III.2. The Rogoff Theorem
First, we reproduce Rogoff’s (1985) proof that it is optimal for society (principal) to select
an agent to head the independent central bank that places a large, but finite weight on
inflation. The optimal degree of central bank independence ε* is defined as that value of ε
that minimizes the expected value of the loss function of society Et—1 LtI.
To solve for the value of ε that minimizes Et—1 LtI, differentiate (3.2) with respect to ε
d Et-1LtI = ∂Πi + ∂Γi
∂ε ∂ε ∂ε
(3.3)
(3.4)
(3.5)
∂γi = χ2(ι -β)2≡ σJ
dε [(i + ε)(i -β )2 + χ]3
∂Π _ -χ2u2
dχ (1+ε)3(1 -β )2
We are now ready to prove:
PROPOSITION 3.1: With a positive natural rate of unemployment, the optimal degree of
central bank independence lies between zero and infinity (For U>0, 0<ε*<∞).
Proof: Note that ε > —1 by assumption. Thus, by inspection of (3.5), ∂ΠI∕∂ε is strictly
negative. Note also, by inspection of (3.4), that ∂ΓI∕∂ε is strictly negative for
[χ + (1 β) ] < g < о zero when ε = 0 and positive for ε > 0.
(1 -β )2 ’
Therefore, ∂Et-1 Lt1∕∂ε is strictly negative for ε ≤ 0. ∂Et-1 Lt1Z∂ε must change from
negative to positive at some sufficiently large value of ε, since as ε approaches positive
infinity, ∂ΓI∕∂ε converges to zero at rate ε-2, whereas ∂ΠI∕∂ε converges to zero at rate
ε-3. Consequently, ε* < ∞7)
The intuition behind this result is the following. From (3.5) it can be seen that increasing
the central bank’s commitment to inflation stabilization decreases the credibility compo-
nent of the social loss function. On the other hand, from (3.4) it follows that having a
more independent central bank increases the stabilization component of the loss
7) As pointed out by Rogoff (1985, p. 1178), it is extremely difficult to write down a closed-form solution
for ɛ*.