New Keynesian Phillips Curve

where , and each small latter variable denotes the log-deviation from the steady state, i.e. .

• IS curve:
  Here, IS means the demand curve, where the current output gap is positively affected by the expected value of future output gap, and is negatively affected by the real interest rate, since holds by the Fisher equation. Also, it is assuming the log utility function, where (elasticity of substitution). If , then we have

• Philips Curve:
  Philips curve shows the relationship between the inflation and the output gap, where the two have a positive correlation, . Also, , meaning that current inflation is determined by the expected inflation.

• Monetary Policy equation:
  MP curve is derived from the Taylor rule, where originally assumed and , since implying that it reacts equally to both inflation and output gap (note that is real neutral interest rate, is a target inflation rate).

• Guess:

Since , we have Thus, from MP, as . Plugging this result back into the original model,

• Verify:

• For demand shock (one unit increase of ): therefore, for demand shock, output gap and inflation moves into the same direction, while policy rate does not react to the current unexpected shock. Here, if , then inflation will also goes to infinity.

• For supply shock (one unit increase of ): therefore, for the supply shock, only inflation increases, while the other two variable does not move.

• For monetary policy shock (one unit increase of ): therefore, for the unexpected MP shock, output gap and inflation will move into the same direction. This shows a possible conflict between reacting to high inflation and the negative output gap.

Since MP curve is no more reduced as the previous model, Here, we reduce the model into two equation form by plugging MP into IS: Then, using the matrix form, Here, denoting , we have Therefore, we have

• For demand shock (one unit increase of ): as the monetary policy reacts for the current IS shock, both output gap and inflation increase less than before.

  • If (reaction coefficient to output gap goes to infinity), then .
  • If (reaction coefficient to inflation), then .
  • If (persistence of inflation), then and .
• For supply shock (one unit increase of ): therefore, for the supply shock, both output gap and inflation increases less than before.

  • If , then , : only output gap stabilized.
  • If , then , : only inflation stabilized.
  • If , then .
• For monetary policy shock (one unit increase of ): therefore, for the unexpected MP shock, output gap and inflation will move into the same direction.

  • If , then .
  • If , then .
  • If , then and .

• : aggregated consumption, where the individual goods are infinitely many and located uniformly along .
• : labor at time

• : total spending on consumption.
• : price of the bond at time , where the return will be . Thus the interest rate is
• : amount of bond buying at time
• : amount of bond bought at time , where the return value is normalized to .
• : labor income
• : tax
• : dividends to be distributed back to households from the profit.

Here, we assume no government, i.e. . Also, note that saving is assumed to be zero, i.e. no investment. Thus, the equilibrium condition is

Lagrangian: F.O.C.
Then,

• Euler Equation:
• Labor supply:

Denote the small letter variable , the log deviation from the steady state.

From Euler Equation: Since the equilibrium condition for the economy is given as then we finally have