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*.
More intriguing information
1. Text of a letter2. Crime as a Social Cost of Poverty and Inequality: A Review Focusing on Developing Countries
3. Social Balance Theory
4. ISSUES AND PROBLEMS OF IMMEDIATE CONCERN
5. Skill and work experience in the European knowledge economy
6. Modelling the Effects of Public Support to Small Firms in the UK - Paradise Gained?
7. The name is absent
8. The role of statin drugs in combating cardiovascular diseases
9. The mental map of Dutch entrepreneurs. Changes in the subjective rating of locations in the Netherlands, 1983-1993-2003
10. The name is absent