187 Then the first-order condition for a minimum loss gives the reaction function of the American central bank: 222 211 2M A B 2G G M=−− −+ (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: 22 111 LG u s=+ (13) 1 LG 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: 22 111 21 2 11 LG (A M 0.5M G 0.5G ) (G T )=−+ −− +− (14) Then the first-order condition for a minimum loss gives the reaction function of the European government: 111122 4G 2A 2T 2M M G=+−+− (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: 1. The Model 188 22 222 LG u s=+ (16) 2 LG 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: 22 222 12 1 22 LG (A M 0.5M G 0.5G ) (G T )=−+ −− +− (17) Then the first-order condition for a minimum loss gives the reaction function of the American government: 222211 4G 2A 2T 2M M G=+−+− (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 12 TT T = = . The solution to this problem is as follows: 11212 6M A 2A 9B 6B 18T=− − − − − (19) 22121 6M A 2A 9B 6B 18T=− − − − − (20) 111 2G A B 2T=++ (21) 222 2G A B 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 1 A causes a decline in European money supply, a decline in Monetary and Fiscal Interaction between Europe and America: Case B 189 American money supply, an increase in European government purchases, and no change in American government purchases. A unit increase in 1 A 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: 111 21 2 uAM0.5MG0.5G=− + −− (1) 222 12 1 uAM0.5MG0.5G=− + −− (2) 11 1 21 2 B M 0.5M G 0.5Gπ= + − + + (3) 22 2 12 1 B M 0.5M G 0.5Gπ= + − + + (4) 111 sGT=− (5) 222 sGT=− (6) And the Nash equilibrium can be described by four equations: 11212 6M A 2A 9B 6B 18T=− − − − − (7) 22121 6M A 2A 9B 6B 18T=− − − − − (8) 111 2G A B 2T=++ (9) 222 2G A B 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 2. Some Numerical Examples 190 - 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 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. 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: 22 111 LM u=π + (11) 22 222 LM u=π + (12) 22 111 LG u s=+ (13) Monetary and Fiscal Interaction between Europe and America: Case B 191 22 222 LG u s=+ (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 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 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 Change in Govt Purchases 0 Change in Govt Purchases 0 Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 2. Some Numerical Examples 192 2) A supply 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 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 a reduction in European money supply of 5 units, a reduction in American money supply of 4 units, an increase in European government purchases of 3 units, and no change in American government purchases. Step four refers to the outside lag. Inflation in Europe stays at 3 percent. Inflation in America stays at zero percent. Unemployment in Europe stays at 3 percent. Unemployment in America stays at zero percent. The structural deficit in Europe goes from zero to 3 percent. And the structural deficit in America stays at zero percent. For an overview see Table 7.8. First consider the effects on Europe. As a result, given a supply shock in Europe, monetary and fiscal interaction has no effects on inflation and unemployment in Europe. And what is more, it causes a structural deficit there. Second consider the effects on America. As a result, monetary and fiscal interaction produces zero inflation, zero unemployment, and a zero structural deficit in America. The initial loss of each policy maker is zero. The supply 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 equally zero. Then policy interaction keeps the loss of the European central bank at 18 units. And what is more, it increases the loss of the European government from 9 to 18 units. On the other hand, policy interaction keeps the loss of the American central bank at zero. Correspondingly, it keeps the loss of the American government at zero. That is to say, in this case, the Nash equilibrium is not Pareto efficient. Monetary and Fiscal Interaction between Europe and America: Case B 193 Table 7.8 Monetary and Fiscal Interaction between Europe and America A Supply Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 Shock in A 1 3 Shock in B 1 3 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 Change in Money Supply − 5 Change in Money Supply − 4 Change in Govt Purchases 3 Change in Govt Purchases 0 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 Structural Deficit 3 Structural Deficit 0 3) A mixed 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 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 9 units, a reduction in American money supply of 6 units, an increase in European government purchases of 3 units, and no change in American government purchases. Step four refers to the outside lag. Inflation in Europe goes from 6 to 3 percent. 2. Some Numerical Examples 194 Inflation in America stays at zero percent. Unemployment in Europe goes from zero to 3 percent. Unemployment in America stays at zero percent. The structural deficit in Europe goes from zero to 3 percent. And the structural deficit in America stays at zero percent. Table 7.9 presents a synopsis. Table 7.9 Monetary and Fiscal Interaction between Europe and America A Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 Shock in A 1 0 Shock in B 1 6 Unemployment 0 Unemployment 0 Inflation 6 Inflation 0 Change in Money Supply − 9 Change in Money Supply − 6 Change in Govt Purchases 3 Change in Govt Purchases 0 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 Structural Deficit 3 Structural Deficit 0 As a result, given a mixed shock in Europe, monetary and fiscal interaction lowers inflation in Europe. On the other hand, it raises unemployment and the structural deficit there. The initial loss of each policy maker is zero. The mixed shock in Europe causes a loss to the European central bank of 36 units, a loss to the European government of zero, a loss to the American central bank of zero, and a loss to the Monetary and Fiscal Interaction between Europe and America: Case B 195 American government of zero. Then policy interaction reduces the loss of the European central bank from 36 to 18 units. On the other hand, it increases the loss of the European government from zero to 18 units. Policy interaction keeps the loss of the American central bank at zero. Correspondingly, it keeps the loss of the American government at zero. The total loss in Europe stays at 36 units. And the total loss in America stays at zero. 4) Another mixed 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 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 a reduction in European money supply of 1 unit, a reduction in American money supply of 2 units, an increase in European government purchases of 3 units, and no change in American government purchases. 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. Inflation in America stays at zero percent. The structural deficit in Europe goes from zero to 3 percent. And the structural deficit in America stays at zero percent. Table 7.10 gives an overview. As a result, given another mixed shock in Europe, monetary and fiscal interaction lowers unemployment in Europe. On the other hand, it raises inflation and the structural deficit there. The initial loss of each policy maker is zero. The mixed shock in Europe causes a loss to the European central bank of 36 units, a loss to the European government of 36 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 36 to 18 units. Correspondingly, it reduces the loss of the European government from 36 to 18 units. Policy interaction keeps the loss of the American central bank at zero. Similarly, it keeps the loss of the American government at zero. 2. Some Numerical Examples 196 Table 7.10 Monetary and Fiscal Interaction between Europe and America Another Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 Shock in A 1 6 Shock in B 1 0 Unemployment 6 Unemployment 0 Inflation 0 Inflation 0 Change in Money Supply − 1 Change in Money Supply − 2 Change in Govt Purchases 3 Change in Govt Purchases 0 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 Structural Deficit 3 Structural Deficit 0 5) A common demand shock. 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 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 of 6 units, as there is in American money supply. There is no change in European government purchases, nor is there in American government purchases. Step four refers to the outside lag. Monetary and Fiscal Interaction between Europe and America: Case B [...]... more, it raises the structural deficit there Given the common mixed shock, policy interaction lowers unemployment in America On the other hand, it raises inflation and the structural deficit there 216 Monetary and Fiscal Interaction between Europe and America: Case B 6) Comparing pure monetary interaction and monetary- fiscal interaction As a result, in case C, the system of pure monetary interaction is... supply shock in Europe In case A, policy interaction achieves zero inflation in Europe On the other hand, it raises unemployment and the structural deficit there In case B, policy interaction has no effect on inflation and unemployment in Europe And what is more, it causes a structural deficit there 11) Comparing pure monetary interaction and monetary- fiscal interaction As a result, in case B, the system... unemployment in Europe On the other hand, it raises inflation and the structural deficit there 2 Some Numerical Examples 203 10) Comparing monetary- fiscal interaction A and monetary- fiscal interaction B First consider a demand shock in Europe In case A, policy interaction achieves zero inflation, zero unemployment, and a zero structural deficit in each of the regions In case B we have the same effects... consider the effects on Europe As a result, given a common mixed shock, monetary and fiscal interaction produces zero inflation in Europe On the other hand, it causes unemployment and a structural deficit there Second consider the effects on America As a result, monetary and fiscal interaction lowers inflation in America On the other hand, it raises unemployment and the structural deficit there 2... supply shock 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 the common supply shock In terms of the model there is an increase in B1 of 3 units, as there is in A1 And there is an increase in B 2 of 3 units, as there is in A 2 Step two refers to the outside lag Inflation in Europe goes... 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 and American goods In terms of the model there is an increase in A1 of 3 units, a decline in B1 of 3 units, an increase in A 2 of 3 units, and a decline in B 2 of 3 units Step two refers to the outside lag Unemployment in Europe goes from zero... total loss in America goes from 27 to zero units 2) A common supply shock 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 the common supply shock In terms of the model there is an increase in B1 of 3 units, as there is in A1 And there is an increase in B 2 of 3 units, as there is in A 2 ... deficit in each of the regions Given a supply shock in Europe, policy interaction has no effect on inflation and unemployment in Europe And what is more, it causes a structural deficit there Given a mixed shock in Europe, policy interaction lowers inflation in Europe On the other hand, it raises unemployment and the structural deficit there Given another type of mixed shock in Europe, policy interaction... 3 Inflation 0 Inflation 3 Structural Deficit 6 Structural Deficit 3 First consider the effects on Europe As a result, given a common supply shock, monetary and fiscal interaction produces zero inflation in Europe On the other hand, it raises unemployment and the structural deficit there Second consider the effects on America As a result, monetary and fiscal interaction has no effect on inflation and. .. pure monetary interaction is superior to the system of monetary and fiscal interaction, see Part Three 204 Chapter 3 Monetary and Fiscal Interaction between Europe and America: Case C 1 The Model This chapter deals with 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 . another mixed shock in Europe, monetary and fiscal interaction lowers unemployment in Europe. On the other hand, it raises inflation and the structural deficit there. The initial loss of each. deficit there. Given another type of mixed shock in Europe, policy interaction lowers unemployment in Europe. On the other hand, it raises inflation and the structural deficit there. Monetary and. and Fiscal Interaction between Europe and America: Case B 203 10) Comparing monetary- fiscal interaction A and monetary- fiscal interaction B. First consider a demand shock in Europe. In case