The ultimate determinants of central bank independence

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 E_t—1 L_t^I.

To solve for the value of ε that minimizes E_t—1 L_tI, differentiate (3.2) with respect to ε

^d Et-1LtI ₌ ∂Πⁱ ₊ ∂Γⁱ
∂ε       ∂ε     ∂ε

∂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, ∂E_t-₁ L_t¹∕∂ε is strictly negative for ε ≤ 0. ∂E_t-₁ L_t¹Z∂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*.

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