FW: Monetary and Fiscal Strategies in the World Economy_2

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FW: Monetary and Fiscal Strategies in the World Economy_2

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1. The Model 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: 2M1 = − 2B1 + M 2 (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: 2M 2 = − 2B2 + M1 (6) Suppose the European central bank lowers European money supply. Then, as a response, the American central bank lowers American money supply. 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: 3M1 = − 4B1 − 2B2 (7) 3M 2 = − 4B2 − 2B1 (8) Equations (7) and (8) show the Nash equilibrium of European money supply and American money supply. As a result there is a unique Nash equilibrium. According to equations (7) and (8), an increase in B1 causes a decline in both
58 Monetary Interaction between Europe and America: Case A European money supply and American money supply. A unit increase in B1 causes a decline in European money supply of 1.33 units and a decline in American money supply of 0.67 units. From equations (1), (7) and (8) follows the equilibrium rate of unemployment in Europe: u1 = A1 + B1 (9) From equations (2), (7) and (8) follows the equilibrium rate of unemployment in America: u 2 = A 2 + B2 (10) From equations (3), (7) and (8) follows the equilibrium rate of inflation in Europe: π1 = 0 (11) And from equations (4), (7) and (8) follows the equilibrium rate of inflation in America: π2 = 0 (12) As a result, given a shock, monetary interaction produces zero inflation in Europe and America.
2. Some Numerical Examples 59 2. Some Numerical Examples For easy reference, the basic model is summarized here: u1 = A1 − M1 + 0.5M 2 (1) u 2 = A 2 − M 2 + 0.5M1 (2) π1 = B1 + M1 − 0.5M 2 (3) π2 = B2 + M 2 − 0.5M1 (4) And the Nash equilibrium can be described by two equations: 3M1 = − 4B1 − 2B2 (5) 3M 2 = − 4B2 − 2B1 (6) 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 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. 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
60 Monetary Interaction between Europe and America: Case A 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. Table 3.1 Monetary Interaction between Europe and America A Demand Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 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 Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 As a result, given a demand shock in Europe, monetary interaction produces 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 L1 = π1 (7) 2 L2 = π2 (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
2. Some Numerical Examples 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 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 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 zero percent. Unemployment in Europe goes from 3 to 6 percent. And unemployment in America stays at zero percent. Table 3.2 gives an overview. First consider the effects on Europe. As a result, given a supply shock in Europe, 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 produces zero inflation and zero unemployment in America. The initial loss of each central bank is zero. The supply shock in Europe causes a loss to the European central bank of 9 units and a loss to the American central bank of zero units. Then monetary interaction reduces the loss of the European central bank from 9 to zero units. And what is more, it keeps the loss of the American central bank at zero units.
62 Monetary Interaction between Europe and America: Case A Table 3.2 Monetary Interaction between Europe and America A Supply Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 3 Shock in A1 3 Shock in B1 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0 −4 −2 Change in Money Supply Change in Money Supply 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 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 8 units and a reduction in American money supply of 4 units. Step four refers to the outside lag. Inflation in Europe goes from 6 to zero percent. Inflation in America stays at zero percent. Unemployment in Europe goes from zero to 6 percent. And unemployment in America stays at zero percent. For a synopsis see Table 3.3. First consider the effects on Europe. As a result, given a mixed shock in Europe, monetary interaction produces zero inflation in Europe. However, as a side effect, it produces unemployment there. Second consider the effects on
2. Some Numerical Examples 63 America. As a result, monetary interaction produces zero inflation and zero unemployment in America. The initial loss of each central bank is zero. The mixed shock in Europe causes a loss to the European central bank of 36 units and a loss to the American central bank of zero units. Then monetary interaction reduces the loss of the European central bank from 36 to zero units. And what is more, it keeps the loss of the American central bank at zero units. Table 3.3 Monetary Interaction between Europe and America A Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 0 Shock in A1 6 Shock in B1 Unemployment 0 Unemployment 0 Inflation 6 Inflation 0 −8 −4 Change in Money Supply Change in Money Supply 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 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 no change in European money supply, nor is there in American money
64 Monetary Interaction between Europe and America: Case A supply. Step four refers to the outside lag. Unemployment in Europe stays at 6 percent. Unemployment in America stays at zero percent. Inflation in Europe stays at zero percent, as does inflation in America. For an overview see Table 3.4. First consider the effects on Europe. As a result, given another mixed shock in Europe, monetary interaction produces zero inflation in Europe. However, as a side effect, it produces unemployment there. Second consider the effects on America. As a result, monetary interaction produces zero inflation and zero unemployment in America. The mixed shock in Europe causes no loss to the European central bank or American central bank. Table 3.4 Monetary Interaction between Europe and America Another Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 6 Shock in A1 0 Shock in B1 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 A1 of 3 units, a decline in B1 of 3 units, an increase in A 2 of 3 units,
2. Some Numerical Examples 65 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 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 3.5 presents a synopsis. Table 3.5 Monetary Interaction between Europe and America A Common Demand Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 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 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
66 Monetary Interaction between Europe and America: Case A 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 B1 of 3 units, as there is in A1 . And there is an increase in B2 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 and American money supply of 6 units each. Step four refers to the outside lag. Inflation in Europe goes from 3 to zero percent, as does inflation in America. Unemployment in Europe goes from 3 to 6 percent, as does unemployment in America. Table 3.6 gives an overview. As a result, given a common supply shock, monetary interaction produces zero inflation in Europe and America. However, as a side effect, it raises unemployment there. 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 equally 9 units. Then monetary interaction reduces the loss of the European central bank from 9 to zero units. Correspondingly, it reduces the loss of the American central bank from 9 to zero units. 7) 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 produces zero inflation in Europe. However, as a side effect, it raises unemployment there. Given a mixed shock in Europe, monetary interaction produces zero inflation in Europe. However, as a side effect, it causes unemployment there. 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 America. However, as a side effect, it raises unemployment there.
2. Some Numerical Examples 67 Table 3.6 Monetary Interaction between Europe and America A Common Supply Shock Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 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 −6 −6 Change in Money Supply Change in Money Supply Unemployment 6 Unemployment 6 Inflation 0 Inflation 0
68 Chapter 2 Monetary Interaction between Europe and America: Case B 1. The Model This chapter deals with case B. The targets of the European central bank are zero inflation and zero unemployment in Europe. Correspondingly, the targets of the American central bank are zero inflation and zero unemployment in America. The model of unemployment and inflation can be characterized by a system of four equations: u1 = A1 − M1 + 0.5M 2 (1) u 2 = A 2 − M 2 + 0.5M1 (2) π1 = B1 + M1 − 0.5M 2 (3) π2 = B2 + M 2 − 0.5M1 (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: 2 2 L1 = π1 + u1 (5) L1 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: M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 68 DOI 10.1007/978-3-642-10476-3_10, © Springer-Verlag Berlin Heidelberg 2010
1. The Model 69 L1 = (B1 + M1 − 0.5M 2 )2 + (A1 − M1 + 0.5M 2 )2 (6) Then the first-order condition for a minimum loss gives the reaction function of the European central bank: 2M1 = A1 − B1 + M 2 (7) Suppose the American central bank lowers American money supply. Then, as a response, the European central bank lowers European money supply. 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: 2 2 L 2 = π2 + u 2 (8) L2 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: L 2 = (B2 + M 2 − 0.5M1 )2 + (A 2 − M 2 + 0.5M1 ) 2 (9) Then the first-order condition for a minimum loss gives the reaction function of the American central bank: 2M 2 = A 2 − B2 + M1 (10) Suppose the European central bank lowers European money supply. Then, as a response, the American central bank lowers American money supply.
70 Monetary Interaction between Europe and America: Case B 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: 3M1 = 2A1 + A 2 − 2B1 − B2 (11) 3M 2 = 2A 2 + A1 − 2B2 − B1 (12) Equations (11) and (12) show the Nash equilibrium of European money supply and American money supply. As a result there is a unique Nash equilibrium. According to equations (11) and (12), an increase in A1 causes an increase in both European money supply and American money supply. A unit increase in A1 causes an increase in European money supply of 0.67 units and an increase in American money supply of 0.33 units. From equations (1), (11) and (12) follows the equilibrium rate of unemployment in Europe: 2u1 = A1 + B1 (13) From equations (2), (11) and (12) follows the equilibrium rate of unemployment in America: 2u 2 = A 2 + B2 (14) From equations (3), (11) and (12) follows the equilibrium rate of inflation in Europe: 2π1 = A1 + B1 (15) And from equations (4), (11) and (12) follows the equilibrium rate of inflation in America: 2 π 2 = A 2 + B2 (16) As a rule, unemployment in Europe and America is not zero. And inflation in Europe and America is not zero either.
2. Some Numerical Examples 71 2. Some Numerical Examples For easy reference, the basic model is reproduced here: u1 = A1 − M1 + 0.5M 2 (1) u 2 = A 2 − M 2 + 0.5M1 (2) π1 = B1 + M1 − 0.5M 2 (3) π2 = B2 + M 2 − 0.5M1 (4) And the Nash equilibrium can be described by two equations: 3M1 = 2A1 + A 2 − 2B1 − B2 (5) 3M 2 = 2A 2 + A1 − 2B2 − B1 (6) It proves useful to study eight 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 - 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 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. And inflation in America stays at zero percent.
72 Monetary Interaction between Europe and America: Case B 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. Table 3.7 Monetary Interaction between Europe and America A Demand Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 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 Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 As a result, given a demand shock in Europe, monetary interaction produces 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 2 L1 = π1 + u1 (7) 2 2 L2 = π2 + u2 (8)
2. Some Numerical Examples 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 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 no change in European money supply or American money supply. 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. And unemployment in America stays at zero percent. Table 3.8 gives an overview. As a result, given a supply shock in Europe, monetary interaction is ineffective. The initial loss of each central bank is zero. The supply 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 keeps the loss of the European central bank at 18 units. And what is more, it keeps the loss of the American central bank at zero units.
74 Monetary Interaction between Europe and America: Case B Table 3.8 Monetary Interaction between Europe and America A Supply Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 3 Shock in A1 3 Shock in B1 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 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 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 raises unemployment there. Second consider the effects on America. As a result,
2. Some Numerical Examples 75 monetary interaction produces zero inflation and zero unemployment in America. The initial loss of each central bank is zero. The mixed shock in Europe causes a loss to the European central bank of 36 units and a loss to the American central bank of zero units. Then monetary interaction reduces the loss of the European central bank from 36 to 18 units. And what is more, it keeps the loss of the American central bank at zero units. Table 3.9 Monetary Interaction between Europe and America A Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 0 Shock in A1 6 Shock in B1 Unemployment 0 Unemployment 0 Inflation 6 Inflation 0 −4 −2 Change in Money Supply Change in Money Supply 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 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 an increase in European money supply of 4 units and an increase in
76 Monetary Interaction between Europe and America: Case B 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 hand, it raises inflation there. Second consider the effects on America. As a result, monetary interaction produces zero inflation and zero unemployment in America. The initial loss of each central bank is zero. The mixed shock in Europe causes a loss to the European central bank of 36 units and a loss to the American central bank of zero units. Then monetary interaction reduces the loss of the European central bank from 36 to 18 units. And what is more, it keeps the loss of the American central bank at zero units. Table 3.10 Monetary Interaction between Europe and America Another Mixed Shock in Europe Europe America Unemployment 0 Unemployment 0 Inflation 0 Inflation 0 6 Shock in A1 0 Shock in B1 Unemployment 6 Unemployment 0 Inflation 0 Inflation 0 Change in Money Supply 4 Change in Money Supply 2 Unemployment 3 Unemployment 0 Inflation 3 Inflation 0