ECON7520: Uncertainty and the Current
Account
Semester 1, 2023
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Motivation: The Great Moderation in the U.S.
Quarterly Real Per Capita U.S. GDP Growth, 1947Q2 - 2017Q4
The volatility of the growth rate of real U.S. GDP per capita
declined significantly after (about) 1984.
This has become known has the Great Moderation.
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Motivation: The Great Moderation in the U.S.
Three proposed explanations for the Great Moderation:
1 Good-luck hypothesis: by chance, the U.S. economy has
been blessed with smaller shocks.
2 Good-policy hypothesis: volatility declined due to
good monetary policy (aggressive low inflation policy of
Volcker and Greenspan Feds)
the phasing out of Regulation Q’s interest rate ceilings for
most types of bank deposits (such ceilings may cause
negative real interest rates on deposits if inflation is high,
causing bank deposits withdrawals, consequently reduced
bank loans, and finally a credit-crunch-induced recession)
3 Structural change hypothesis: the amplitude of business
cycles was reduced by structural change, particularly in
inventory management and in the financial sector.
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Motivation: The Great Moderation in the U.S.
We will not examine the merits of these explanations.
Instead, we are curious about possible connections of the
Great Moderation with the deterioration of the U.S. CA
balance from the 1980s.
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Motivation: The Great Moderation in the U.S.
U.S. CA-to-GDP Ratio, 1947Q1 - 2017Q4
1947-1983: on average, CA surplus of 0.34% of GDP;
1984-2017: on average, CA deficit of 2.8% of GDP.
Any relationship with the volatility of the GDP growth?
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Questions in Today’s Lecture
Questions:
Is the timing of the Great Moderation and the emergence
of protracted current account deficits pure coincidence?
Is there a casual connection between the two?
We will explore the effects of changes in output uncertainty on
the trade balance and the current account.
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ECON7520
Open Economy Model with Uncertainty
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Uncertainty and Household’s Saving
How does uncertainty affect the current account?
Potentially through the household saving channel.
How does uncertainty affect household’s saving?
Through a motive for precautionary saving.
1 High uncertainty about the future
People save more. The CA improves.
2 Low uncertainty about the future (Great Moderation)
People save less. The CA deteriorates.
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Setup
Two-period small open economy: periods 1 and 2.
The single consumption good is perishable.
The single asset traded in the financial market is a bond
(measured in units of the consumption good).
There is a representative household (HH) endowed with
B0 = 0 units of the bond at the beginning of period 1.
units of the good as described on the next slide.
Interest Rates:
r0 for the initial bond holdings,
r1 = 0 for the bonds held at the end of period 1.
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Endowment
Endowment with units of the good: The household is
endowed with:
In period 1: Q1 = Q units of the goods.
In period 2: Q2 =
{
Q + σ with probability 1
2
(good state),
Q − σ with probability 1
2
(bad state).
We assume σ ≥ 0.
If σ > 0 then the amount of endowment in period 2 is
uncertain.
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Expected Utility Assumption
The HH cares about expected utility from consumption,
U(C1,C2(·)) = lnC1 + E [lnC2] .
If there are a good state (G) and a bad state (B) in period
2, the expected utility is
U(C1,C2(·)) = lnC1 + [piG lnC2(G) + piB lnC2(B)] ,
where
piG is the probability of the good state,
piB is the probability of the bad state,
C2(G) is consumption in the good state,
C2(B) is consumption in the bad state.
We assume piG = piB =
1
2 so that
U(C1,C2(·)) = lnC1 +
[
1
2
lnC2(G) +
1
2
lnC2(B)
]
.
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Household’s Budget Constraint
The HH’s budget constraints are
C1 + B1 = (1+ r0) · 0+ Q
C2(G) = (1+ 0) · B1 + Q + σ
C2(B) = (1+ 0) · B1 + Q − σ
In the good state where Q2 = Q + σ, therefore
C2(G) = 2Q + σ − C1. (1)
In the bad state where Q2 = Q − σ, therefore
C2(B) = 2Q − σ − C1. (2)
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Utility Maximization
The household’s utility maximization problem is:
max
C1,C2(G),C2(B)
(
lnC1 +
[
1
2
lnC2(G) +
1
2
lnC2(B)
])
subject to (1), (2) (3)
Substituting (1) and (2) into (3), we obtain
max
C1
(
lnC1 +
[
1
2
ln (2Q + σ − C1) +
1
2
ln (2Q − σ − C1)
])
(4)
The only one unknown variable in (4) is C1.
The household picks C1 to maximize her expected utility.
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Math Review: Maximization
Necessary and Sufficient Conditions for a Maximum
Necessary condition: If x∗ is a local maximum of f then
f
′(x∗) = 0.
If f ′′(x) < 0 for all x then f is strictly concave.
Sufficient condition: If f is strictly concave and f ′(x∗) = 0
then x∗ is a global maximum of f .
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Utility Maximization
Let
f (C1) = lnC1 +
[
1
2
ln (2Q + σ − C1) +
1
2
ln (2Q − σ − C1)
]
Then
f
′(C1) =
And
f
′′(C1) =
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Utility Maximization
The objective function f of (4) is strictly concave.
Therefore, C1 such that f
′(C1) = 0 maximizes the HH’s
expected utility.
f ′(C1) = 0 is equivalent to
1
C1
− 1
2
[
1
2Q + σ − C1
+
1
2Q − σ − C1
]
= 0.
or
1
C1
=
1
2
[
1
2Q + σ − C1
+
1
2Q − σ − C1
]
(5)
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Case w/o Uncertainty
We start with the case w/o uncertainty.
Suppose that σ = 0 (thus no uncertainty) then (5) implies
1
C1
=
1
2
[
1
2Q − C1
+
1
2Q − C1
]
and thus
C1 = Q.
Then (1) and (2) imply that
C2(G) = C2(B) = 2Q − C1 = Q.
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Case w/o Uncertainty
Therefore, C1 = C2(G) = C2(B) = Q — consumption in
periods 1 and 2 is the same.
TB1 = Q − C1 = 0.
CA1 = r0B0 + TB1 = TB1 = 0.
Intuitively, the HH does not need to save or borrow in order
to smooth consumption because the endowment stream is
already perfectly smooth.
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Case with Uncertainty
Recall the FOC (5):
1
C1
=
1
2
[
1
2Q + σ − C1
+
1
2Q − σ − C1
]
This equation can be rewritten as
1
C1
=
1
2
[
4Q − 2C1
(2Q − C1)2 − σ2
]
We can derive:
4Q2 − 4QC1 + C21 − σ2 = C1(2Q − C1).
By rewriting, we obtain:
2C21 − 6QC1 + 4Q2 − σ2 = 0 (6)
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Case with Uncertainty
Solution Formula for a Quadratic Equation
The solution(s) to the quadratic equation ax2 + bx + c = 0 are
given by
x =
−b ±
√
b2 − 4ac
2a
.
If we apply the above formula to (6), we obtain
C1 =
6Q ±
√
36Q2 − 8(4Q2 − σ2)
4
or
C1 =
3Q ±
√
Q2 + 2σ2
2
. (7)
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Case with Uncertainty
The solution C1 =
3Q+
√
Q2+2σ2
2 is not feasible.
To see this, first note that
3Q +
√
Q2 + 2σ2
2
>
3Q +
√
Q2
2
= 2Q.
Recall that
C2(G) = 2Q + σ − C1, (1)
C2(B) = 2Q − σ − C1. (2)
If C2 ≥ 0 in both states then C1 is capped:
C1 ≤ 2Q − σ
Therefore, the only feasible solution in (7) is
C1 =
3Q −
√
Q2 + 2σ2
2
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Case with Uncertainty
From
C1 =
3Q −
√
Q2 + 2σ2
2
and
C2(G) = 2Q + σ − C1, (1)
C2(B) = 2Q − σ − C1 (2)
we obtain
C2 =

2Q + σ −
3Q−
√
Q2+2σ2
2 if Q2 = Q + σ (G state)
2Q − σ − 3Q−
√
Q2+2σ2
2
if Q2 = Q − σ (B state)
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Analysis
The first period consumption is less than the one in the
case w/o uncertainty:
C1 =
3Q −
√
Q2 + 2σ2
2
<
3Q −
√
Q2
2
= Q
Furthermore, the optimal C1 is a decreasing function of σ,
C1 =
3Q −
√
Q2 + 2σ2
2
= Q −
√
Q2 + 2σ2 −Q
2︸ ︷︷ ︸
precautionary saving
If σ = 0, then precautionary savings are zero.
σ ↑→ precautionary saving ↑→ C1 ↓
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Analysis
Note that
C2(B) = Q − σ︸ ︷︷ ︸
Q2
+
√
Q2 + 2σ2 −Q
2︸ ︷︷ ︸
precautionary saving
When σ increases, Q2 decreases in the bad state.
The household tries to avoid low consumption in the bad
state by increasing her precautionary saving.
This has an implication for the CA as well.
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Implication for CA
Note that
TB1 = Q − C1(σ)
where C1(σ) is a decreasing function of σ.
Because CA1 = r0B0 + TB1 and B0 = 0 by assumption,
CA1 = TB1 = Q −C1(σ).
In response to an increase in uncertainty σ, the HH uses
the trade balance as a vehicle to save in period 1.
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Implication for CA
Therefore:
1 High uncertainty about the future (σ large)
People save more. The CA improves.
2 Low uncertainty about the future (σ small; Great
Moderation)
People save less. The CA deteriorates.
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The Great Moderation in the U.S.
Quarterly Real Per Capita U.S. GDP Growth, 1947Q2 - 2017Q4
The volatility of the growth rate of real U.S. GDP per capita
declined significantly after (about) 1984.
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Motivation: The Great Moderation in the U.S.
U.S. CA-to-GDP Ratio, 1947Q1 - 2017Q4
1947-1983: on average, CA surplus of 0.34% of GDP;
1984-2017: on average, CA deficit of 2.8% of GDP.
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Incomplete Markets
Before we conclude, we point out that the prediction that
an increase in uncertainty results in evelated precautionary
saving depends on the assumption that financial markets
are incomplete.
The HH would like to buy an asset that pays more in state B
than in state G.
However, the only financial instrument (the bond) available
to the HH pays the same in both states — markets are
incomplete.
Tis means that the HH faces an uninsurable income risk.
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Incomplete Markets
If the following state contingent claims are for purchase
in period 1 then markets are complete:
1 An asset that in period 2 pays one unit of the good in the
good and zero in the bad state.
2 An asset that in period 2 pays one unit of the good in the
bad and zero in the good state.
Under complete markets,
the HH ins able to insure against output volatility without
resorting to precautionary saving.
the positive relationship between the level of uncertainty
and the CA disappears.
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Incomplete Markets
Thus our result that a decrease in uncertainty leads to a
deterioration of the CA relies on the assumption that
financial markets are incomplete.
Are real-world financial markets incomplete? This is not
unreasonable to assume:
More than two states G and B: policymaking, weather,
natural catshrophes, epidemics, technicolocal innovations
etc. all create uncertainty and (possibly infinitely?) many
states.
It is thus conceivable that the available assets do not span
all possible states.
Also, local policy such as prohibition of trade in derivatives
may cause markets to be incomplete.
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Summary and Conclusion
Today, we studied:
1 The Great Moderation.
2 A small open economy model with uncertainty.
3 Precautionary savings.
Next week, we will study external adjustments in open
economies.