ECO419:
INTERNATIONAL
MACROECONOMICS
Joseph Steinberg
Associate Professor
Economics Department
University of Toronto
Today’s agenda
• Recap from last time
• Developed model of international borrowing and lending in a “small
open endowment economy”
• Analyzed response of model economy to external shocks, namely
to GDP
• Uncertainty and international borrowing/lending (Ch 6)
• The “Great Moderation”
• Risk aversion, prudence, and precautionary saving
• Implications of uncertainty for current account/trade balance
• Uncertainty and growth
Graphical depiction of equilibrium
Graphical depiction of equilibrium
TB1=Q1-C1<0
Graphical depiction of equilibrium
TB2=Q2-C2>0
TB1=Q1-C1<0
The Euler Equation
• Solution to model characterized by Euler equation
U1 (C1 , C2 ) = (1 + r )U 2 (C1 , C2 )
u '(C1 ) =  (1 + r )u '(C2 )
• Intuition: Must be indifferent between marginal
reallocation of consumption across time
• Could give up one unit of consumption today in exchange for (1+r)
units tomorrow
• This reallocation would lower lifetime utility by U1 (C1 , C2 ) and raise it
by (1 + r )U 2 (C1 , C2 )
• This option is always affordable (satisfies the intertemporal BC), so
if the net effect was positive, the household should have chosen
that allocation to begin with!
The Great Moderation
• U.S. macroeconomic volatility fell dramatically after early
1980s
• Called “Great Moderation”
• Some macroeconomists claimed we had “solved” business cycles
• U.S. trade balance/current account started to fall about
the same time volatility did
• Cross-country data shows similar story
• Perri and Fogli (2015) show changes in volatility correlated with
changes in NIIP
• Countries with increased volatility saw NIIP rise
• Countries with reduced volatility saw NIIP fall
• Can we make sense of this relationship in our model?
YoY growth rate of U.S. GDP per capita
Source: IM via U.S. Bureau of Economic Analysis
U.S. trade balance as pct. Of GDP
Source: IM via U.S. Bureau of Economic Analysis
Volatility and NFA across countries
Source: Perri and Fogli (2015)
Volatility and NFA across countries
Source: Perri and Fogli (2015)
Uncertainty in a small open economy
• Same model structure as last time, with two differences
1. GDP endowment in period 2 is uncertain
Q1 = Q,
2.
Q + 
Q2 = 
Q − 
with prob 0.5
with prob 0.5
Households have expected utility preferences
U (C1 , C2 ) = u (C1 ) + E u (C2 ) 
• To keep things simple, we assume
• no discounting: 𝛽 = 1
• Interest rate is zero: 𝑟 = 0
• initial NFA is zero: 𝐵0 = 0
Uncertainty in a small open economy
• The parameter σ is the standard deviation of period-2
output
• Increasing σ is analogous to increasing the volatility of the
economy’s GDP
• Note also that the expected value of period-2 output is
E Q2  = 0.5  (Q +  ) + 0.5  (Q −  ) = Q
• So increasing σ does not affect the average level of period-
2 output
• Thus, it does not affect the expected present value of period-2 output
• If there were no uncertainty, TB1 would be zero because there would
be no incentive to borrow or lend… autarky smooths consumption
perfectly
Uncertainty in a small open economy
• We cannot use intertemporal BC approach to solving for
this version of the small open economy model because
there are two versions of period-2 budget constraint
good state : C2 = Q +  + B1
bad state : C2 = Q −  + B1
• Use period-1 BC to substitute out B1:
𝐵1 = 𝑄 − 𝐶1
↓
good state: 𝐶2 = 2𝑄 + 𝜎 − 𝐶1
bad state: 𝐶2 = 2𝑄 − 𝜎 − 𝐶1
Equilibrium with uncertainty
• Household’s lifetime utility maximization problem:
max C1 u (C1 ) + 0.5u (2Q +  − C1 ) + 0.5u (2Q −  − C1 )
• Solution characterized by Euler equation:
u '(C1 ) = 0.5u '(2Q +  − C1 ) + 0.5u '(2Q −  − C1 )
= E u '(C2 )
• To go further, must put some structure on the period utility
function little-u…
Solution with quadratic utility
• Suppose that 𝑢 𝐶 = −0.5 𝐶ҧ − 𝐶
2
• Then marginal utility given by 𝑢 ′ 𝐶 = 𝐶 − 𝐶ҧ
• Euler equation:
𝐶1 − 𝐶ሜ = 0.5 2𝑄 + 𝜎 − 𝐶1 − 𝐶ሜ + 0.5 2𝑄 − 𝜎 − 𝐶1 − 𝐶ሜ
• Simplifies to
C1 = Q
• What does this imply about the trade balance in period 1?
• It is balanced! TB1=0
• No borrowing or lending
• Autarky allocation
• Implies 𝐶2 = 𝑄2 in both good and bad states of the world
Solution with quadratic utility
• Note that this solution obtains regardless of σ
• Suppose that σ is very high… wouldn’t you be willing to
save a little bit to insure against the bad state of the
world?
• Recall that
good state : C2 = Q +  + B1
bad state : C2 = Q −  + B1
• So if we save a little bit in period 1, consumption rises in
both states in period 2
• We call this precautionary saving
Aside: risk aversion and prudence
• Intuitive idea: given two
options
• 1) certain payoff x
• 2) uncertain payoff y with E[y]>x
• People may prefer x to y,
especially if y is very risky
• Decreasing marginal utility:
loss from bad outcome larger
than gain from good outcome
Aside: risk aversion and prudence
• You may have seen Arrow-Pratt coefficients of absolute
and relative risk aversion:
−u ''(C )
−Cu ''(C )
A(C ) =
, R(C ) =
u '(C )
u '(C )
• Relative risk aversion interpretation:
du '(C ) / dC
u '(C )
du(C ) / u (C )
=
dC / C
%u '(C )
=
%C
= elasticity of MU w.r.t C
R(C ) = −C
Aside: risk aversion and prudence
• Quadratic utility actually does exhibit risk aversion by
these measures
R(C ) =
−Cu ''(C )
−1
C
= −C
=
0
u '(C )
C −C C −C
• Concave function, so has diminishing marginal utility
• Thus, risk aversion is not enough to deliver a
precautionary saving motive
Aside: risk aversion and prudence
• Coefficient of relative prudence:
−Cu '''(C )
P(C ) =
u ''(C )
• Note u '''(C ) = 0 for quadratic utility, hence no prudence, and
no precautionary saving!
1−
• What about power utility: u (C ) = C / (1 −  )
•  is coefficient of relative risk aversion, hence “CRRA
utility”
u '(C ) = C − , u ''(C ) = − C − −1 , u '''(C ) =  ( + 1)C − − 2
−C   ( + 1)C − − 2 ( + 1)C − −1
P(C ) =
=
=  +1
−  −1
−  −1
− C
C
Solution with log utility
• Log utility U (c) = log(c) is special case of power utility with
 = 1 (risk averse AND prudent)
• Euler equation:

1 1
1
1
= 
+

C1 2  2Q +  − C1 2Q −  − C1 
• Does not have analytical solution, however we can easily
figure out the qualitative implications
Solution with log utility, continued
• Suppose that, just as with quadratic utility, the household
chose not to do any precautionary saving, i.e., C1 = Q
• Plug this into the Euler equation:

1 1
1
1
= 
+
Q 2  2Q +  − Q 2Q −  − Q 
1 1
1 
= 
+
2  Q +  Q −  

1
Q −
Q +
= 
+
2  (Q +  )(Q −  ) (Q +  )(Q −  ) 
=
1  2Q 
2  Q 2 −  2 
Solution with log utility, continued
• Simplify further:
Q2
1= 2
Q − 2
• Plainly not true as long as   0 : contradiction! Proves C1  Q
• Note that in actuality we have
Q2
1 2
Q − 2
• LHS less than RHS: marginal util in first period less than
expected marginal util in second period
• Household would be better off by saving, i.e., setting
C1  Q, thereby setting TB1  0 and B1  0
Changing the volatility of Q2
 → 0 precautionary saving motive
disappears… inequality on previous slide gets closer and
closer to equality
• Conversely, as  →  inequality becomes more severe,
hinting that more precautionary saving is optimal
• If σ fell from a high value to a low value, we would expect
to see less precautionary saving, i.e., we would expect
the trade balance to fall
• Or, if we compared two countries with different values of
σ, the higher-σ country should have a higher trade
balance, all else equal
• And this is what we saw in the data!
• Note that as
Incorporating growth and uncertainty
• Fast-growing countries like China are also more volatile
than developed countries
Productivity volatility in France and China
Incorporating growth and uncertainty
• Fast-growing countries like China are also more volatile
than developed countries
• One on hand, one might expect rapid growth to lead such
countries to borrow to smooth consumption
• Not what we saw in the data: fast-growing countries tend
to lend, not borrow!
Fast-growing countries borrow, not lend!
Y-axis:
sum of
current
account
deficits as
pct of 1980
GDP
X-axis:
Annualized average
productivity growth
between 1980 and 2000
Incorporating growth and uncertainty
• Fast-growing countries like China are also more volatile
•
•
•
•
than developed countries
One on hand, one might expect rapid growth to lead such
countries to borrow to smooth consumption
Not what we saw in the data: fast-growing countries tend
to lend, not borrow!
On the other hand, though, more volatility should lead to
more precautionary saving
Growth and uncertainty have opposing impacts on
equilibrium borrowing/lending behavior… could this help
resolve the puzzle?
Uncertainty AND growth in an SOE
• Two-period model
• Period 1: Q1 = 1, B0 = 0
2
• Period 2: Q2 ~ N (  ,  ),   1
• World interest rate r = 0
• CARA (constant absolute risk aversion) preferences:


max C1 ,C2 , B1 −e − C1 + E  −e − C2 
subject to
C𝐶11=+1𝐵
+1B=
1 1
c𝐶1 2+ =
B1𝑄=2 Q
+2 1
Uncertainty AND growth in an SOE
• Euler equation:
e− C1 = E e− C2 
• Substitute budget constraints:
C2 = Q2 + 1 − C1
 e −C1 = E e − (Q2 +1−C1 ) 
• Rearrange:
e−C1 = E eC1 e −1e − Q2  = eC1 e−1E e − Q2 
Uncertainty AND growth in an SOE
• Note about normal variables: if X is normally distributed
with mean  and variance  2 , then
E e  = e
X
 + 2 / 2
• Noting that −Q2 ~ N ( −  ,  ), we have
2
e
− C1
= e e E e
C1 −1
− Q2
C1 −1 −  + 2 /2
 = e e e
• Now we can just rearrange:
−C1 = C1 − 1 −  +  2 / 2
2C1 = 1 +  −  2 / 2
1+   2
C1 =
−
2
4
Uncertainty AND growth in an SOE
• Solution:
1+   2
C1 =
−
,
2
4
1−   2
TB1 = Q1 − C1 =
+
2
4
• Deterministic (𝜎 2 ) world mirrors IM Chapter 3 logic: higher
μ (faster growth) means more larger trade deficit (more
borrowing)
• No-growth (𝜇 = 1) world mirrors IM Ch. 6: more volatility
(higher σ) means larger trade surplus (more saving)
• With growth and uncertainty, it depends on how much
uncertainty there is and the speed of growth
• Even with very rapid growth in mean income, if income is
also very volatile the country may want to save