88
2) A common supply shock. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to the common
supply shock. In terms of the model there is an increase in
1
B of 3 units, as there
is in
1
A . And there is an increase in
2
B of 3 units, as there is in
2
A . Step two
refers to the outside lag. Inflation in Europe goes from zero to 3 percent, as does
inflation in America. Unemployment in Europe goes from zero to 3 percent, as
does unemployment in America.

Step three refers to the policy response. According to the Nash equilibrium
there is a reduction in European money supply of 4 units and a reduction in
American money supply of 2 units. Step four refers to the outside lag. Inflation in
Europe goes from 3 to zero percent. Inflation in America stays at 3 percent.
Unemployment in Europe goes from 3 to 6 percent. And unemployment in
America stays at 3 percent. Table 3.16 gives an overview. Table 3.16
Monetary Interaction between Europe and America
A Common Supply Shock

Europe America

Monetary Interaction between Europe and America: Case C

89
First consider the effects on Europe. As a result, given a common supply
shock, monetary interaction produces zero inflation in Europe. However, as a
side effect, it raises unemployment there. Second consider the effects on
America. As a result, monetary interaction has no effect on inflation and
unemployment in America. The initial loss of each central bank is zero. The
common supply shock causes a loss to the European central bank of 9 units and a
loss to the American central bank of 18 units. Then monetary interaction reduces
the loss of the European central bank from 9 to zero units. On the other hand, it
keeps the loss of the American central bank at 18 units.

3) A common mixed shock. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to the common
mixed shock. In terms of the model there is an increase in
1
B of 6 units and an
increase in
2
B of equally 6 units. Step two refers to the outside lag. Inflation in
Europe goes from zero to 6 percent, as does inflation in America. Unemployment
in Europe stays at zero percent, as does unemployment in America.

Step three refers to the policy response. According to the Nash equilibrium
there is a reduction in European money supply of 10 units and a reduction in
American money supply of 8 units. Step four refers to the outside lag. Inflation in
Europe goes from 6 to zero percent. Inflation in America goes from 6 to 3
percent. Unemployment in Europe goes from zero to 6 percent. And
unemployment in America goes from zero to 3 percent. For a synopsis see Table

Shock in B
1

6
Shock in B
2

6
Unemployment 0 Unemployment 0
Inflation 6 Inflation 6
Change in Money Supply
− 10
Change in Money Supply
− 8
Unemployment 6 Unemployment 3
Inflation 0 Inflation 3
4) Another common mixed shock. In each of the regions, let initial
unemployment be zero, and let initial inflation be zero as well. Step one refers to
the common mixed shock. In terms of the model there is an increase in
1
A of 6
units and an increase in
2
A of equally 6 units. Step two refers to the outside lag.
Unemployment in Europe goes from zero to 6 percent, as does unemployment in
America. Inflation in Europe stays at zero percent, as does inflation in America.


1

6
Shock in A
2

6
Shock in B
1

0
Shock in B
2

0
Unemployment 6 Unemployment 6
Inflation 0 Inflation 0
Change in Money Supply 2 Change in Money Supply 4
Unemployment 6 Unemployment 3
Inflation 0 Inflation 3
5) Summary. Given a common demand shock, monetary interaction produces
zero inflation and zero unemployment in each of the regions. Given a common
supply shock, monetary interaction produces zero inflation in Europe. And what
is more, monetary interaction has no effect on inflation and unemployment in
America. Given a common mixed shock, monetary interaction produces zero
inflation in Europe. And what is more, monetary interaction lowers inflation in
America. On the other hand, it raises unemployment there.

inflation in Europe, zero inflation in America, and zero unemployment in
America. This chapter deals with case A, and the next chapters deal with cases B
and C.

The policy makers are the European central bank and the American central
bank. The targets of monetary cooperation are zero inflation in Europe and
America. The instruments of monetary cooperation are European money supply
and American money supply. There are two targets and two instruments. We
assume that the European central bank and the American central bank agree on a
common loss function: 22
12
L =π +π
(5)

L is the loss caused by inflation in Europe and America. We assume equal
weights in the loss function. The specific target of monetary cooperation is to
minimize the loss, given the inflation functions in Europe and America. Taking

M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 92
DOI 10.1007/978-3-642-10476-3_12, © Springer-Verlag Berlin Heidelberg 2010

93
account of equations (3) and (4), the loss function under monetary cooperation
can be written as follows: 22

1
B causes a reduction in both European and
American money supply. Obviously, the cooperative equilibrium is identical to
the corresponding Nash equilibrium. That is to say, monetary cooperation case A
is equivalent to monetary interaction case A. For some numerical examples see
Chapter 1.

Monetary Cooperation between Europe and America: Case A

94
Chapter 5
Monetary Cooperation
between Europe and America: Case B
The model of unemployment and inflation can be represented by a system of
four equations: 111 2
uAM0.5M=− + (1)

222 1
uAM0.5M=− + (2)

11 1 2
B M 0.5Mπ= + − (3)

22 2 1

(6) M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 94
DOI 10.1007/978-3-642-10476-3_13, © Springer-Verlag Berlin Heidelberg 2010

95
Then the first-order conditions for a minimum loss are: 11212 2
5M 2A A 2B B 4M=−−++ (7)

221211
5M 2A A 2B B 4M=−−++ (8)

Equation (7) shows the first-order condition with respect to European money
supply. And equation (8) shows the first-order condition with respect to
American money supply.

The cooperative equilibrium is determined by the first-order conditions for a
minimum loss. The solution to this problem is as follows: 11212
3M 2A A 2B B=+−− (9)

22121
3M 2A A 2B B=+−− (10)


B M 0.5Mπ= + − (3)

22 2 1
BM0.5Mπ= + − (4)

The policy makers are the European central bank and the American central
bank. The targets of monetary cooperation are zero inflation in Europe, zero
inflation in America, and zero unemployment in America. The instruments of
monetary cooperation are European money supply and American money supply.
There are three targets but only two instruments, so what is needed is a loss
function. We assume that the European central bank and the American central
bank agree on a common loss function: 222
122
L0.50.5u=π + π +
(5)

L is the loss caused by inflation in Europe, inflation in America, and
unemployment in America. We assume equal weights in the loss function. The
specific target of monetary cooperation is to minimize the loss, given the
inflation functions and the unemployment function. Taking account of equations
(2), (3) and (4), the loss function under monetary cooperation can be written as
follows:

M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 96
DOI 10.1007/978-3-642-10476-3_14, © Springer-Verlag Berlin Heidelberg 2010

97

minimum loss. The solution to this problem is as follows: 12 12
3M A 4B B=− − (9)

2212
3M 2A 2B 2B=−− (10)

Equations (9) and (10) show the cooperative equilibrium of European money
supply and American money supply. As a result there is a unique cooperative
equilibrium. Obviously, the cooperative equilibrium is identical to the
corresponding Nash equilibrium. That is to say, monetary cooperation case C is
equivalent to monetary interaction case C. For some numerical examples see
Chapter 3. Monetary Cooperation between Europe and America: Case C
Part Four Fiscal Policies
in Europe and America Absence of a Deficit Target 101
Chapter 1


222 1
BG0.5Gπ= + + (4)

Here
1
u denotes the rate of unemployment in Europe,
2
u is the rate of
unemployment in America,
1
π is the rate of inflation in Europe,
2
π is the rate of
inflation in America,
1
G is European government purchases,
2
G is American
government purchases,
1
A is some other factors bearing on the rate of
unemployment in Europe,
2
A is some other factors bearing on the rate of
unemployment in America,
1
B is some other factors bearing on the rate of
inflation in Europe, and
2

European government purchases lowers European unemployment. On the other
hand, it raises European inflation. Correspondingly, an increase in American
government purchases lowers American unemployment. On the other hand, it
raises American inflation. Then consider the spillover effects. According to the
model, an increase in European government purchases lowers American
unemployment and raises American inflation. Similarly, an increase in American
government purchases lowers European unemployment and raises European
inflation.

According to the model, a unit increase in European government purchases
lowers European unemployment by 1 percentage point. On the other hand, it
raises European inflation by 1 percentage point. And what is more, a unit
increase in European government purchases lowers American unemployment by
0.5 percentage points and raises American inflation by 0.5 percentage points. For
instance, let European unemployment be 2 percent, and let European inflation be
2 percent as well. Further, let American unemployment be 2 percent, and let
American inflation be 2 percent as well. Now consider a unit increase in
European government purchases. Then European unemployment goes from 2 to
1 percent. On the other hand, European inflation goes from 2 to 3 percent. And
Fiscal Interaction between Europe and America

103
what is more, American unemployment goes from 2 to 1.5 percent, and
American inflation goes from 2 to 2.5 percent.

The target of the European government is zero unemployment in Europe.
The instrument of the European government is European government purchases.
By equation (1), the reaction function of the European government is:
Equations (7) and (8) show the Nash equilibrium of European government
purchases and American government purchases. As a result there is a unique
Nash equilibrium. According to equations (7) and (8), an increase in
1
A causes
an increase in European government purchases and a decline in American
government purchases. A unit increase in
1
A causes an increase in European
government purchases of 1.33 units and a decline in American government
1. The Model

104
purchases of 0.67 units. As a result, given a shock, fiscal interaction produces
zero unemployment in Europe and America.

2. Some Numerical Examples
For easy reference, the basic model is summarized here: 111 2
uAG0.5G=−− (1)

222 1

A of 3 units and a decline in
1
B of equally 3 units. Step two refers
Fiscal Interaction between Europe and America

105
to the outside lag. Unemployment in Europe goes from zero to 3 percent.
Unemployment in America stays at zero percent. Inflation in Europe goes from
zero to – 3 percent. And inflation in America stays at zero percent.

Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European government purchases of 4 units and a reduction
in American government purchases of 2 units. Step four refers to the outside lag.
Unemployment in Europe goes from 3 to zero percent. Unemployment in
America stays at zero percent. Inflation in Europe goes from – 3 to zero percent.
And inflation in America stays at zero percent. Table 4.1 presents a synopsis. Table 4.1
Fiscal Interaction between Europe and America
A Demand Shock in Europe

Europe America

Unemployment 0 Unemployment 0
Inflation 0 Inflation 0
Shock in A
1

3

22
Lu=
(8)

The initial loss of the European government is zero, as is the initial loss of the
American government. The demand shock in Europe causes a loss to the
European government of 9 units and a loss to the American government of zero
units. Then fiscal interaction reduces the loss of the European government from 9
to zero units. And what is more, fiscal interaction keeps the loss of the American
government at zero units.

2) A supply shock in Europe. In each of the regions let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to the supply
shock in Europe. In terms of the model there is an increase in
1
B of 3 units and
an increase in
1
A of equally 3 units. Step two refers to the outside lag. Inflation
in Europe goes from zero to 3 percent. Inflation in America stays at zero percent.
Unemployment in Europe goes from zero to 3 percent. And unemployment in
America stays at zero percent.

Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European government purchases of 4 units and a reduction
in American government purchases of 2 units. Step four refers to the outside lag.
Unemployment in Europe goes from 3 to zero percent. Unemployment in
America stays at zero percent. Inflation in Europe goes from 3 to 6 percent. And
inflation in America stays at zero percent. Table 4.2 gives an overview.


Inflation 3 Inflation 0
Change in Govt Purchases 4 Change in Govt Purchases
− 2
Unemployment 0 Unemployment 0
Inflation 6 Inflation 0
3) A mixed shock in Europe. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to the mixed
shock in Europe. In terms of the model there is an increase in
1
A of 6 units. Step
two refers to the outside lag. Unemployment in Europe goes from zero to 6
percent. Unemployment in America stays at zero percent. Inflation in Europe
stays at zero percent, as does inflation in America.

Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European government purchases of 8 units and a reduction
in American government purchases of 4 units. Step four refers to the outside lag.
Unemployment in Europe goes from 6 to zero percent. Unemployment in
America stays at zero percent. Inflation in Europe goes from zero to 6 percent.
And inflation in America stays at zero percent. For a synopsis see Table 4.3.

First consider the effects on Europe. As a result, given a mixed shock in
Europe, fiscal interaction produces zero unemployment in Europe. However, as a
side effect, it produces inflation there. Second consider the effects on America.
2. Some Numerical Examples

108

4) Another mixed shock in Europe. In each of the regions, let initial
unemployment be zero, and let initial inflation be zero as well. Step one refers to
the mixed shock in Europe. In terms of the model there is an increase in
1
B of 6
units. Step two refers to the outside lag. Inflation in Europe goes from zero to 6
percent. Inflation in America stays at zero percent. Unemployment in Europe
stays at zero percent, as does unemployment in America.

Step three refers to the policy response. According to the Nash equilibrium
there is no change in European government purchases, nor is there in American
Fiscal Interaction between Europe and America

109
government purchases. Step four refers to the outside lag. Inflation in Europe
stays at 6 percent. Inflation in America stays at zero percent. Unemployment in
Europe stays at zero percent, as does unemployment in America. For an
overview see Table 4.4.

First consider the effects on Europe. As a result, given another mixed shock
in Europe, fiscal interaction produces zero unemployment in Europe. However,
as a side effect, it produces inflation there. Second consider the effects on
America. As a result, fiscal interaction produces zero unemployment and zero
inflation in America. The mixed shock in Europe causes no loss to the European
government or American government.
1
B of 3 units, an increase in
2
A of 3 units,
2. Some Numerical Examples

110
and a decline in
2
B of 3 units. Step two refers to the outside lag. Unemployment
in Europe goes from zero to 3 percent, as does unemployment in America.
Inflation in Europe goes from zero to – 3 percent, as does inflation in America.

Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European government purchases and American
government purchases of 2 units each. Step four refers to the outside lag.
Unemployment in Europe goes from 3 to zero percent, as does unemployment in
America. Inflation in Europe goes from – 3 to zero percent, as does inflation in
America. Table 4.5 presents a synopsis. Table 4.5
Fiscal Interaction between Europe and America
A Common Demand Shock

Europe America

Unemployment 0 Unemployment 0
Inflation 0 Inflation 0
Shock in A

Then fiscal interaction reduces the loss of the European government from 9 to
Fiscal Interaction between Europe and America

111
zero units. Correspondingly, it reduces the loss of the American government
from 9 to zero units.

6) A common supply shock. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to the common
supply shock. In terms of the model there is an increase in
1
B of 3 units, as there
is in
1
A . And there is an increase in
2
B of 3 units, as there is in
2
A . Step two
refers to the outside lag. Inflation in Europe goes from zero to 3 percent, as does
inflation in America. Unemployment in Europe goes from zero to 3 percent, as
does unemployment in America.

Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European government purchases and American
government purchases of 2 units each. Step four refers to the outside lag.
Unemployment in Europe goes from 3 to zero percent, as does unemployment in
America. Inflation in Europe goes from 3 to 6 percent, as does inflation in
America. Table 4.6 gives an overview.


Shock in A
1

3
Shock in A
2

3
Shock in B
1

3
Shock in B
2

3
Unemployment 3 Unemployment 3
Inflation 3 Inflation 3
Change in Govt Purchases 2 Change in Govt Purchases 2
Unemployment 0 Unemployment 0
Inflation 6 Inflation 6 Fiscal Interaction between Europe and America

113
Chapter 2
Fiscal Cooperation
between Europe and America



L is the loss caused by unemployment in Europe and America. We assume equal
weights in the loss function. The specific target of fiscal cooperation is to
minimize the loss, given the unemployment functions in Europe and America.
Taking account of equations (1) and (2), the loss function under fiscal
cooperation can be written as follows: 22
11 2 2 2 1
L (A G 0.5G ) (A G 0.5G )=−− +−−
(6)

Then the first-order conditions for a minimum loss are: 1122
5G 4A 2A 4G=+− (7)

M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 113
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114

2211
5G 4A 2A 4G=+− (8)

Equation (7) shows the first-order condition with respect to European
government purchases. And equation (8) shows the first-order condition with
respect to American government purchases.