Active Central Bank Policy

8.2 Active Central Bank Policy

8.2.1 Demand Shock

If the central bank adopts an active policy stance it will be able to respond to demand shocks as far as it can estimate the size of these shocks. With demand shocks only, there is no trade-off between output and inflation stabilisation so no

explicit optimization is required. Under perfect information σ 2 p ,σ 2 y the monetary policy authority is able to perfectly offset the effects of the shock, which leads to a situation where both the output gap and inflation rate are perfectly stabilised around the steady state of the economy. In this case the bank interest rate follows:

(91) Under imperfect information, the role of stabilisation policy becomes more

i t = δE t |I t n t

complicated as the central bank must form it’s own estimate of the size of the shock before enacting this policy. In this case as the forecast includes the interest rate to be set by the central bank. Since this forecast represents the best available information at the time when decisions have to be made, it leads the private sector to ”trust” in the action of the central bank with all of the fluctuations which occur in practice due to mistakes in the way the shock was forecast. In this case the Phillips curve is the same as that which would prevail without any noise.

(92) By construction, all of the losses that accrue in this case are attributable to

π t = βE t π t +1 +kˆ y t +W y ε t +W π ω t

data errors. Essentially, the central bank responds to all the loss that it can respond to, leaving the error in observations to have some residual demand shock on the economy. By construction these are given by the noise terms themselves.

Noise to observable variables is introduced in an ex-post manner, which means that after the policy acts on the economy, the observed outcome features the noise levels assumed earlier. In that sense, the action of policy can be as consisting of th

8.2.2 Supply Shock

In the case where the central policy-maker faces supply shock, the problem becomes non-trivial as it becomes necessary to find the optimal trade-off between inflation and the output gap. The welfare function is derived in a later section, but by jumping forwards it is possible to present a form of this objective function and by postulating a benevolent policymaker who acts to optimize this concept of social welfare allows the closed form solution to the behavior of the economy to be solved. Take the welfare function to be:

X ∞  − θ 2α

t +s −   Loss t =E 0 β

(93) Essentially, take the simplified version of the above function to be:

X ∞ L t =E 2

2 t +s +Bˆ y t +s + Cn t +s y ˆ t +s + Dm t +s y ˆ t +s (94)

s =0

Solving this equation gives an equation for the optimal output gap/inflation trade off that the central bank is prepared to accept. Rational agents in the economy recognize the objectives and optimization problem of the central bank, so fully expect it to choose a value on the optimal trade-off path as given by the maximization problem of the central bank. This implies the following trade-off between the output gap and the inflation rate:

Ak

The set of coefficients {A, B, C, D} correspond to exogenously given values in the utility function. For ease of exposition, the policy is stated in terms of these newly defined coefficients. Essentially, this trade-off can be restated as a single coefficient, relating the response of the interest rate to the supply shock. While the maximum of the welfare function is obtained by setting the response to the demand shock to be complete, for the supply shock a partial response is preferred as this spreads the welfare consequences of the supply shock onto both the output gap and the inflation rate.

The class of rules considered here are those corresponding to discretionary optimization. The primary purpose of this is to avoid the additional complexity The class of rules considered here are those corresponding to discretionary optimization. The primary purpose of this is to avoid the additional complexity

(96) Note, that policy considered here is entirely discretionary and postulating the

i t =φ 1 E t |I t n t +φ 2 E t |I t m t

above rule as a description of policy rules out a commitment solution. Given this relationship between the interest rate set by the central bank and the underlying shocks. This has implications for the form of the distribution of the economic observables and the underlying shocks as follows:

 t    y ˆ t ′  ˜N [(0) , (Σ)] π ′ t

Here is the form of the variance-covariance matrix: 

and partitioning it as

the partitions themselves are equal to:

φ  σ 2 φ n 2 0 1− 1 σ 2 k 2 1− 1 2 2 Σ 11 =

k 2 φ 1 2 2 1−  2 δ σ n (1 − φ 2 ) σ µ

Again, as shown previously, the expectations of the underlying shocks can

be calculated and a Phillips curve can be derived in the usual way.

Unlike the previous case, it has not been possible to compute a closed form solution for the expectations above. Numerical solutions remain viable however. These are presented in the sections that follow.