Optimal Monetary Policy in the new Keynesian model. The two equations for the AD curve and the Phillips curve are

Transcription

1 Economics 05 K. Kletzer Spring 05 Optimal Monetary Policy in the new Keynesian model The two equations for the AD curve and the Phillips curve are y t E t y t+ σ (i t E t π t+ δ)+g t (AD) and π t E t π t+ + κy t + u t (PC) Please note that the AD curve is written a little differently, using δ. The original derivation of the IS curve starts with the condition, u (C t )(+r t ) u (C t+ ). For the utility function, we took logarithms and found that u (C) C σ σ c t+ c t + σ (r t δ) where (+δ) and c logc. We substituted the Fisher equation, r t i t E t π t+,andthe equilibrium condition, y c,togettheadcurve. We will hypothesize that the Fed wants to keep the inflation rate and the output gap equal to zero. Because it can face a tradeoff between keeping inflation low and keeping unemployment low, the Fed weights the cost of higher inflation against the cost of larger negative output gaps. The cost of inflation and the output gap for the Fed today is represented by the function y t + π t. The Fed also faces tradeoffs between keeping inflation and the output gap near zero today and keeping them near zero in the future. The total cost for the Fed is L y t + π t + Et y t+ + π t+ + Et y t+ + π t The Fed wants to makes this loss as close to zero as possible. This is equivalent to trying to maximize L. Let s derive the optimal monetary policy when the Fed cannot commit its future policies. This means the Fed maximizes L with respect to inflation, π t, and the output gap, y t, taking future inflation and output gaps as given. (a) First, maximize y t + π t given the Phillips curve using the Lagrangian, L y t + π t + λ (πt (E t π t+ + κy t + u t )).

2 Keep E t π t+ and u t as exogenous (ie. given). Show that you get the simple relationship between inflation and the output gap, π t κ y t. (b) Because these are both endogenous variables, this is a reduced-form equation. To find how inflation and the ouput gap respond to cost-push shocks, u t, we need to use the Phillips curve again. Start with π t E t π t+ + κy t + u t and replace y t with y t κ π t and rearrange your algebra until you get π t E t π t+ + () To solve this equation, write out the version of it for time t +as E t π t+ E t π t+ + E t u t+ and substitute this into equation. If you keep substituting you get π t u t + E t u t+ + E t u t T by assuming that E t π t+t goes to zero as T grows to infinity. This assumption means that the economy does not go into hyperinflations or hyperdeflations. Your next step is use the equation for u t where u t ρu t + ε t and ε t is white noise (identically and independently distributed with zero mean). The expectations of u are found by iterating the equation for u t as u t+ ρu t + ε t+ u t+ ρu t+ + ε t+ ρ (ρu t + ε t+ )+ε t+ ρ u t + ρε t+ + ε t+ and so on. Taking expecations, E t u t+ ρu t + E t ε t+ ρu t andsoon. E t u t+ ρ u t + ρe t ε t+ + E t ε t+ ρ u t

3 Substitute these and you will get the equation for inflation, π t u t + ρu t + u t + ρ ρ You know how to solve the sum + ρ u t z +a + a +... a. Applying this, inflation is found to be π t u t ρ ρ + κ You can now use the reduced-form relationship between the output gap and inflation, π t κ y t,toget y t κ ρ + κ These are the inflation rate and output gap under the optimal monetary policy. Notice that a cost-push shock (a positive u t ) raises inflation and reduces the output gap. This is a trade-off between inflation and unemployment. Further, notice that inflation and output gaps vary only with cost-push shocks, u t, and do not depend on g t. (c) We now return to the AD equation to find the optimal interest rate for the Fed. To do this, we rearrange as y t E t y t+ σ (i t E t π t+ r n t )+g t i t E t π t+ r n t + σ (E ty t+ y t + g t ) Then substitute the reduced-form equation for the output gap as a function of inflation, y t κ π t and y t+ κ π t+ into the AD equation to get i t δ + σ κ σ g t + E t (π t+ )+ κ σ σ π t. In part (b), we found that π t ρ + κ We use this to find an expression for the E t (π t+ ) as follows: π t+ u t+ ρ + κ

4 (ρu t + ε t+ ) ρ + κ ρ u t + ρ + κ ρ + κ ρπ t + ε t+. ρ + κ Taking the expectation, we get E t (π t+ )ρπ t. By substituting, we get the monetary policy (MP) rule: i t δ + σ κ σ g t + ρπ t + κ σ σ π t rt n + ρ +( ρ) κ π t. σ In this expression, the natural rate of interest was written in for the sum δ+ σ g t. This gives us an explanation for temporary and permanent changes in the natural rate of interest. A fiscal expansion (increase in g t )raises the natural rate of interest. A permanent fiscal expansion is a permanent rise in g t. (d) Commitment means the Fed chooses a rule that it will not change in the future. The derivation of the MP rule in parts (a) to (c) assumed that the Fed chooses its MP rule for one period at a time. That rule is followed at every date, but that does not mean that it was chosen optimally for more than one date at a time. With commitment, the Fed chooses its optimal rule for all times. This means that the Fed chooses a rule internalizing the future effects of that rule on the current inflation. To think about how commitment works, reconsider how we found the optimal monetary policy under discretion. We sought solutions for π t and y t taking expected future inflation as given. For commitment, we want to find the optimal solution for π t, π t+ π t+,...and for y t, y t+, y t+,...using the Phillips curve, π t E t π t+ + κy t + u t taking account of the effects of future inflation, E t π t+, on current inflation π t. Under discretion, the Fed ignores these costs, but under commitment it takes the extra cost of future inflation into account and chooses a more aggressive anti-inflation policy. The marginal cost of letting y t turn negative is κ y tand the marginal π cost of letting inflation rise is t ρ. The optimum for the Fed sets these equal. The reduced-form equation with Fed commitment is π t ρ κ y t. Without doing the math, the solution for inflation and the output gap are π t ρ + κ ( ρ) ε t+ 4

5 and κ y t ( ρ) ρ + κ ( ρ) This is just the same as shrinking the weight on the output gap to ( ρ). Compared to discretion, inflation is lower and the output gap is a larger negative number with a committed Fed. Under commitment, unemployment rises more in the short run, but decreases faster because inflation falls faster. The Fed is able to take better advantage of the tradeoff between unemployment and inflation if it can commit to its policy rule. 5