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FW: Monetary and Fiscal Strategies in the World Economy_6

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FW: Monetary and Fiscal Strategies in the World Economy_6

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  1. 1. The Model 187 Then the first-order condition for a minimum loss gives the reaction function of the American central bank: 2M 2 = A 2 − B2 − 2G 2 − 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: 2 2 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 + 0.5M 2 − G1 − 0.5G 2 )2 + (G1 − T1 )2 (14) Then the first-order condition for a minimum loss gives the reaction function of the European government: 4G1 = 2A1 + 2T1 − 2M1 + M 2 − G 2 (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:
  2. 188 Monetary and Fiscal Interaction between Europe and America: Case B LG 2 = u 2 + s 2 2 (16) 2 LG 2 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: LG 2 = (A 2 − M 2 + 0.5M1 − G 2 − 0.5G1 )2 + (G 2 − T2 ) 2 (17) Then the first-order condition for a minimum loss gives the reaction function of the American government: 4G 2 = 2A 2 + 2T2 − 2M 2 + 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 = T1 = T2 . The solution to this problem is as follows: 6M1 = − A1 − 2A 2 − 9B1 − 6B2 − 18T (19) 6M 2 = − A 2 − 2A1 − 9B2 − 6B1 − 18T (20) 2G1 = A1 + B1 + 2T (21) 2G 2 = A 2 + 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
  3. 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 + 0.5M 2 − G1 − 0.5G 2 (1) u 2 = A 2 − M 2 + 0.5M1 − G 2 − 0.5G1 (2) π1 = B1 + M1 − 0.5M 2 + G1 + 0.5G 2 (3) π2 = B2 + M 2 − 0.5M1 + G 2 + 0.5G1 (4) s1 = G1 − T1 (5) s 2 = G 2 − T2 (6) And the Nash equilibrium can be described by four equations: 6M1 = − A1 − 2A 2 − 9B1 − 6B2 − 18T (7) 6M 2 = − A 2 − 2A1 − 9B2 − 6B1 − 18T (8) 2G1 = A1 + B1 + 2T (9) 2G 2 = A 2 + 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
  4. 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: 2 2 LM1 = π1 + u1 (11) LM 2 = π2 + u 2 2 (12) 2 2 2 LG1 = u1 + s1 (13)
  5. 2. Some Numerical Examples 191 LG 2 = u 2 + s 2 (14) 2 2 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 3 Shock in A1 −3 Shock in B1 Unemployment 3 Unemployment 0 −3 Inflation 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
  6. 192 Monetary and Fiscal Interaction between Europe and America: Case B 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 B1 of 3 units and an increase in A1 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.
  7. 2. Some Numerical Examples 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 3 Shock in A1 3 Shock in B1 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 −5 −4 Change in Money Supply Change in Money Supply 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 B1 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.
  8. 194 Monetary and Fiscal Interaction between Europe and America: Case B 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 0 Shock in A1 6 Shock in B1 Unemployment 0 Unemployment 0 Inflation 6 Inflation 0 −9 −6 Change in Money Supply Change in Money Supply 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
  9. 2. Some Numerical Examples 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 A1 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.
  10. 196 Monetary and Fiscal Interaction between Europe and America: Case B 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 6 Shock in A1 0 Shock in B1 Unemployment 6 Unemployment 0 Inflation 0 Inflation 0 −1 −2 Change in Money Supply Change in Money Supply 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 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 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.
  11. 2. Some Numerical Examples 197 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. And the structural deficit in Europe stays at zero percent, as does the structural deficit in America. For a synopsis see Table 7.11. Table 7.11 Monetary and Fiscal Interaction between Europe and America A Common Demand Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 3 3 Shock in A1 Shock in A2 −3 −3 Shock in B1 Shock in B2 Unemployment 3 Unemployment 3 −3 −3 Inflation Inflation Change in Money Supply 6 Change in Money Supply 6 Change in Govt Purchases 0 Change in Govt Purchases 0 Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 As a result, given a common demand shock, monetary and fiscal interaction produces zero inflation, zero unemployment, and a zero structural deficit in each of the regions. The initial loss of each policy maker is zero. The common demand shock causes a loss to the European central bank of 18 units, a loss to the American central bank of 18 units, a loss to the European government of 9 units, and a loss
  12. 198 Monetary and Fiscal Interaction between Europe and America: Case B to the American government of 9 units. Then policy 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. Policy interaction reduces the loss of the European government from 9 to zero units. Similarly, 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, 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 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 9 units, as there is in American money supply. There is an increase in European government purchases of 3 units, as there is in American government purchases. Step four refers to the outside lag. Inflation in Europe stays at 3 percent, as does inflation in America. Unemployment in Europe stays at 3 percent, as does unemployment in America. And the structural deficit in Europe goes from zero to 3 percent, as does the structural deficit in America. For an overview see Table 7.12. As a result, given a common supply shock, monetary and fiscal interaction has no effect on inflation and unemployment. Over and above that, it raises the structural deficits. The initial loss of each policy maker is zero. The common supply shock causes a loss to the European central bank of 18 units, a loss to the American central bank of 18 units, a loss to the European government of 9 units, and a loss to the American government of 9 units. Then policy interaction keeps the loss of the European central bank at 18 units. Correspondingly, it keeps the loss of the American central bank at 18 units. And what is more, policy interaction increases the loss of the European government from 9 to 18 units. Similarly, it increases the loss of the American government from 9 to 18 units. That is to say, in this case, the Nash equilibrium is not Pareto efficient.
  13. 2. Some Numerical Examples 199 Table 7.12 Monetary and Fiscal Interaction between Europe and America A Common Supply Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 3 3 Shock in A1 Shock in A2 3 3 Shock in B1 Shock in B2 Unemployment 3 Unemployment 3 Inflation 3 Inflation 3 −9 −9 Change in Money Supply Change in Money Supply Change in Govt Purchases 3 Change in Govt Purchases 3 Unemployment 3 Unemployment 3 Inflation 3 Inflation 3 Structural Deficit 3 Structural Deficit 3 7) A common mixed 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 mixed shock. In terms of the model there is an increase in B1 of 6 units and an increase in B2 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 15 units, as there is in American money supply. There is an increase in European government purchases of 3 units, as there is in American government purchases. Step four refers to the outside lag. Inflation in Europe goes from 6 to 3 percent, as does inflation in
  14. 200 Monetary and Fiscal Interaction between Europe and America: Case B America. Unemployment in Europe goes from zero to 3 percent, as does unemployment in America. And the structural deficit in Europe goes from zero to 3 percent, as does the structural deficit in America. Table 7.13 presents a synopsis. Table 7.13 Monetary and Fiscal Interaction between Europe and America A Common Mixed Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 0 0 Shock in A1 Shock in A2 6 6 Shock in B1 Shock in B2 Unemployment 0 Unemployment 0 Inflation 6 Inflation 6 Change in Money Supply − 15 Change in Money Supply − 15 Change in Govt Purchases 3 Change in Govt Purchases 3 Unemployment 3 Unemployment 3 Inflation 3 Inflation 3 Structural Deficit 3 Structural Deficit 3 As a result, given a common mixed shock, monetary and fiscal interaction lowers inflation. On the other hand, it raises unemployment and the structural deficits. The initial loss of each policy maker is zero. The common mixed shock causes a loss to the European central bank of 36 units, a loss to the American central bank of 36 units, a loss to the European government of zero, and a loss to
  15. 2. Some Numerical Examples 201 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 American central bank from 36 to 18 units. On the other hand, policy interaction increases the loss of the European government from zero to 18 units. Similarly, it increases the loss of the American government from zero to 18 units. The total loss in Europe stays at 36 units. And the same applies to the total loss in America. 8) Another common mixed 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 mixed shock. In terms of the model there is an increase in A1 of 6 units and an increase in A 2 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. Step three refers to the policy response. According to the Nash equilibrium there is a reduction in European money supply of 3 units, as there is in American money supply. There is an increase in European government purchases of 3 units, as there is in American government purchases. Step four refers to the outside lag. Unemployment in Europe goes from 6 to 3 percent, as does unemployment in America. Inflation in Europe goes from zero to 3 percent, as does inflation in America. And the structural deficit in Europe goes from zero to 3 percent, as does the structural deficit in America. Table 7.14 gives an overview. As a result, given another common mixed shock, monetary and fiscal interaction lowers unemployment. On the other hand, it causes inflation and structural deficits. The initial loss of each policy maker is zero. The common mixed shock causes a loss to the European central bank of 36 units, a loss to the American central bank of 36 units, a loss to the European government of 36 units, and a loss to the American government of equally 36 units. Then policy interaction reduces the loss of the European central bank from 36 to 18 units. Correspondingly, it reduces the loss of the American central bank from 36 to 18 units. Policy interaction reduces the loss of the European government from 36 to 18 units. Similarly, it reduces the loss of the American government from 36 to 18
  16. 202 Monetary and Fiscal Interaction between Europe and America: Case B units. The total loss in Europe goes from 72 to 36 units. And the same applies to the total loss in America. Table 7.14 Monetary and Fiscal Interaction between Europe and America Another Common Mixed Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 Structural Deficit 0 Structural Deficit 0 6 6 Shock in A1 Shock in A2 0 0 Shock in B1 Shock in B2 Unemployment 6 Unemployment 6 Inflation 0 Inflation 0 −3 −3 Change in Money Supply Change in Money Supply Change in Govt Purchases 3 Change in Govt Purchases 3 Unemployment 3 Unemployment 3 Inflation 3 Inflation 3 Structural Deficit 3 Structural Deficit 3 9) Summary. Given a demand shock in Europe, policy interaction achieves zero inflation, zero unemployment, and a zero structural 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 lowers unemployment in Europe. On the other hand, it raises inflation and the structural deficit there.
  17. 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. Second consider a 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 of pure monetary interaction is superior to the system of monetary and fiscal interaction, see Part Three.
  18. 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 zero unemployment in America. The targets of the European government are zero unemployment and a zero structural deficit in Europe. And the targets of the American government are zero unemployment and a zero structural deficit in America. The model of unemployment, inflation, and the structural deficit can be represented by a system of six equations: u1 = A1 − M1 + 0.5M 2 − G1 − 0.5G 2 (1) u 2 = A 2 − M 2 + 0.5M1 − G 2 − 0.5G1 (2) π1 = B1 + M1 − 0.5M 2 + G1 + 0.5G 2 (3) π2 = B2 + M 2 − 0.5M1 + G 2 + 0.5G1 (4) s1 = G1 − T1 (5) s 2 = G 2 − T2 (6) 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: 2M1 = − 2B1 − 2G1 − G 2 + M 2 (7) Suppose the American central bank lowers American money supply. Then, as a response, the European central bank lowers European money supply. M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 204 DOI 10.1007/978-3-642-10476-3_23, © Springer-Verlag Berlin Heidelberg 2010
  19. 1. The Model 205 The targets of the American central bank are zero inflation and zero unemployment in America. The instrument of the American central bank is American money supply. There are two targets but only one instrument, so what is needed is a loss function. We assume that the American central bank has a quadratic loss function: LM 2 = π2 + u 2 2 (8) 2 LM 2 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: LM 2 = (B2 + M 2 − 0.5M1 + G 2 + 0.5G1 ) 2 (9) + (A 2 − M 2 + 0.5M1 − G 2 − 0.5G1 ) 2 Then the first-order condition for a minimum loss gives the reaction function of the American central bank: 2M 2 = A 2 − B2 − 2G 2 − G1 + M1 (10) 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: 2 2 LG1 = u1 + s1 (11)
  20. 206 Monetary and Fiscal Interaction between Europe and America: Case B 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 + 0.5M 2 − G1 − 0.5G 2 )2 + (G1 − T1 )2 (12) Then the first-order condition for a minimum loss gives the reaction function of the European government: 4G1 = 2A1 + 2T1 − 2M1 + M 2 − G 2 (13) 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: LG 2 = u 2 + s 2 2 (14) 2 LG 2 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: LG 2 = (A 2 − M 2 + 0.5M1 − G 2 − 0.5G1 )2 + (G 2 − T2 ) 2 (15) Then the first-order condition for a minimum loss gives the reaction function of the American government: 4G 2 = 2A 2 + 2T2 − 2M 2 + M1 − G1 (16)
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