment increases. [if adjustment costs not too large
and productivity shocks sufficiently persistent]. Investment is still key for a
countercyclical trade balance unless φ is very small (roughly smaller than 1).
Implies a positive (contemporaneous) correlation between Home terms-of-trade
and (Home) net exports as in the data. After the shock, both terms-of-trade
and net exports improve (the J-curve?)
Main findings...
Compared to the one good model, because domestic and foreign goods are im-perfect substitutes, less tendency for strong negative cross-country correlationof investment (see (3)). International transmission through relative prices alsoreduces the volatility of investment within a country. These are good newscompared to the one good model.
Some international transmission through international trade - Home productiv-ity shock increases both Home and Foreign output (as Foreign TOT improves).Goes in the right direction for the "Quantity Puzzle" but effects are moderate.Risk sharing still implies strong cross-country consumption correlation (higherthan the correlation of output).
Of course, the new mechanisms rely on elasticities of substitution that are nottoo high.
New anomalies...
- ’Consumption Real Exchange Rate Anomaly’
- Terms-of-trade and real exchange rate are not volatile enough. In particular,TOT are 6 times more volatile in the data than in the benchmark calibration(unless φ is very low; but in that case quantities traded and net exports are notvolatile enough).
- International transmission relies on countercyclical terms-of-trade: weak evi-dence in the data. But need to condition for productivity shocks
Corsetti, Dedola, Leduc (2007) argues that for the US, US productivityshocks appreciates the US terms-of-trade. See also Kollmann (2007)
Acemoglu and Ventura (2002) provide evidence that higher productivitydepreciates the terms-of-trade in the long term
Robustness checks
Large spill-overs - higher direct transmission through shocks
Low elasticity of substitution between domestic and foreign goods: should make
trade link more important
High elasticity of substitution between domestic and foreign goods: should
make trade link less important
Government spending shocks: should lead to stronger output comovements
The role of incomplete markets. Does it help?
In two good model, still cross-country correlation of consumption to high and’consumption real exchange rate anomaly’.
Incomplete markets help in some dimensions but overall does not makes a bigdifference in many models....
Unless:
- Highly persistent shocks that generate strong wealth effects (Kollmann (1996),Baxter and Crucini (1995, IER) in one-good models, Corsetti, Dedola, Leduc(2007) in a two good model).
- Assumptions quite extreme about the frictions on financial markets but even inthat case terms-of-trade adjustment is a substitute for risk-sharing on financialmarkets (Cole and Obstfeld (1991)).
Corsetti, Dedola, Leduc (2007) [CDL]
Starting point = the consumption-real exchange rate anomaly ("Backus-Smith
puzzle")
Complete markets models predict (assuming symmetric countries):
uHc,t =PH,t
PF,tuFc,t
with CRRA:
(CH,t
CF,t
)−σ
=PH,t
PF,t= RERt
This is at odds with the data. For the US, CDL argues that changes in con-
sumption (relative to ROW) are positively related with a US real exchange rate
appreciation.
Key idea
Incomplete markets set-up (either bond economy as above or financial autarky)
that implies large enough wealth effect to inverse the standard international
transmission mechanism.
Wealth effects can generate an appreciation of the TOT following a Home
productivity shock. Will generate increase in Home consumption together with
an appreciation of the Home RER.
How does it work?
Suppose high productivity shock at Home. If output increases faster than
demand for Home goods (holding prices constant), then the Home terms-of-
trade will depreciate as in standard complete markets model. But suppose
that wealth effects are strong: higher Home productivity at Home increases a
lot the wealth of Home households; Aggregate Home consumption (demand)
will increase much faster than Foreign consumption. As Home consumption is
biased towards local goods, this increase more the demand for Home goods.
If this increase in demand is large enough compared to the increase in Home
output (the reason why wealth effects must be very large), excess demand for
Home goods and the Home TOT will appreciate (despite larger supply of Home
goods). This in turn reinforces the wealth effect.
When is such a mechanism possible?
Two cases:
1) either low elasticity of substitution between Home and Foreign tradable
goods. In this case, following a productivity increase, the Home terms of
trade and the RER appreciate, hurting foreign consumers. With a low price
elasticity, a terms-of-trade depreciation that reduces domestic wealth relative
to the rest of the world would actually result in a drop of the world demand for
domestic goods (because of Home bias in consumption domestic tradables are
mainly demanded by domestic households). For the world markets to clear, a
larger supply of domestic tradables must be matched by an appreciation of the
country’s terms of trade, driving up domestic wealth and demand.
2) high elasticity but highly persistent shocks (almost unit root) such that apositive Home productivity shock raises faster in the short-run demand for Homegoods (than supply) which results also in an appreciation of the Home terms-of-trade. Equivalently, because shocks are highly persistent, higher output todaymeans even more output later on (due to K accumulation). Home consumerswants to smooth this increase in consumption over time and raise consumptionnow. If this effect is sufficently large, in the SR demand for Home goods (drivenby investment and Home consumption) increase faster than Home output. TOTappreciates on impact (and depreciates in the the LR).
Note that in the middle range of elasticities of substitution, we are too closefrom the Cole and Obstfeld case: terms-of-trade act as a substitute for risk-sharing and market incompleteness does not imply wealth effects large enough.
For simplicity I will consider only the first case in a simple endowment economyunder financial autarky (see CDL, part 3.1).
A two country/two good endowment economy under financial autarky
- Same simple set-up with two periods as earlier
- Two countries Home (H) and Foreign (F ).
- Countries perfectly symmetric ex-ante
- But relax complete markets and assume financial autarky
I abstract from time indices for simplicity
Intratemporal allocation across goods (abstracting from time/state indices)
cii = a
(piPi
)−φ
Ci cij = (1− a)
(pj
Pi
)−φ
Ci
Demand of Home over Foreign goods (with market clearing conditions)
q−φΩa
[(PFPH
)φ−1PFCF
PHCH
]=yHyF
Ωu(x) is a continuous function of (u, x) with Ωu(x) =1+x(1−u
u)
x+(1−uu
)
We log-linearize the model around the symmetric equilibrium where y equalunity, and use Jonesian hats (x ≡ log(x/x)) to denote the log deviation of avariable x from its mean value x.
The international transmission mechanism (financial autarky)
Terms-of-trade and relative output
Home country’s real exchange rate RER ≡ PHPF
: RER = PHPF
= (2a− 1)q.
Assume symmetric countries, relative demand of Home over Foreign goods
becomes:
yHyF
= q−φΩa
[(PFPH
)φ−1PFCF
PHCH
]
⇒ y = −φq + (2a− 1) (φ− 1)RER+ (2a− 1) PC
⇒ y = −[φ(1− (2a− 1)2 + (2a− 1)2
]q + (2a− 1) PC
where PC = PHCH − PFCF denotes relative consumption expenditures.
Terms-of-trade and relative output
Under financial autarky, consumption expenditures = incomes: PiCi = piyi ⇒
PC = q + y
y = −[φ(1− (2a− 1)2) + (2a− 1)2
]q
︸ ︷︷ ︸substitution effect
+ (2a− 1) PC︸ ︷︷ ︸income effect
(1− (2a− 1))y = −(1− (2a− 1))[φ(1 + (2a− 1))− (2a− 1)]q
y = − [2aφ− (2a− 1)] q ⇒ q = −y
1− 2a(1− φ)
=⇒ With φ high enough (think close to unity), the international transmission
of supply (productivity) shocks is positive : a positive supply shock at Home
worsens the Home terms-of-trade and transfers income to the other country.
Terms-of-trade fluctuations provide risk-sharing.
Cole and Obstfeld (1991): an equivalence result
PC = PHCH − PFCF = q + y =2a(φ− 1)
1− 2a(1− φ)y
⇒ CH − CF =2aφ− 1
1− 2a(1− φ)y
Remind that under complete markets, we had:
(CH − CF
)CM=
(2a− 1)
σλy =
(2a− 1)
σφ(1− (2a− 1)2
)+ (2a− 1)2
y
Note that allocations coincide when a = 1/2 (no consumption Home bias) and
φ = 1 (also true for φ = σ = 1 and any a)
Cole and Obstfeld (1991): an equivalence result
=⇒ Hence despite financial autarky, the allocation is efficient (identical to the
complete markets one). At the heart of Cole and Obstfeld (1991): terms-of-
trade provide automatic insurance to supply shocks. Financial markets do not
matter
An important negative result (Cole and Obstfeld (1991)): in a two-country/two-
good model, terms-of-trade movements act as substitute for risk-sharing: the
adjustment of relative prices transmit shock from one country to the other:
countries with lower productivity enjoy higher value of output (improved terms-
of-trade). Dampens the wealth effect associated to productivity shocks (damp-
ens the mechanism through which market incompleteness matters)
Does incomplete markets matter in a two good model?
If the asset structure matters, we have to move away from the Cole and Ob-
stfeld world with terms-of-trade adjustment that exactly offsets the impact of
productivity shocks.
1) Either terms-of-trade effects are small (because goods are close substitutes)
and in that case, we are close to the one good model. Can reconcile the
"quantity puzzle" with highly persistent shocks
2) Or terms-of-trade movement are destabilizing = negative transmission
of productivity shocks: Home terms-of-trade appreciate following a positive
Home productivity shock. This might happen with a combination of the fol-
lowing ingredients: incomplete financial markets and sufficiently low elasticity
of substitution between Home and Foreign goods
Terms-of-trade and relative output
Under financial autarky, as shown earlier:
q = −y
1− 2a(1− φ)
=⇒ With incomplete markets (here financial autarky), the international trans-
mission of supply (productivity) shocks can be negative : a positive supply
shock at Home can appreciate Home terms-of-trade. This happens for a suffi-
ciently low elasticity of substitution: φ < 1− 1/2a. Intuition?
Terms-of-trade and relative output
With φ < 1 − 1/2a, holding expenditures constant(PC
), relative demand
responds very little to fall in the relative price of Home goods as it is hard
to substitute Home for Foreign goods. Income effects dominate in driving the
relative price: following a positive Home productivity shock, Home consumers
have higher income that is spent mostly on Home goods; this increases the
demand for these goods, increasing their price. TOT appreciates. In other
words, the relative demand for Home goods is upward-sloping with respect to
the relative price: as the income effect dominates, for a positive supply shock
at Home to be matched by an increase in world demand, the Home terms of
trade needs to appreciate.
Terms-of-trade and relative output
For φ > 1− 1/2a, terms-of-trade depreciate following a positive productivity
shock: as shown below, if φ < 1/2a, the fall in Home terms-of-trade is so large
that a positive Home productivity shock benefit more to Foreign consumers.
When φ > 1/2a, qualitatively similar responses as in the complete market case
(but might be different with K accumulation).
Relative consumption and RER
PC = PHCH − PFCF = q + y = 2a(1− φ)q =2a(1− φ)
2a− 1RER
⇒ CH − CF =1− 2aφ
2a− 1RER
Remind that under complete markets:
(CH − CF
)CM= −
1
σRER
Under financial autarky, if φ < 1/2a, Home real exchange rate and Home
relative consumption moves in the same direction = solution to the consumption
real exchange rate anomaly
Contrary to CM, with Home Bias, the response of relative consumption to
relative output is not always positive. In response to a positive Home supply
shock, consumption can grow less at Home than abroad: this happens when
1− 1/2a < φ < 1/2a. In that case, the adverse movement of Home terms of
trade cause ‘immiserizing growth’.
Brief summary: 3 cases
High elasticity case:
*φ > 1/2a: a positive Home output shock depreciates the Home terms of
trade, to the benefit of Foreign consumers (positive transmission). Relative
consumption and the real exchange rate are positively correlated. Both this
correlation, and international price movements, have the same sign as under
complete markets. This is the conventional view of transmission.
Low elasticity cases:
* φ < 1− 1/2a: the international transmission is negative. A positive Home
output shocks appreciates the Home terms of trade, and the real exchange rate.
Home consumption rises relative to Foreign consumption. Relative consumption
- RER correlation has the opposite sign with respect to the case of complete
markets.
*1−1/2a < φ < 1/2a: transmission is positive: actually, the fall in the Home
terms of trade is so large, that Foreign consumers benefit from a Home supply
shocks more than domestic consumers. Foreign consumption rises relative to
domestic consumption, as the Home real exchange rate depreciates.
Discussion of the quantitative model and key implications
CDL then set-up a dynamic model with incomplete markets (bond economy)
that incorporates the features presented above.
They use distribution services to generate a low trade elasticity. Needs η units of
non-tradable goods (distribution sector) to sell output. Drives a wedge between
producer and consumer prices (but still needs a low preference parameter φ to
generate appreciation of terms-of-trade following a productivity shock).
If pH = producer price of tradables, then consumer price of tradables from
country i = pi = pi+ηpN where pN =price of one unit of distrbution service
(non-tradable)
Then, intratemporal allocation across goods in country H:
cHH = a
(pHPH
)−φ
Ci cHF = (1− a)
(pFPH
)−φ
CH
⇒
(cHH
cHF
)=
(a
1− a
)(pHpF
)−φ
=(a
1− a
)(pH + ηpNpF + ηpN
)−φ
Then denoting µ = ηpNpH
, the distribution margin in setady-stade, the log-
linearization of the previous equation gives:
cHH − cHF = −φ(1− µ)(pH − pF
)= −φ(1− µ) TOT
with φ ≈ 0.8 and µ ≈ 0.5, then φ(1 − µ) ≈ 0.4. Equivalent ot a low
pass-through of exchange rates fluctuations.
According to CDL, a low trade elasticity:
1) helps to solve the "consumption-real exchange rate anomaly" and to generate
high volatility of relative prices (but low volatility of quantities traded cf. last
week).
2) Can generate appreciation of Home terms-of-trade in response to innovations
in Home productivity [here done under financial autarky but still holds in the
bond (dynamic) economy]
But...
* Large debate on the value of this elasticity: micro evidence versus macro
evidence (see also Imbs and Mejean (2008)).
* Is the appreciation of Home terms of trade following improvement in produc-
tivity a robust stylized fact? Relies on very strong wealth effects, strong home
bias in preferences (fairly closed economy) and large economy (account for a
large share of world demand).
About 2) see in particular discussions by Basu and Kollmann of "Productivity,
External Balance and Exchange Rates: Evidence on the Transmission Mecha-
nism Among G7 Countries" by CDL. Might be true for the US, hard to believe
that is a general mechanism (not that CDL focus on the US).
