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FW: Monetary and Fiscal Strategies in the World Economy_1

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FW: Monetary and Fiscal Strategies in the World Economy_1

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  1. 22 Fiscal Policy Table 1.3 Fiscal Policy A Demand Shock Unemployment 0 Inflation 0 −2 Shock in A 2 Shock in B −2 Unemployment 2 Inflation Change in Govt Purchases 2 Unemployment 0 Inflation 0 Table 1.4 Fiscal Policy A Supply Shock Unemployment 0 Inflation 0 Shock in A 2 Shock in B 2 Unemployment 2 Inflation 2 Change in Govt Purchases 2 Unemployment 0 Inflation 4
  2. 23 Chapter 3 Monetary and Fiscal Interaction An increase in money supply lowers unemployment. On the other hand, it raises inflation. Correspondingly, an increase in government purchases lowers unemployment. On the other hand, it raises inflation. The target of the central bank is zero inflation. By contrast, the target of the government is zero unemployment. The model of unemployment and inflation can be represented by a system of two equations: u = A − αM − β G (1) π = B + αεM + βεG (2) Of course this is a reduced form. Here u denotes the rate of unemployment, π is the rate of inflation, M is money supply, G is government purchases, α is the monetary policy multiplier with respect to unemployment, αε is the monetary policy multiplier with respect to inflation, β is the fiscal policy multiplier with respect to unemployment, βε is the fiscal policy multiplier with respect to inflation, A is some other factors bearing on the rate of unemployment, and B is some other factors bearing on the rate of inflation. The endogenous variables are the rate of unemployment and the rate of inflation. According to equation (1), the rate of unemployment is a positive function of A, a negative function of money supply, and a negative function of government purchases. According to equation (2), the rate of inflation is a positive function of B, a positive function of money supply, and a positive function of government purchases. A unit increase in A raises the rate of unemployment by 1 percentage point. A unit increase in B raises the rate of inflation by 1 percentage point. A unit increase in money supply lowers the rate of unemployment by α percentage points. On the other hand, it raises the rate of inflation by αε percentage points. A unit increase in government purchases lowers the rate of unemployment by β M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 23 DOI 10.1007/978-3-642-10476-3_4, © Springer-Verlag Berlin Heidelberg 2010
  3. Monetary and Fiscal Interaction 24 percentage points. On the other hand, it raises the rate of inflation by βε percentage points. The target of the central bank is zero inflation. The instrument of the central bank is money supply. By equation (2), the reaction function of the central bank is: αεM = − B − βεG (3) Suppose the government raises its purchases. Then, as a response, the central bank lowers money supply. The target of the government is zero unemployment. The instrument of the government is government purchases. By equation (1), the reaction function of the government is: βG = A − αM (4) Suppose the central bank lowers money supply. Then, as a response, the government raises its purchases. The Nash equilibrium is determined by the reaction functions of the central bank and the government. From the reaction function of the central bank follows: β dM =− (5) α dG And from the reaction function of the government follows: α dG =− (6) β dM That is to say, the reaction curves do not intersect. As an important result, there is no Nash equilibrium.
  4. 25 Chapter 4 Monetary and Fiscal Cooperation 1. The Model An increase in money supply lowers unemployment. On the other hand, it raises inflation. Correspondingly, an increase in government purchases lowers unemployment. On the other hand, it raises inflation. The policy makers are the central bank and the government. The targets of policy cooperation are zero inflation and zero unemployment. The model of unemployment and inflation can be characterized by a system of two equations: u = A − αM − β G (1) π = B + αεM + βεG (2) Of course this is a reduced form. Here u denotes the rate of unemployment, π is the rate of inflation, M is money supply, G is government purchases, α is the monetary policy multiplier with respect to unemployment, αε is the monetary policy multiplier with respect to inflation, β is the fiscal policy multiplier with respect to unemployment, βε is the fiscal policy multiplier with respect to inflation, A is some other factors bearing on the rate of unemployment, and B is some other factors bearing on the rate of inflation. The endogenous variables are the rate of unemployment and the rate of inflation. According to equation (1), the rate of unemployment is a positive function of A, a negative function of money supply, and a negative function of government purchases. According to equation (2), the rate of inflation is a positive function of B, a positive function of money supply, and a positive function of government purchases. A unit increase in A raises the rate of unemployment by 1 percentage point. A unit increase in B raises the rate of inflation by 1 percentage point. A unit increase in money supply lowers the rate of unemployment by α percentage M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 25 DOI 10.1007/978-3-642-10476-3_5, © Springer-Verlag Berlin Heidelberg 2010
  5. 26 Monetary and Fiscal Cooperation points. On the other hand, it raises the rate of inflation by αε percentage points. A unit increase in government purchases lowers the rate of unemployment by β percentage points. On the other hand, it raises the rate of inflation by βε percentage points. The policy makers are the central bank and the government. The targets of policy cooperation are zero inflation and zero unemployment. The instruments of policy cooperation are money supply and government purchases. Thus there are two targets and two instruments. We assume that the policy makers agree on a common loss function: L = π2 + u 2 (3) L is the loss caused by inflation and unemployment. For ease of exposition we assume equal weights in the loss function. The specific target of policy cooperation is to minimize the loss, given the inflation function and the unemployment function. Taking account of equations (1) and (2), the loss function under policy cooperation can be written as follows: L = (B + αεM + βεG)2 + (A − αM − βG)2 (4) Then the first-order conditions for a minimum loss are: (1 + ε 2 )αM = A − εB − (1 + ε 2 )βG (5) (1 + ε 2 )βG = A − εB − (1 + ε 2 )αM (6) Equation (5) shows the first-order condition with respect to money supply. And equation (6) shows the first-order condition with respect to government purchases. Obviously, equations (5) and (6) are identical. There are two endogenous variables, money supply and government purchases. On the other hand, there is only one independent equation. Thus there is an infinite number of solutions. The cooperative equilibrium is determined by the first-order conditions for a minimum loss. The solution to this problem is as follows:
  6. 1. The Model 27 A − εB αM + β G = (7) 1 + ε2 Equation (7) yields the optimum combinations of money supply and government purchases. As a result, monetary and fiscal cooperation can reduce the loss caused by inflation and unemployment. From equations (1) and (7) follows the optimum rate of unemployment: ε2 A + εB u= (8) 1 + ε2 And from equations (2) and (7) follows the optimum rate of inflation: εA + B π= (9) 1 + ε2 Unemployment is not zero, nor is inflation.
  7. 28 Monetary and Fiscal Cooperation 2. Some Numerical Examples For ease of exposition we assume that monetary and fiscal policy multipliers are unity α = β = ε = 1 . On this assumption, the model of unemployment and inflation can be written as follows: u = A−M−G (1) π = B+ M +G (2) A unit increase in A raises the rate of unemployment by 1 percentage point. A unit increase in B raises the rate of inflation by 1 percentage point. A unit increase in money supply lowers the rate of unemployment by 1 percentage point. On the other hand, it raises the rate of inflation by 1 percentage point. A unit increase in government purchases lowers the rate of unemployment by 1 percentage point. On the other hand, it raises the rate of inflation by 1 percentage point. The model can be solved this way: 2M + 2G = A − B (3) 2u = A + B (4) 2π = A + B (5) Equation (3) shows the optimum combinations of money supply and government purchases, equation (4) shows the optimum rate of unemployment, and equation (5) shows the optimum rate of inflation. It proves useful to study three distinct cases: - a demand shock - a supply shock - a mixed shock. 1) A demand shock. Let initial unemployment be zero, and let initial inflation be zero as well. Step one refers to a decline in aggregate demand. In terms of the model there is an increase in A of 2 units and a decline in B of equally 2 units.
  8. 2. Some Numerical Examples 29 Step two refers to the outside lag. Unemployment goes from zero to 2 percent. And inflation goes from zero to – 2 percent. Step three refers to the policy response. According to the model, a first solution is an increase in money supply of 2 units and an increase in government purchases of zero units. Step four refers to the outside lag. Unemployment goes from 2 to zero percent. And inflation goes from – 2 to zero percent. Table 1.5 presents a synopsis. As a result, given a demand shock, monetary and fiscal cooperation achieves both zero inflation and zero unemployment. A second solution is an increase in money supply of 1 unit and an increase in government purchases of equally 1 unit. A third solution is an increase in money supply of zero units and an increase in government purchases of 2 units. And so on. The loss function under policy cooperation is: L = π2 + u 2 (6) The initial loss is zero. The demand shock causes a loss of 8 units. Then policy cooperation brings the loss down to zero again. Table 1.5 Cooperation between Central Bank and Government A Demand Shock Unemployment 0 Inflation 0 −2 Shock in A 2 Shock in B −2 Unemployment 2 Inflation Change in Money Supply 2 Change in Govt Purchases 0 Unemployment 0 Inflation 0 2) A supply shock. Let initial unemployment and inflation be zero each. Step one refers to the supply shock. In terms of the model there is an increase in B of
  9. 30 Monetary and Fiscal Cooperation 2 units and an increase in A of equally 2 units. Step two refers to the outside lag. Inflation goes from zero to 2 percent. And unemployment goes from zero to 2 percent as well. Step three refers to the policy response. According to the model, a first solution is to keep money supply and government purchases constant. Step four refers to the outside lag. Obviously, inflation stays at 2 percent, and unemployment stays at 2 percent as well. Table 1.6 gives an overview. As a result, given a supply shock, monetary and fiscal cooperation is ineffective. The initial loss is zero. The supply shock causes a loss of 8 units. Then policy cooperation keeps the loss at 8 units. Table 1.6 Cooperation between Central Bank and Government A Supply Shock Unemployment 0 Inflation 0 Shock in A 2 Shock in B 2 Unemployment 2 Inflation 2 Change in Money Supply 0 Change in Govt Purchases 0 Unemployment 2 Inflation 2 3) A mixed shock. Let initial unemployment and inflation be zero each. Step one refers to the mixed shock. In terms of the model there is an increase in B of 4 units. Step two refers to the outside lag. Inflation goes from zero to 4 percent. And unemployment stays at zero percent. Step three refers to the policy response. According to the model, a first solution is a reduction in money supply of 2 units and a reduction in government purchases of zero units. Step four refers to the outside lag. Inflation goes from 4 to 2 percent. And unemployment goes from zero to 2 percent. For a synopsis see Table 1.7. As a result, given a mixed shock, monetary and fiscal cooperation lowers inflation. On the other hand, it raises unemployment. A second solution is a
  10. 2. Some Numerical Examples 31 reduction in money supply of 1 unit and a reduction in government purchases of equally 1 unit. A third solution is a reduction in money supply of zero units and a reduction in government purchases of 2 units. And so on. The initial loss is zero. The mixed shock causes a loss of 16 units. Then policy cooperation brings the loss down to 8 units. Table 1.7 Cooperation between Central Bank and Government A Mixed Shock Unemployment 0 Inflation 0 Shock in A 0 Shock in B 4 Unemployment 0 Inflation 4 −2 Change in Money Supply Change in Govt Purchases 0 Unemployment 2 Inflation 2 4) Summary. Given a demand shock, policy cooperation achieves both zero inflation and zero unemployment. Given a supply shock, policy cooperation is ineffective. Given a mixed shock, policy cooperation reduces the loss to a certain extent. 5) Comparing policy interaction and policy cooperation. Under policy interaction there is no Nash equilibrium. By contrast, policy cooperation can reduce the loss caused by inflation and unemployment. Judging from this point of view, policy cooperation seems to be superior to policy interaction.
  11. Part Two The Closed Economy Presence of a Deficit Target
  12. 35 Chapter 1 Fiscal Policy 1. The Model An increase in government purchases lowers unemployment. On the other hand, it raises inflation. And what is more, it raises the structural deficit. The targets of the government are zero unemployment and a zero structural deficit. The model of unemployment, inflation, and the structural deficit can be represented by a system of three equations: A−G u= (1) Y B+G π= (2) Y G−T s= (3) Y Here u denotes the rate of unemployment, π is the rate of inflation, s is the structural deficit ratio, G is government purchases, T is tax revenue at full- employment output, G − T is the structural deficit, A is some other factors bearing on the rate of unemployment, B is some other factors bearing on the rate of inflation, and Y is full-employment output. The endogenous variables are the rate of unemployment, the rate of inflation, and the structural deficit ratio. According to equation (1), the rate of unemployment is a positive function of A and a negative function of government purchases. According to equation (2), the rate of inflation is a positive function of B and a positive function of government purchases. According to equation (3), the structural deficit ratio is a positive function of government purchases. M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 35 DOI 10.1007/978-3-642-10476-3_6, © Springer-Verlag Berlin Heidelberg 2010
  13. 36 Fiscal Policy To simplify notation we assume that full-employment output is unity. On this assumption, the model can be written as follows: u = A−G (4) π = B+G (5) s = G−T (6) A unit increase in government purchases lowers the rate of unemployment by 1 percentage point. On the other hand, it raises the rate of inflation by 1 percentage point. And what is more, it raises the structural deficit ratio by 1 percentage point. For instance, let initial unemployment be 2 percent, let initial inflation be 2 percent, and let the initial structural deficit be 2 percent as well. Now consider a unit increase in government purchases. Then unemployment goes from 2 to 1 percent. On the other hand, inflation goes from 2 to 3 percent. And what is more, the structural deficit goes from 2 to 3 percent as well. The targets of the government are zero unemployment and a zero structural deficit. The instrument of the government is government purchases. There are two targets but only one instrument, so what is needed is a loss function. We assume that the government has a quadratic loss function: L2 = u 2 + s2 (7) L 2 is the loss to the government caused by unemployment and the structural deficit. We assume equal weights in the loss function. The specific target of the government is to minimize the loss, given the unemployment function and the structural deficit function. Taking account of equations (4) and (6), the loss function of the government can be written as follows: L 2 = (A − G) 2 + (G − T) 2 (8) Then the first-order condition for a minimum loss is: 2G = A + T (9)
  14. 1. The Model 37 Here G is the optimum level of government purchases. An increase in A requires an increase in government purchases. And an increase in B requires no change in government purchases. From equations (4) and (9) follows the optimum rate of unemployment: 2u = A − T (10) From equations (5) and (9) follows the optimum rate of inflation: 2π = A + 2B + T (11) And from equations (6) and (9) follows the optimum structural deficit ratio: 2s = A − T (12) Unemployment is not zero. And the same holds for inflation and the structural deficit.
  15. 38 Fiscal Policy 2. Some Numerical Examples For easy reference, the basic model is summarized here: u = A−G (1) π = B+G (2) s = G−T (3) And the optimum level of government purchases is: 2G = A + T (4) It proves useful to study two distinct cases: - a demand shock - a supply shock. 1) A demand shock. 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 aggregate demand. In terms of the model there is an increase in A of 6 units and a decline in B of equally 6 units. Step two refers to the outside lag. Unemployment goes from zero to 6 percent. Inflation goes from zero to – 6 percent. And the structural deficit stays at zero percent. Step three refers to the policy response. What is needed, according to the model, is an increase in government purchases of 3 units. Step four refers to the outside lag. Unemployment goes from 6 to 3 percent. The structural deficit goes from zero to 3 percent. And inflation goes from – 6 to – 3 percent. Table 2.1 presents a synopsis. As a result, given a demand shock, fiscal policy lowers unemployment and deflation. On the other hand, it raises the structural deficit. The loss function of the government is: L2 = u 2 + s2 (5)
  16. 2. Some Numerical Examples 39 The initial loss is zero. The demand shock causes a loss of 36 units. Then fiscal policy brings the loss down to 18 units. Table 2.1 Fiscal Policy A Demand Shock Unemployment 0 Inflation 0 Structural Deficit 0 −6 Shock in A 6 Shock in B −6 Unemployment 6 Inflation Structural Deficit 0 Change in Govt Purchases 3 −3 Unemployment 3 Inflation Structural Deficit 3 2) A supply shock. 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 terms of the model there is an increase in B of 6 units and an increase in A of equally 6 units. Step two refers to the outside lag. Inflation goes from zero to 6 percent. Unemployment goes from zero to 6 percent as well. And the structural deficit stays at zero percent. Step three refers to the policy response. What is needed, according to the model, is an increase in government purchases of 3 units. Step four refers to the outside lag. Unemployment goes from 6 to 3 percent. The structural deficit goes from zero to 3 percent. And inflation goes from 6 to 9 percent. Table 2.2 gives an overview. As a result, given a supply shock, fiscal policy lowers unemployment. On the other hand, it raises the structural deficit. And what is more, it raises inflation.
  17. 40 Fiscal Policy The initial loss is zero. The supply shock causes a loss of 36 units. Then fiscal policy reduces the loss to 18 units. 3) Summary. Given a demand shock, fiscal policy can reduce the loss to a certain extent. And the same is true of a supply shock. Table 2.2 Fiscal Policy A Supply Shock Unemployment 0 Inflation 0 Structural Deficit 0 Shock in A 6 Shock in B 6 Unemployment 6 Inflation 6 Structural Deficit 0 Change in Govt Purchases 3 Unemployment 3 Inflation 9 Structural Deficit 3
  18. 41 Chapter 2 Monetary and Fiscal Interaction 1. The Model An increase in money supply lowers unemployment. On the other hand, it raises inflation. However, it has no effect on the structural deficit. Corres- pondingly, an increase in government purchases lowers unemployment. On the other hand, it raises inflation. And what is more, it raises the structural deficit. The target of the central bank is zero inflation. By contrast, the targets of the government are zero unemployment and a zero structural deficit. The model of unemployment, inflation, and the structural deficit can be characterized by a system of three equations: u = A−M−G (1) π = B+M +G (2) s = G−T (3) Here u denotes the rate of unemployment, π is the rate of inflation, s is the structural deficit ratio, M is money supply, G is government purchases, T is tax revenue at full-employment output, G − T is the structural deficit, A is some other factors bearing on the rate of unemployment, and B is some other factors bearing on the rate of inflation. The endogenous variables are the rate of unemployment, the rate of inflation, and the structural deficit ratio. According to equation (1), the rate of unemployment is a positive function of A, a negative function of money supply, and a negative function of government purchases. According to equation (2), the rate of inflation is a positive function of B, a positive function of money supply, and a positive function of government purchases. According to equation (3), the structural deficit ratio is a positive function of government purchases. A unit increase in money supply lowers the rate of unemployment by 1 percentage point. On the other hand, it raises the rate M. Carlberg, Monetary and Fiscal Strategies in the World Economy, 41 DOI 10.1007/978-3-642-10476-3_7, © Springer-Verlag Berlin Heidelberg 2010
  19. Monetary and Fiscal Interaction 42 of inflation by 1 percentage point. However, it has no effect on the structural deficit ratio. A unit increase in government purchases lowers the rate of unemployment by 1 percentage point. On the other hand, it raises the rate of inflation by 1 percentage point. And what is more, it raises the structural deficit ratio by 1 percentage point. The target of the central bank is zero inflation. The instrument of the central bank is money supply. By equation (2), the reaction function of the central bank is: M = − B−G (4) Suppose the government raises government purchases. Then, as a response, the central bank lowers money supply. The targets of the government are zero unemployment and a zero structural deficit. The instrument of the government is government purchases. There are two targets but only one instrument, so what is needed is a loss function. We assume that the government has a quadratic loss function: L2 = u 2 + s2 (5) L 2 is the loss to the government caused by unemployment and the structural deficit. We assume equal weights in the loss function. The specific target of the government is to minimize the loss, given the unemployment function and the structural deficit function. Taking account of equations (1) and (3), the loss function of the government can be written as follows: L 2 = (A − M − G) 2 + (G − T) 2 (6) Then the first-order condition for a minimum loss gives the reaction function of the government: 2G = A + T − M (7) Suppose the central bank lowers money supply. Then, as a response, the government raises government purchases.
  20. 1. The Model 43 The Nash equilibrium is determined by the reaction functions of the central bank and the government. The solution to this problem is as follows: M = − A − 2B − T (8) G = A+B+T (9) Equations (8) and (9) show the Nash equilibrium of money supply and government purchases. As a result there is a unique Nash equilibrium. An increase in A causes a decline in money supply and an increase in government purchases. And the same applies to an increase in B. A unit increase in A causes a decline in money supply of 1 unit and an increase in government purchases of equally 1 unit. A unit increase in B causes a decline in money supply of 2 units and an increase in government purchases of 1 unit. From equations (1), (8) and (9) follows the equilibrium rate of unemploy- ment: u =A+B (10) From equations (2), (8) and (9) follows the equilibrium rate of inflation: π=0 (11) And from equations (3) and (9) follows the equilibrium structural deficit ratio: s=A+B (12) Inflation is zero. By contrast, unemployment is not zero, nor is the structural deficit.
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