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Lecture Notes 1
Harris Dellas
Definitions of the foreign accounts and some identities (further details and description of the
variables in the supplement to note 1)
The current account (CA) is an important window linking any economy to the rest of the world
CAt = T Bt + N F P
GN P = GDP + N F P
= C + I + G + TB + NFP
= C + I + G + CA
=C +S+T
CA = S − I +
T
− G}
| {z
budget deficit (surplus)
Derivation of the CA using the country’s budget constraint Suppose that the only
international asset is a one period bond that pays an interest of r
Yt + (1 + r)Bt−1 = Ct + It + Gt + Bt
where Y is GDP. B > 0 positive means the country is lending to the rest of the world while
B < 0 means it is borrowing.
Yt − Ct − Gt − It +rBt−1 = Bt − Bt−1
|
{z
}
(1)
CAt = T Bt + N F P = Bt − Bt−1
(2)
T Bt
(3)
Equation emphasizes trade as the key determinant of the CA while equation 1 savings and
investment. Naturally, they are equivalent.
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Lecture Objective: The determination of macroeconomic variables: Y, C, I, L, CA,
B, q, ..
We will work through a succession of models. We will start with the simplest possible macroeconomic environment and then continue adding new features, expanding the number of variables
to be determined. While we will rely mostly on a two period model, we will occasionally use a
multi-period version1 . We will use these models to determine the properties of the equilibrium
aggregate quantities –in particular of the current account– and prices and then always ask the
same question. Namely, how the equilibrium values are affected by changes in the economic
environment (shocks to productivity, preferences, economic policy,..).
• Model 1
1. Small open economy
2. Endowments (no production)
3. A single, perishable good
• Model 2
1. Small open economy
2. Production
3. A single, non-perishable good (investment)
• Model 3
1.
2. Two (large) economies
3. Endowments (no production)
4. A single, perishable good
• Model 4
1. Two (large) economies
2. Production
3. A single, non-perishable good (investment)
• Model 5
1. Two (large) economies
2. Endowments (no production)
3. (Two) Country specific, perishable goods
1
Obstfeld and Rogoff and Uribe contain multiperiod versions of our models.
2
Assumptions shared across models:
• Abstract from monetary (nominal) considerations
• No uncertainty
• Representative agent
• Two periods
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1.1
MODEL 1: Small open economy, endowment
A useful starting point: The closed economy (one country) model
Preferences
u(c1 ) + βu(c2 )
The budget constraints
Y1 = C1 + B1
Y2 + (1 + r)B1 = C2 + B2
where B is savings. The world ends after the second period so B2 = 0
The optimization problem
max U
c1 ,c2 ,B1
subject to
ω = Y1 +
Y2
c2
= c1 +
1+r
1+r
The FOC is
uc1 = β(1 + r)uc2
The assumption of perishability implies that savings cannot take the form of stored (invested)
goods but only the form of a ”paper” claim.
The representative agent assumption means that there can be no net borrowers or lenders in
the economy.
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Consequently, the equilibrium aggregate quantities are
B1 = 0
c1 = Y1
c2 = Y2
Even if the quantity of loans in equilibrium is zero, there is a price for loans, r (r : B = 0. It is
determined by the FOC
uc1 (Y1 ) = β(1 + r)uc2 (Y2 )
r=
uc1 (Y1 )
−1
βuc2 (Y2 )
The General Equilibrium of the model determines the endogenous variables as a function of the
exogenous variables:
{Y1 , Y2 }
→
{c1 , c2 , B1 , r}
Comparative statics
Y1 ↑
dr
dY1
c1 ↑
dc1 = dY1
dc2 →
= uc1 < 0
dr
dY2
> 0 people want to borrow against high future income. Given a fixed supply of output at
present (Y1 , this desire is discouraged by having a higher interest rate.
dr
dβ
<0
β ↑ When people become more patient (β ↑), they want to save more. The interest
rate must decrease to discourage this desire.
1.2
Small Open Economy
The assumption of small means inability to affect international prices. In our case this means
that r is given, that is, the country can borrow as much as its budget cnstraint permits it without
any effect on r.
Yt + (1 + r)Bt−1 = Ct + Bt
CAt = Bt − Bt−1 = Yt − Ct +rt−1 Bt−1
| {z }
TB
4
r
Specialize this budget constraint to the case of two periods
B0 (1 + r) + Y1 = C1 + B1
(1 + r)B1 + Y2 = C2 + B2
or combined together,
B0 (1 + r) + Y1 +
C2
Y2
= C1 +
1+r
1+r
B2 = 0 is the transversality condition.
Implications of B0 6= 0.
Let us now solve for the equilibrium. The first order condition with regard to B1 is
uc1 = β(1 + r)uc2
Solving the FOC for c2 gives the first equation below. Plugging this equation into the budget
constraint in the first period produces an equation for c1 (the second equation below)
c2 = c(r, c1 , Y1 , Y2 , B0 )
c1 = c(r, Y1 , Y2 , B0 )
Hence, we have been able to determine the equilibrium quantities of c1 and c2 as a function of
output, initial assets and the world interest rate.
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Using the budget constraints we have the following expressions for the CA
CA1 = B1 − B0 = Y1 − C1 + rB0
CA2 = B2 − B1 = −B1 = Y2 − C2 + rB1
CA1 + CA2 = B1 − B0 − B1 = −B0
if B0 = 0
→
CA1 + CA2 = 0
In order to have stationarity in consumption we impose the condition2 β(1 + r) = 1
A key property of the model: The FOC implies uc1 = uc2 ⇒ c1 = c2
The small open economy can thus achieve perfect consumption smoothing over time independent
of the path of Y . We shall see later that this is no longer the case when this economy is allowed
to refuse to pay its debts.
Question: How do changes in the economic environment affect the CA?
A temporary increase in current output (Y1 ↑, Y2 →)
CA = B1 − B0 = Y1 − C1 + rB0 →
dCA
dC1
=1−
dY1
dY1
1
Determine dC
dY1 by differentiating the FOC and the overall budget constraint (equation 1.2) and
combining to get
(1 + r)Uc2 c2
dc1
=
<1
dY1
uc1 c1 + (1 + r)uc2 c2
Consequently,
dCA
dY1
> 0, that is, the CA1 improves
A temporary increase in future output (Y2 ↑, Y1 →)
Similarly, we can establish 0 <
rB0 we have
dC1
dY2
< 1 and thus from the definition of the CA, CA = Y1 − C1 +
dC1
dCA1
=−
<0
dY2
dY2
That is, the CA1 deteriorates
A permanent increase in output
Y1 ↑
2
Y2 ↑: dY1 = dY2 = dY
Try to figure out what happens in an economy with an infinite time horizon when this condition is violated.
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From the FOC we have that: uc1 c1 dc1 = uc2 c2 dc2 ⇒ dc1 = dc2 = dc. Differentiating the budget
constraint then gives dc = dY .
dC1
=1
dY1
dCA
=0
dY
The big picture: The permanent income hypothesis tells us how changes in income (output)
affect consumption-savings decisions and shape the response of the CA.
Y1 ↑, Y2 →→ B ↑ CA ↑
Y2 ↑, Y1 →→ B ↓ CA ↓
Y1 ↑, Y2 ↑ dY1 = dY2 → B → CA →
Exercise: Think of the effects on the CA of the effect of a current increase in ouput that is
expected to be followed by an even larger effect in the future (positive output growth)
Y1 ↑, Y2 ↑ dY2 = dY1
Exersise: You may redo the analysis in a model which also contains a labor-leisure choice (so
that utility is u(c, n)) and a production function (y=f(n) where n is labor).
Government Spending and the Current Account: Twin Deficits
Y1 = C1 + G1 + B1
Y2 + (1 + r)B1 = C2 + G2
C2
G2
Y2
= C1 +
+ G1 +
Y1 +
1+r
1+r
1+r
Y1 = C1 + T1 + B
Y2 + (1 + r)B = C2 + T2
Combining these two budget constraints, we get
C1 +
C2
T2
Y2
+ T1 +
= Y1 +
1+r
1+r
1+r
Y1 +
Y2
C2
G2
= C1 +
+ G1 +
1+r
1+r
1+r
or
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Note that it does not make any difference whether we assume a balanced budget T1 = G1 ,
T2 = G2 or debt financed deficits, G1 = T1 + B g , G2 + (1 + r)B g = T2 . This is due to Ricardian
equivalence.
The social planning problem is
max U (C1 ) + βU (C2 ) = U (C1 ) + βU ((1 + r)Y1 + Y2 − (1 + r)C1 − (1 + r)G1 − G2 )
C1
which leads to the first order Euler condition
UC1 = β(1 + r)UC2
We now compute
leads to
d(CA)
dG1 .
Again assume that β(1 + r). Total differentiation of the Euler equation
dC1 dC1
(1 + r)UC2 C2
<1
=−
< 0 and dG1
UC1 C1 + (1 + r)UC2 C2
dG1 Y1 = C 1 + G 1 + B
we have CA = B − B0 so that dCA=dB. Then
0=
dB
dB
dC1
dC1
+1+
=⇒
= −1 −
<0
dG1
dG1
dG1
dG1
CA1 ↓ (−), CA2 ↑ (+), CA1 + CA2 = 0
What about the effect of an expected future increase in government spending? Differentiating
the Euler equation with regard to G2 we can see that
dC1
<0
dG2
Differentiation of the resource constraint (eq. 1.2 gives
0=
dC1
dB
dB
CA1 +
+0+
=⇒
>0
CA2 −
dG2
dG2
dG2
Again, the permanent income hypothesis can be used to understand the effects of changes in
the level of government spending on savings and the current account. Recall that government
spending is completely useless (from the point of view of the consumers) in the economy. For
the households, an increase in government spending is equivalent a reduction in their output
(income). Hence, with temporary increase in G savings decreases and the CA deteriorates.
Exercise: (a)The effects on the CA of a permanent increase in G. (b) The effects of a change in
dG2
G that satisfies: dG1 + (1+r)
= 0.
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2
MODEL 2: Small open economy, Investment
In the previous section, a country could transfer resources over time only by borrowing from or
lending to the rest of the world. In this section we revisit the study of the current account in a
model where there is an additional means of saving, namely physical capital. For simplicity, we
assume that all domestic capital is owned by domestic residents3
Specification of the model
• Production technology: Y = F (K)
• Capital Accumulation: Kt+1 = It + Kt , depreciation, δ = 0
• Government Budget constraint: Tt = Gt
• Resource (Budget) Constraint:
F (Kt ) + (1 + r)Bt + Kt = Ct + Tt + Bt+1 + Kt+1
Combining these equations:
Yt + (1 + r)Bt = Ct + Gt + It + Bt+1
The current account is:
CAt = Bt+1 − Bt = Yt + rBt − (Ct + It + Gt )
{z
}
|
absorption
Savings are:
St = Yt + rBt − Ct − Tt
Substituting for Yt in the previous equation gives
CAt = St − It + Tt − Gt
A 2 period model.
B0 = 0
Y1 = F (K1 )
Y2 = F (K2 ) = F (I1 + K1 )
Y1 = C1 + I1 + G1 + B
Y2 + (1 + r)B = C2 + I2 + G2
I2 = K3 − K2
K3 = 0(transversality)
3
If there is foreign ownership we need to take this into account in defining the NFAP of the country and hence
the current account.
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Combining these equations, we get
C1 + I1 + G1 +
Y2
(F (I1 + K1 )
C2 + I2 + G2
= Y1 +
= F (K1 ) +
1+r
1+r
1+r
The problem is then
max U (C1 )+βU (C2 ) = U (C1 )+βU ((1 + r)(F (K1 ) − C1 − G1 − I1 ) + F (I1 + K1 ) − G2 + I1 + K1 )
which leads to the first order conditions
/C1 : UC1 = β(1 + r)UC2
/I1 : βUC2 (−(1 + r) + FK2 + 1) = 0 =⇒ FK2 = r
where FK2 = F 0 (K2 ).
Note that differentiating the 2nd FOC gives F 00 dK2 = dr so that dK2 /dr = 1/F 00 < 0.
Figure 1: Capital choice
r
F 0 (K2 )
K2
I1
K2
=⇒ r
Main points:
Because r is exogenous (given by the world) desired investment and capital stock are independent
of demand conditions (preferences).
Separation of savings from investment decisions under the following conditions
1. Small Open Economy
2. Perfect Capital mobility
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3. (No borrowing limits)
The Feldstein–Horioka puzzle: A strong positive correlation between national savings and investment rates. If capital is very mobile across countries, then the correlation between savings
and investment should be close to zero, as the preceding analysis shows.
Exercise: Write the production function as Y=AF(K). Examine the effects of a current temporary, expected future and a permanent positive productivity shock (A ↑). What is the response
of S and I and the CA.
3
MODEL 3: Two -large- country model, endowments
Now we will g
Y1 = C 1 + B
(4)
Y2 + (1 + r)B = C2
Y1?
Y2?
=
C1?
+ (1 + r)B =
C2?
?
(5)
+B
?
B + B? = 0
(6)
(7)
(8)
(9)
Combining (4)–(8), we obtain the market clearing conditions for the two outputs
Y1 + Y1? = C1 + C1?
Y2 +
Y2?
= C2 +
C2?
(10)
(11)
Combining (4)–(7)gives the intertemporal budget constraint for each country
Y1 + Y2 /(1 + r) = C1 + C2 /(1 + r)
(12)
Y1? + Y2? /(1 + r) = C1? + C2? /(1 + r)
(13)
Finally the FOCs for the optimal choice of consumption in each country are
UC1 = β(1 + r)UC2 =⇒ C2 = C2 (C1 , r)
(14)
UC? 1 = β(1 + r)UC? 2 =⇒ C2? = C2? (C1? , r)
(15)
Equations 10-13 are 6 equations in 5 unknowns, {C1 , C2 , C1? , C2? , r}.
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To solve for the equilibrium of the model we use Walras law: If there are N markets, need to
consider equilibrium (D = S) in N-1 markets. The N-th will clear when the N-1 clear.
Exercise: Assume U (C) = log(C), Solve the model. Determine {C1 , C2 , C1? , C2? , r} for arbitrary
{Y1 , Y2 , Y1? , Y2? }. That is, find
Ct = Ct (Y1 , Y2 , Y1? , Y2? ) for t = 1, 2
r = Rt (Y1 , Y2 , Y1? , Y2? )
B = B(Y1 , Y2 , Y1? , Y2? )
Compute
dCt dr
dY , dY
,
dB
dY .
Figure 2: Variation in Y1 , endowmentworldeconomy
r
r
r↓
r
B + B? = 0
B?
B
• What is the main difference between a small and a large economy? In the former case r is
exogenous. In the large economy: Smaller variations in the current account as a result of
economic disturbances than in the small open economy due to the “dampening effect” of
the induced change in r. Small open economies exhibit greater macroeconomic volatility.
• Growth and welfare. Is a country hurt by an increase in trading partners’ growth rates?
In Fig. 2 an increase in Y2 /Y1 shifts the Home saving curve upwards, increasing world
interest rates and making the foreign country (borrower) worse off.
• Empirical implication: War and the CA. War waging countries run a current account
(”borrow from abroad to finance the war”). The experience of Japan and Sweden around
WWI.
Japan’s CA experience during the Russian War, 1904-05 (no disruption of world financial
markets, relative certainty about the winner)
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4
MODEL 4: Two country model, investment
• 2 country model with investment: S = B + I
• 2 periods.
• Domestic technology: Y = AF (K)
• Foreign technology: Y ? = A? F (K ? )
• Period t = 1: K1 + A1 F (K1 ) = C1 + K2 + B and K2 = I1 + K1 imply: Y1 = C1 + I1 + B
• Period t = 2: K2 + A2 F (K2 ) + (1 + r)B = C2
where r is the interest rate on the one period international bonds.
Note also that K1 is predetermined and that we have assumed a zero rate of capital depreciation.
The representative individual maximizes:
max U (C1 ) + βU ((1 + r)(A1 F (K1 ) − C1 − I1 ) + A2 F (K1 + I1 ) + K1 + I1 )
C1 ,I1
The first order conditions are then given by
/I1 : βU 0 (C2 ) −(1 + r) + A2 F 0 (K2 ) + 1 = 0
(16)
/C1 : U 0 (C1 ) = β(1 + r)U 0 (C2 )
(17)
The first order condition for I1 reduces to
r = A2 F 0 (K2 )
This equation represents an arbitrage condition between the two possible investments (I1 , B):
r = A2 F 0 (K1 + I1 ) ⇐⇒ I1 = I(r , A2 , K1 )
−
+
−
Solving the model
The second first order condition above expresses the intertemporal allocation of consumption,
from which we get
C2 = C2 (r , C1 )
+
+
The intertemporal budget constraint of the agent writes
K1 + A1 F (K1 ) − C1 − (I1 + K1 ) =
13
C2 − K1 − I1 − A2 F (K1 + I1 )
1+r
which can be used to solve for period 1 consumption as
C1 = C1 (r, K1 , A1 , A2 )
From the first period resource constraint, A1 F (K1 ) = C1 + I1 + B, one can get B as
B = B(r, K1 , A1 , A2 ) = CA
Plaguing all these equation in the budget constraint allows to solve for the equilibrium r, that
is, r satisfies A1 F (K1 ) = C1 + I1 + B
If B > 0, S > I, CA > 0, home lends its excess savings to the rest of the world.
Comparative exercises
An increase in current domestic productivity (3)
A1 ↑, A2 →, and {A?1 , A?2 } →:
• I1 curve does not shift (I1 →)
• Y1 − C1 : A1 ↑=⇒ Y1 = A1 F (K1 ) ↑. By the permanent income hypothesis, we have
∆C1 < ∆Y1 , hence Y1 − C1 shifts to the right.
The world interest rate decreases. The foreign CA worsens further.
Figure 3: An increase in A1
Y2? − C2?
Y1 − C 1
Y1 ↑
r
B?
B?
I1?
I1
An increase in expected future domestic productivity can be analyzed similarly (4). Starting
from a zero initial CA, the effect on the CA is negative (consider the foreign CA).
The non-separation of investment from savings
14
Figure 4: An increase in A2
Y1 − C 1
r0
A2 ↑, r ↑
r
I1
Question: Can the model account for movements in real world interest rates? For instance, why
rates were high in 80s? How?
An increase in the expected profitability of investment. It increases r and may increase or
decrease I. Under log utility, world I goes down
15