International Macroeconomics
Session 12
Nicolas Coeurdacier - [email protected]
Master EPP - Fall 2014
International real business cycles with multiple goods
- Two country/two good endowment economy
- Two country/two good RBC model with complete markets
Backus, Kehoe, Kydland (AER, 1994); Heathcote and Perri (Handbook, 2013).
- The role of incomplete markets in a two good model.
Revisiting the role-of-terms of trade adjustment. Cole and Obstfeld (1991).
Corsetti, Dedola and Leduc (2007)
Relaxing the assumption of a single perfectly tradable good?
So far, model of intertemporal trade through trade in securities: just one good,
no interesting intratemporal trade!
Model that has nothing to say about...
- International transmission through trade in goods
- Variations in relative prices.
- About the real exchange rate
Relaxing the assumption of a single perfectly tradable good?
Types of multi-good models
- Trade models : Ricardian / H0S / Intra-industry trade models
- Traded - Non-traded goods models
Business cycle literature has focused on:
- Specialization (Ricardian model of trade) (Backus, Kehoe and Kydland, 1994)
- Traded - Non-Traded (Backus and Smith, 1993)
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 consumption 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
complete 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)
Abstract from time indices for simplicity.
Aggregate consumption index
(φ−1)/φ φ/(φ−1)
1/φ
+ (1 − a)
(cHF )
φ/(φ−1)
a1/φ (cF F )(φ−1)/φ + (1 − a)1/φ (cF H )(φ−1)/φ
a1/φ (cHH )(φ−1)/φ
CH =
CF =
with φ elasticity of substitution between the two goods, and cij = consumption
of good j by country i, i = H, F
Home bias in preferences a ≥ 12 . With a = 12 , identical preferences ⇒ a
homogenous to goods market integration. With “Cobb-Douglas” preferences,
(φ = 1), a = share of consumption spending devoted to local good
Price indices
PH =
PF =
a (pH )1−φ + (1 − a) (pF )1−φ
(1 − a) (pH
)1−φ
+ a (pF
1/(1−φ)
1/(1−φ)
1−φ
)
Terms of trade: q = relative price of Home goods over Foreign goods
pH
q≡
pF
Real Exchange Rate
PF
RER =
=
PH
(1 − a)q 1−φ + a
aq 1−φ + (1 − a)
1/(1−φ)
Intertemporal Utility
Ui = u(C0,i) + βE0u(C1,i) = u(C0,i) + β
s
π(s)u(C1,i(s))
Ct,i=aggregate consumption at date t in country i and 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 −
s
pb(s)bi(s)
P1,iC1,i(s) = p1,iy1,i(s) + bi(s)
λi0
βλi1(s)
Resource constraints (at both dates and in all states at t=1)
F
cH
H + cH = yH
;
H
cF
F + cF = yF
with yi endowment of country i
Asset market clearing condition
bi(s) = 0
i
Euler equation for Arrow-Debreu Securities
u′(C0,i) = λi0P0,i
⇒
u′(C0,i)
pb(s) P
0,i
=
u′(C1,i(s))
βπ(s) P
1,i
π(s)u′(C1,i(s)) = λi1(s)P1,i
⇒ pb(s) =
u′(C1,i)/P1,i
βπ(s) u′(C )/P
0,i
0,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′(C1,F )/P1,F
= ′
⇒
u′(C0,H )/P0,H
u (C0,F )/P0,F
C1,H −σ
P1,H /P1,F
=
C0,H
P0,H /P0,F
C1,F −σ
C0,F
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 addressing the problem of (too) high cross-country consumption correlations (’Quantity
Puzzle’)
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 real
exchange rate fluctuations, CRRA preferences & separability between leisure
and consumption.
Remark 1: If countries are symmetric ex-ante (q0 = 1):
P1,F
C1,F −σ
=
= RER
C1,H
P1,H
Remark 2: extension to a multi-period model?
"The consumption-real exchange rate anomaly"
in the table real exchange rate is RERt =PF/PH . Source: Corsetti et al.
Intratemporal allocation across goods (abstracting from time/state indices)
pi
cii = a
Pi
−φ
Ci
pj −φ
cij = (1 − a)
Ci
Pi
Demand of Home over Foreign goods (with market clearing conditions)
P
C
y
q−φΩa ( F )φ F = H
PH CH
yF
Ωu(x) is a continuous function of (u, x) with Ωu(x) =
1+x( 1−u
u )
x+( 1−u
u )
Remark: if a = 1/2, then Ω1/2(x) = 1 for all x and yyH = q−φ. Terms of
F
trade decreases with an elasticity 1/φ with respect to increase in output at
Home 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 ≡ yyH to denote relative outputs in both countries.
F
We log-linearize the model around the symmetric equilibrium where y equal
unity, and use Jonesian hats (x ≡ log(x/x̄)) 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
Assume symmetric countries:
CH
CF
−σ
PH
yH
PF φ−1/σ
−φ
=
⇒
= q Ωa (
)
PF
yF
PH
(1)
PF
Home country’s real exchange rate RER ≡ P
:
H
PH
RER = −
= −(2a − 1)q.
PF
(2)
Log-linearizing (1) and using (2) implies:
y = −φq − (2a − 1) φ −
1 − (2a − 1)2
1
RER = −λq
σ
(2a−1)2
+ σ .
where λ ≡ φ
Note that λ > 0 as 1/2 < a < 1.
A relative increase in the supply of the home good (ŷ > 0) is always associated
with a worsening of the terms of trade (q̂ < 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 (productivity) shocks is always positive : a positive supply shock at Home unambiguously worsens Home terms-of-trade.
The international transmission mechanism
Relative consumption and relative output
1 PH
(2a − 1)
(2a − 1)
CH − CF = −
=−
q ⇒ CH − CF =
y
σ PF
σ
σλ
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 movements cause ‘immiserizing growth’. In response to a positive supply shocks,
domestic consumption will never fall either in absolute level, or relative to Foreign 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 = pH yH − PH CH = −NXF ; NXF = pF yF − PF CF
1
q
1
NXH =
qy − PH CH − PF CF =
(1 − λ) − (2a − 1)(1 − )
2
2
σ
1
NXH = q(1 − a) 1 − φ + (2a − 1) ( − φ)
σ
y
1
(1 − a) φ − 1 + (2a − 1) (φ − )
NXH =
λ
σ
Net exports are procyclical unless the elasticity of substitution (φ) between
Home and Foreign goods is very low. What does it mean? [recall that we do
not have investment]
Terms-of-trade volatility and trade 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
In the data, terms-of-trade are very volatile around 2 times more volatile than
output (low φ?). But so are (real) 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?
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 will introduce a wedge between marginal utilities of consumption 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.
A benchmark two-country/two good IRBC model with production
Two symmetric countries, Home (H) and Foreign (F ), each with a representative 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:
∞
β tu(Cit, lit) =
Ui = E0
∞
π(st)β tu(Cit, lit)
t=0 st
t=0
where Ci,t is i’s aggregate consumption and lit is labor supplied by the representative household in country i
examples: u(Cit, lit) =
1−σ
Ci,t
[ 1−σ
1+ω
li,t
− ν 1+ω ]
1−σ
BKK: u(Cit, lit) =
(Citµ (1−lit)1−µ)
1−σ
Consumption and price index
CH,t =
a1/φ
(φ−1)/φ
H
cH,t
+ (1 − a)1/φ
(φ−1)/φ φ/(φ−1)
H
cF,t
CF,t =
a1/φ
(φ−1)/φ
F
cF,t
+ (1 − a)1/φ
(φ−1)/φ φ/(φ−1)
F
cH,t
where cij,t is country i′s consumption of the good produced by j; 12 < a < 1.
PH,t =
PF,t =
a pH,t
1−φ
+ (1 − a) pF,t
(1 − a) pH,t
1−φ
+ a pF,t
1−φ 1/(1−φ)
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
(φ−1)/φ
iH
H,t
+ (1 − a)1/φ
(φ−1)/φI φ/(φ−1)
H
iF,t
IF,t =
a1/φ
(φ−1)/φ
iF
F,t
+ (1 − a)1/φ
(φ−1)/φ φ/(φ−1)
F
iH,t
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
Household decisions
∞
E0
β tu(Cit, lit)
t=0
∞
=
π(st)β tu(Cit, lit)
t=0 st
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 :
Q(st, st+1)BH (st, st+1)
Pi,t(Ci,t + Iit) +
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
Aggregate consumption (with λH,t =Lagrange-multiplier of BC):
uc(Cit, lit) = λH,tPit
Intra-temporal allocation across goods due to CES preferences:
−φ
p
H,t
cH
CH,t,
H,t = a
PH,t
−φ
p
F,t
cH
CH,t
F,t = (1 − a)
PH,t
Household decisions
Relative demand:
cH
H,t
pH,t −φ
a
=
H
1 − a pF,t
cF,t
Standard CES demand:
- increase in a raise the relative demand for local goods
p
- increase in pH,t lowers (relative) demand of local goods with a constant price
F,t
elasticity φ.
Household decisions
Labor supply decision:
wH,t
ul(Cit, lit)
)
−
=(
uc(Cit, lit)
PH,t
Example: u(Cit, lit) =
1−σ
Ci,t
[ 1−σ
1+ω
li,t
− ν 1+ω ]
w
ω = C −σ ( H,t )
⇒ νli,t
i,t P
H,t
- Increase in real wage increases labor supply with an elasticity 1/ω
- Increase in consumption (or increase in wealth) lowers marginal utility of
consumption and labor supply (wealth effect of the labor supply)
Household decisions
Euler equations for Arrow-Debreu securities:
λH,tQ(st, st+1) = βπ(st+1)λH,t+1
uc(Cit, lit)
uc(Cit+1, lit+1)
t
t+1
⇒
Q(s , st+1) = βπ(s )
Pit
Pit+1
Symmetric expressions hold for the country F household
Fundamental Risk-sharing Condition
Last equation for country H and F gives (assuming that countries are ex-ante
symmetric : λH,0 = λF,0):
PH,t
uc(CHt, lHt)
1
λH,t = λF,t for all t ⇒
=
=
uc(CF t, lF t)
PF,t
RERt
This is the fundamental risk-sharing condition with real exchange rate fluctuations; appears in most two-country models with perfect risk-sharing (complete
markets)
Remark: With separable utility:
CF,t −σ
PF,t
=
= RERt
CH,t
PH,t
Fundamental Risk-Sharing Condition
Remark: With separable utility:
CF,t −σ
PF,t
=
= RERt
CH,t
PH,t
This is the well-know ’Kollman-Backus-Smith’ relationship that links the real
exchange rate to the ratio of consumption. Note if a = 1/2, then RERt = 1,
we are back to the risk-sharing condition in a one good economy. With different
consumption baskets, the real exchange rate introduces a wedge between the
ratio of marginal utilities. Intuition?
Good news? Not really since relative consumption and real exchange rate are
not correlated in the data (see below) = the consumption-real exchange rate
anomaly
Market clearing conditions
Market-clearing in goods market:
F
H
F
cH
H,t + cH,t + iH,t + iH,t = yH,t ,
H
F
H
cF
F,t + cF,t + iF,t + iF,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. Factors of
production are paid their marginal productivity
wi,t = pi,t
∂yi,t
∂yi,t
; ri,t = pi,t
∂li,t
∂ki,t
where pi,t is the price of the country i good, wi,t is the wage in country i and
ri,t the rental rate of capital.
Due to the Cobb-Douglas technology, a share (1 − θ) of output is paid to
workers and a share θ to capital:
wi,tli,t = (1 − θ)pi,tyi,t,
ri,tki,t = θpi,tyi,t
Firms’ decisions
Investment decisions have two dimensions: firms choose aggregate investment 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 following first-order conditions:
iH
H,t
pH,t −φ
= a
IH,t
PH,t
iH
pF,t −φ
F,t
= (1 − a)
IH,t
PH,t
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 = Et̺it,t+1[pi,t+1θzi,t+1(li,t+1)1−θ (ki,t+1)θ−1 + (1 − δ)Pi,t+1] (3)
λ
where ̺it,t+1 = β λi,t+1 is the pricing kernel used by the firm at date t to value
i,t
date t+1 payoffs (that are expressed in units of the country i final consumption
good)
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
1−σ
Instantaneous utility: u(Cit, lit) =
(Citµ (1−lit)1−µ)
1−σ
Calibration:
US vs RoW (apart from technology shocks which are calibrated to US vs.
Europe). Parameters similar to BKK 1992 are set to their benchmark value.
No adjustment costs.
New parameters: 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
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
Note: Terms-of-trade are defined as
p = pF /pH . Think of RER.
Comments
Let us consider a transitory productivity shock in the Home country. The
increase in the Home production induces a decrease in the relative price of the
domestic good: Home terms-of trade depreciates (like in the simple endowment
economy). The real exchange of the Home country depreciates. It induces an
increase in the demand for the Home good in both countries: « expenditures
switching effect ». The higher the elasticity of substitution between goods, the
lower the depreciation of the real exchange rate (or TOT).
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
pH /pF and (Home) net exports as in the data. After the shock, both termsof-trade and net exports improve (the J-curve?)
Because domestic and foreign goods are imperfect substitutes, less tendency
for strong negative cross-country correlation of investment (see Eq. (3)). International transmission through relative prices also reduces the volatility of
investment within a country. These are good news compared to the one good
model.
Some international transmission through international trade - Home productivity 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 (much
higher than the correlation of output) but this relies on a specific calibration
(see below)
Good news about international comovements and the quantity puzzle
Heathcote and Perri (2013).
BKK cannot reproduce international co-movements and fail on the quantity
puzzle for 3 joint reasons:
- too high elasticity of substitution φ
- international spillovers: good productivity shock in H is good news about
future productivity in F and both countries want to consume more today.
Tend to equalize consumption
- non-separability between leisure and consumption: ties together comovement
in labor and consumption. If both countries work more in response to a shock,
marginal utility of consumption in both countries rise = additional force towards
equalizing consumption.
Good news about international comovements and the quantity puzzle
Heathcote and Perri (2013).
Relaxing the three altogether can bring us very close to the data (at least for
quantities).
Many authors have interpreted international business cycle co-movement as
evidence against efficiency.
Under reasonable parameters, observed co-movement consistent with efficiency.
Not much of a "Quantity Puzzle".
ρ =shock persistence; σ = elasticity of sub. (φ); ψ = spillover parameter. Source
H&P(2013)
Key intuition. Heathcote and Perri (2013)
Country H (resp. F ) produces aluminium (resp. bricks), also major consumer
of it (home bias). If aluminium/bricks poor substitutes (and EIS high), then,
when aluminum production ↑, efficient not too transfer any aluminum, rather
H imports more bricks from F and consumption ↑ more than output. Opposite
happens when bricks ↑.
So corr(cH , cF ) < corr(yH , yF ), corr(nx, y) < 0
Stable mix between aluminium and bricks more valuable than stable consumption. To make countries not willing to smooth consumption, relative prices
(terms-of-trade) adjust more.
New anomalies
As in the endomwent economy case, complete markets models predict (assuming symmetric countries):
uH
c,t =
with CRRA:
PH,t F
uc,t
PF,t
CF,t −σ
PF,t
=
= RERt
CH,t
PH,t
This is at odds with the data = the consumption-real exchange rate anomaly
("Kollmann-Backus-Smith puzzle").
New anomalies
- In BKK, terms-of-trade and real exchange rate are not volatile enough compared to the data in the benchmark calibration (especially TOT which are 6
times more volatile in the data!)
Remark: Terms-of-trade vs trade quantities volatility. Low
are also very volatile (even though net exports is not)
φ helps but real quantities traded
- 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). International transmission relies on countercyclical terms-oftrade: weak evidence in the data. But need to condition for productivity shocks
Corsetti, Dedola, Leduc (2007) argues that for the US, US productivity shocks appreciates
the US terms-of-trade. See also Kollmann (2007)
Acemoglu and Ventura (2002) provide evidence that higher productivity depreciates the
terms-of-trade in the long term
Does incomplete markets matter in a two good model?
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 (dampens the
mechanism through which market incompleteness matters). In the Cole and
Obstfeld economy, it is possible that consumption under financial autarky is
strictly identical to the one under complete markets!
Cole and Obstfeld (1991) versus Corsetti, Dedola, Leduc (2007) [CDL]
If the asset structure matters, we have to move away from the Cole and Obstfeld 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 (CDL (2007)). This might happen with a combination
of the following ingredients: incomplete financial markets and low elasticity of
substitution between Home and Foreign goods (also possible with high elasticity
of substitution and highly persistent shocks (=strong wealth effects))
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)
pi
cii = a
Pi
−φ
Ci
pj −φ
cij = (1 − a)
Ci
Pi
Demand of Home over Foreign goods (with market clearing conditions)
P
P C
y
q−φΩa ( F )φ−1 F F = H
PH
PH CH
yF
Ωu(x) is a continuous function of (u, x) with Ωu(x) =
1+x( 1−u
u )
x+( 1−u
u )
We log-linearize the model around the symmetric equilibrium where y equal
unity, and use Jonesian hats (x ≡ log(x/x̄)) to denote the log deviation of a
variable x from its mean value x.
The international transmission mechanism (financial autarky)
Terms-of-trade and relative output
H : RER = PH = (2a − 1)q.
Home country’s real exchange rate RER ≡ P
P
P
F
F
Assume symmetric countries, relative demand of Home over Foreign goods
becomes:
yH
yF
PF φ−1 PF CF
=
(
)
PH
PH CH
⇒ y = −φq + (2a − 1) (φ − 1)RER + (2a − 1) P C
⇒ y = − φ(1 − (2a − 1)2 + (2a − 1)2 q + (2a − 1) P C
q−φΩa
where P C = PH CH − PF CF 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 + (2a − 1) P C
substitution effect
income effect
(1 − (2a − 1))y = −(1 − (2a − 1))[φ(1 + (2a − 1)) − (2a − 1)]q
y
y = − [2aφ − (2a − 1)] q ⇒ q = −
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
P C = PH CH − PF CF = q + y =
⇒ CH − CF =
2a(φ − 1)
y
1 − 2a(1 − φ)
2aφ − 1
y
1 − 2a(1 − φ)
Remind that under complete markets, we had:
CH − CF
CM
(2a − 1)
(2a − 1)
=
y=
y
2
2
σλ
σφ 1 − (2a − 1) + (2a − 1)
Note that allocations coincide when a = 1/2 (no consumption Home bias) and
φ = 1 (or for φ = σ = 1 and any a). This is Cole and Obstfeld (1991)’result.
If the asset structure matters, we have to move away from the Cole and Obstfeld
world.
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 transmission of supply (productivity) shocks can be negative : a positive supply
shock at Home can appreciate Home terms-of-trade. This happens for a sufficiently low elasticity of substitution: φ < 1 − 1/2a. Intuition?
Terms-of-trade and relative output
With φ < 1 − 1/2a, holding expenditures constant P C , 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
2a(1 − φ)
P C = PH CH − PF CF = q + y = 2a(1 − φ)q = −
RER
2a − 1
1 − 2aφ
⇒ CH − CF = −
RER
2a − 1
Remind that under complete markets:
1
RER
σ
Under financial autarky, if φ < 1/2a, Home real exchange rate and Home
relative consumption moves in opposite direction = solution to the consumption
real exchange rate anomaly
CH − CF
CM
=
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.
How does incomplete markets matter?
Suppose high productivity shock at Home. If output increases faster than
demand for Home goods (holding prices constant), then the Home terms-oftrade 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 a
positive Home productivity shock raises faster in the short-run demand for Home
goods (than supply) which results also in an appreciation of the Home terms-oftrade. Equivalently, because shocks are highly persistent, higher output today
means even more output later on (due to K accumulation). Home consumers
wants to smooth this increase in consumption over time and raise consumption
now. If this effect is sufficiently large, in the SR demand for Home goods (driven
by investment and Home consumption) increase faster than Home output. TOT
appreciates on impact (and depreciates in the the LR).
Note that in the middle range of elasticities of substitution, we are too close
from the Cole and Obstfeld’s world: terms-of-trade act as a substitute for risksharing and market incompleteness does not imply wealth effects large enough.
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 distribution service
(non-tradable)
N
Then denoting µ = ηp
pH , the distribution margin in steady-stade, the loglinearization of the previous equation gives:
cHH − cHF = −φ(1 − µ) pH − pF = −φ(1 − µ)q
with φ ≈ 0.8 and µ ≈ 0.5, then φ(1 − µ) ≈ 0.4. Equivalent of 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).
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 productivity 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 Mechanism 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).