57
central bank is zero inflation in America. In case B the targets of the European
central bank are zero inflation and zero unemployment in Europe. And the targets
of the American central bank are zero inflation and zero unemployment in
America. In case C the European central bank has a single target, that is zero
inflation in Europe. By contrast, the American central bank has two conflicting
targets, that is zero inflation and zero unemployment in America. This chapter
deals with case A, and the next chapters deal with cases B and C.

The target of the European central bank is zero inflation in Europe. The
instrument of the European central bank is European money supply. By equation
(3), the reaction function of the European central bank is: 112
2M 2B M=− + (5)

Suppose the American central bank lowers American money supply. Then, as a
response, the European central bank lowers European money supply.

The target of the American central bank is zero inflation in America. The
instrument of the American central bank is American money supply. By equation
(4), the reaction function of the American central bank is: 221
2M 2B M=− + (6)

Suppose the European central bank lowers European money supply. Then, as a
response, the American central bank lowers American money supply.

111
uAB=+ (9)

From equations (2), (7) and (8) follows the equilibrium rate of unemployment in
America: 222
uAB=+ (10)

From equations (3), (7) and (8) follows the equilibrium rate of inflation in
Europe: 1
0π= (11)

And from equations (4), (7) and (8) follows the equilibrium rate of inflation in
America: 2
0π= (12)

As a result, given a shock, monetary interaction produces zero inflation in
Europe and America.



It proves useful to study six distinct cases:
- a demand shock in Europe
- a supply shock in Europe
- a mixed shock in Europe
- another mixed shock in Europe
- a common demand shock
- a common supply shock.

1) A demand 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
a decline in the demand for European goods. In terms of the model there is an
increase in
1
A of 3 units and a decline in
1
B of equally 3 units. Step two refers
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 money supply of 4 units and an increase in
2. Some Numerical Examples

60
American money supply 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 3.1 presents a synopsis.

zero inflation and zero unemployment in each of the regions. The loss functions
of the European central bank and the American central bank are respectively: 2
11
L =π (7)

2
22
L =π (8)

The initial loss of the European central bank is zero, as is the initial loss of the
American central bank. The demand shock in Europe causes a loss to the
European central bank of 9 units and a loss to the American central bank of zero
Monetary Interaction between Europe and America: Case A

61
units. Then monetary interaction reduces the loss of the European central bank
from 9 to zero units. And what is more, monetary interaction keeps the loss of the
American central bank 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

Inflation 0 Inflation 0
Shock in A
1

3
Shock in B
1

3
Unemployment 3 Unemployment 0
Inflation 3 Inflation 0
Change in Money Supply
− 4
Change in Money Supply
− 2
Unemployment 6 Unemployment 0
Inflation 0 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
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


0
Shock in B
1

6
Unemployment 0 Unemployment 0
Inflation 6 Inflation 0
Change in Money Supply
− 8
Change in Money Supply
− 4
Unemployment 6 Unemployment 0
Inflation 0 Inflation 0
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
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 no change in European money supply, nor is there in American money
2. Some Numerical Examples


Unemployment 6 Unemployment 0
Inflation 0 Inflation 0
Change in Money Supply 0 Change in Money Supply 0
Unemployment 6 Unemployment 0
Inflation 0 Inflation 0
5) A common demand shock. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to a decline in the
demand for European and American goods. In terms of the model there is an
increase in
1
A of 3 units, a decline in
1
B of 3 units, an increase in
2
A of 3 units,
Monetary Interaction between Europe and America: Case A

65
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 money supply and American money supply of 6
units each. Step four refers to the outside lag. Unemployment in Europe goes

Inflation
− 3
Inflation
− 3
Change in Money Supply 6 Change in Money Supply 6
Unemployment 0 Unemployment 0
Inflation 0 Inflation 0
As a result, given a common demand shock, monetary interaction produces
zero inflation and zero unemployment in each of the regions. The initial loss of
each central bank is zero. The common demand shock causes a loss to the
European central bank of 9 units and a loss to the American central bank of
equally 9 units. Then monetary interaction reduces the loss of the European
2. Some Numerical Examples

66
central bank from 9 to zero units. Correspondingly, it reduces the loss of the
American central bank 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

unemployment there.

Monetary Interaction between Europe and America: Case A

67
Table 3.6
Monetary Interaction between Europe and America
A Common Supply Shock

Europe America

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

3
Shock in A
2

3
Shock in B
1

3
Shock in B
2

3
Unemployment 3 Unemployment 3

222 1
uAM0.5M=− + (2)

11 1 2
B M 0.5Mπ= + − (3)

22 2 1
BM0.5Mπ= + − (4)

The targets of the European central bank are zero inflation and zero
unemployment in Europe. The instrument of the European central bank is
European money supply. There are two targets but only one instrument, so what
is needed is a loss function. We assume that the European central bank has a
quadratic loss function: 22
111
Lu=π +
(5)

1
L is the loss to the European central bank caused by inflation and
unemployment in Europe. We assume equal weights in the loss function. The
specific target of the European central bank is to minimize its loss, given the
inflation function and the unemployment function. Taking account of equations
(1) and (3), the loss function of the European central bank can be written as
follows:
2
L is the loss to the American central bank caused by inflation and
unemployment in America. We assume equal weights in the loss function. The
specific target of the American central bank is to minimize its loss, given the
inflation function and the unemployment function. Taking account of equations
(2) and (4), the loss function of the American central bank can be written as
follows: 22
222 1 22 1
L (B M 0.5M ) (A M 0.5M )=+− +−+
(9)

Then the first-order condition for a minimum loss gives the reaction function of
the American central bank: 2221
2M A B M=−+ (10)

Suppose the European central bank lowers European money supply. Then, as a
response, the American central bank lowers American money supply.

1. The Model

70
The Nash equilibrium is determined by the reaction functions of the
European central bank and the American central bank. The solution to this
problem is as follows:

222
2u A B=+ (14)

From equations (3), (11) and (12) follows the equilibrium rate of inflation in
Europe: 111
2ABπ= + (15)

And from equations (4), (11) and (12) follows the equilibrium rate of inflation in
America: 222
2ABπ= + (16)

As a rule, unemployment in Europe and America is not zero. And inflation in
Europe and America is not zero either.
Monetary Interaction between Europe and America: Case B

71
2. Some Numerical Examples
For easy reference, the basic model is reproduced here: 111 2

a common demand shock
-
a common supply shock
-
a common mixed shock
-
another common mixed shock.

1) A demand 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
a decline in the demand for European goods. In terms of the model there is an
increase in
1
A of 3 units and a decline in
1
B of equally 3 units. Step two refers
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.

2. Some Numerical Examples

72
Step three refers to the policy response. According to the Nash equilibrium
there is an increase in European money supply of 4 units and an increase in
American money supply 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 3.7 presents a synopsis.


of the European central bank and the American central bank are respectively: 22
111
Lu=π + (7)

22
222
Lu=π + (8)

Monetary Interaction between Europe and America: Case B

73
The initial loss of the European central bank is zero, as is the initial loss of the
American central bank. The demand shock in Europe causes a loss to the
European central bank of 18 units and a loss to the American central bank of zero
units. Then monetary interaction reduces the loss of the European central bank
from 18 to zero units. And what is more, monetary interaction keeps the loss of
the American central bank 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.

Shock in B
1

3
Unemployment 3 Unemployment 0
Inflation 3 Inflation 0
Change in Money Supply 0 Change in Money Supply 0
Unemployment 3 Unemployment 0
Inflation 3 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
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 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 6 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. For a synopsis see Table 3.9.

First consider the effects on Europe. As a result, given a mixed shock in
Europe, monetary interaction lowers inflation in Europe. On the other hand, it

Change in Money Supply
− 4
Change in Money Supply
− 2
Unemployment 3 Unemployment 0
Inflation 3 Inflation 0
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
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 money supply of 4 units and an increase in
2. Some Numerical Examples

76
American money supply of 2 units. Step four refers to the outside lag.
Unemployment in Europe goes from 6 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. For an overview see Table 3.10.

First consider the effects on Europe. As a result, given another mixed shock
in Europe, monetary interaction lowers unemployment in Europe. On the other

Monetary Interaction between Europe and America: Case B

77
5) A common demand shock. In each of the regions, let initial unemployment
be zero, and let initial inflation be zero as well. Step one refers to a decline in the
demand for European and American goods. In terms of the model there is an
increase in
1
A of 3 units, a decline in
1
B of 3 units, an increase in
2
A of 3 units,
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 money supply and American money supply of 6
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 3.11 presents a
synopsis.
Unemployment 0 Unemployment 0
Inflation 0 Inflation 0
2. Some Numerical Examples

78
As a result, given a common demand shock, monetary interaction produces
zero inflation and zero unemployment in each of the regions. The initial loss of
each central bank is zero. The common demand shock causes a loss to the
European central bank of 18 units and a loss to the American central bank of
equally 18 units. Then monetary interaction reduces the loss of the European
central bank from 18 to zero units. Correspondingly, it reduces the loss of the
American central bank from 18 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

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 Money Supply 0 Change in Money Supply 0
Unemployment 3 Unemployment 3
Inflation 3 Inflation 3
7) 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.


0
Shock in A
2

0
Shock in B
1

6
Shock in B
2

6
Unemployment 0 Unemployment 0
Inflation 6 Inflation 6
Change in Money Supply
− 6
Change in Money Supply
− 6
Unemployment 3 Unemployment 3
Inflation 3 Inflation 3
8) 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


Unemployment 0 Unemployment 0
Inflation 0 Inflation 0
Shock in A
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 6 Change in Money Supply 6
Unemployment 3 Unemployment 3
Inflation 3 Inflation 3
9) Summary. Given a demand shock in Europe, monetary interaction
produces zero inflation and zero unemployment in each of the regions. Given a
supply shock in Europe, monetary interaction is ineffective. Given a mixed shock