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1. The Model 187
Then the first-order condition for a minimum loss gives the reaction function of the American central bank:
2M2 = A2 −B2 −2G2 −G1 + M1 (12)
Suppose the European central bank lowers European money supply. Then, as a response, the American central bank lowers American money supply.
The targets of the European government are zero unemployment and a zero structural deficit in Europe. The instrument of the European government is European government purchases. There are two targets but only one instrument, so what is needed is a loss function. We assume that the European government has a quadratic loss function:
LG1 = u1 +s1 (13)
LG1 is the loss to the European government caused by unemployment and the structural deficit in Europe. We assume equal weights in the loss function. The
specific target of the European government is to minimize its loss, given the unemployment function and the structural deficit function. Taking account of equations (1) and (5), the loss function of the European government can be written as follows:
LG1 = (A1 −M1 +0.5M2 −G1 −0.5G2)2 +(G1 −T )2 (14)
Then the first-order condition for a minimum loss gives the reaction function of the European government:
4G1 = 2A1 + 2T −2M1 + M2 −G2 (15)
The targets of the American government are zero unemployment and a zero structural deficit in America. The instrument of the American government is American government purchases. There are two targets but only one instrument, so what is needed is a loss function. We assume that the American government has a quadratic loss function:
188 Monetary and Fiscal Interaction between Europe and America: Case B
LG2 = u2 +s2 (16)
LG2 is the loss to the American government caused by unemployment and the structural deficit in America. We assume equal weights in the loss function. The
specific target of the American government is to minimize its loss, given the unemployment function and the structural deficit function. Taking account of equations (2) and (6), the loss function of the American government can be written as follows:
LG2 = (A2 −M2 +0.5M1 −G2 −0.5G1)2 +(G2 −T2)2 (17)
Then the first-order condition for a minimum loss gives the reaction function of the American government:
4G2 = 2A2 + 2T2 −2M2 + M1 −G1 (18)
Suppose the European government raises European government purchases. Then, as a response, the European central bank lowers European money supply, the American central bank lowers American money supply, and the American government lowers American government purchases.
The Nash equilibrium is determined by the reaction functions of the European central bank, the American central bank, the European government,
and the American government. We assume T = T = T2. The solution to this problem is as follows:
6M1 = − A1 −2A2 −9B1 −6B2 −18T (19) 6M2 = − A2 −2A1 −9B2 −6B1 −18T (20) 2G1 = A1 +B1 +2T (21) 2G2 = A2 + B2 + 2T (22)
Equations (19) to (22) show the Nash equilibrium of European money supply, American money supply, European government purchases, and American government purchases. As a result there is a unique Nash equilibrium. An increase in A1 causes a decline in European money supply, a decline in
2. Some Numerical Examples 189
American money supply, an increase in European government purchases, and no
change in American government purchases. A unit increase in A1 causes a decline in European money supply of 0.17 units, a decline in American money
supply of 0.33 units, and an increase in European government purchases of 0.5 units.
2. Some Numerical Examples
For easy reference, the basic model is reproduced here:
u1 = A1 −M1 +0.5M2 −G1 −0.5G2 (1) u2 = A2 −M2 +0.5M1 −G2 −0.5G1 (2) π1 = B1 + M1 −0.5M2 +G1 +0.5G2 (3) π2 = B2 + M2 −0.5M1 +G2 +0.5G1 (4) s1 = G1 −T (5) s2 = G2 −T2 (6)
And the Nash equilibrium can be described by four equations:
6M1 = − A1 −2A2 −9B1 −6B2 −18T (7) 6M2 = − A2 −2A1 −9B2 −6B1 −18T (8) 2G1 = A1 +B1 +2T (9) 2G2 = A2 + B2 + 2T (10)
It proves useful to study eight distinct cases: - a demand shock in Europe
- a supply shock in Europe - a mixed shock in Europe
190 Monetary and Fiscal Interaction between Europe and America: Case B
- another mixed shock in Europe - 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, let initial inflation be zero, and let the initial structural deficit 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 A1 of 3 units and a decline in B1 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. Inflation in America stays at zero percent. The structural deficit in Europe stays at zero percent, as does the structural deficit 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, an increase in American money supply of 2 units, no change in European government purchases, and no change in American government purchases. 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. Inflation in America stays at zero percent. The structural deficit in Europe stays at zero percent, as does the structural deficit in America. For a synopsis see Table 7.7.
As a result, given a demand shock in Europe, monetary and fiscal interaction produces zero inflation, zero unemployment, and a zero structural deficit in each of the regions.
The loss functions of the European central bank, the American central bank, the European government, and the American government are respectively:
LM1 = π1 + u1 (11) LM2 = π2 +u2 (12) LG1 = u1 +s1 (13)
2. Some Numerical Examples 191
LG2 = u2 +s2 (14)
The initial loss of each policy maker is zero. The demand shock in Europe causes a loss to the European central bank of 18 units, a loss to the European government of 9 units, a loss to the American central bank of zero, and a loss to the American government of zero. Then policy interaction reduces the loss of the European central bank from 18 to zero units. Correspondingly, it reduces the loss of the European government from 9 to zero units. Policy interaction keeps the loss of the American central bank at zero. Similarly, it keeps the loss of the American government at zero.
Table 7.7
Monetary and Fiscal Interaction between Europe and America A Demand Shock in Europe
Europe America
Unemployment 0 Inflation 0 Structural Deficit 0
Shock in A1 3 Shock in B1 − 3 Unemployment 3
Inflation − 3 Change in Money Supply 4 Change in Govt Purchases 0 Unemployment 0 Inflation 0
Structural Deficit 0
Unemployment 0 Inflation 0 Structural Deficit 0
Unemployment 0 Inflation 0 Change in Money Supply 2 Change in Govt Purchases 0 Unemployment 0 Inflation 0
Structural Deficit 0
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